Merge branch 'master' of http://182.92.203.7:2001/r/testbookLayout

zhongshujie

2024-10-16 218a387c6191311ed29b1aa81612e0aa3b4fa9dc

 src/books/mathBook/view/components/chapter005.vue

@@ -1,25 +1,52 @@
<template>
  <div class="chapter" num="6">


  <div class="chapter" num="5">
    <!-- 第五单元首页 -->
    <div class="page-box" page="160">
      <div v-if="showPageList.indexOf(160) > -1">


        <div class="padding-116">第五单元首页</div>
        <h1 id="a009"><img class="img-0" alt="" src="../../assets/images/dy5.jpg" /></h1>
        <div class="padding-116">
          <p>
            中华优秀传统文化源远流长、博大精深，是中华文明的智慧结晶.成语“周而复始”出自《汉书·礼乐志》，“精健日月，星辰度理，阴阳五行，周而复始”.在现实世界中，许多事物的运动变化会呈现循环往复、周而复始的规律，我们称这种变化规律为周期性.例如，表针旋转、车轮滚动、物体简谐振动等.这些有规律的变化现象都可用三角函数来刻画.
          </p>
          <p>
            本单元我们将在已学函数概念的基础上，利用函数的思想和方法来学习三角函数的相关内容.三角函数是研究自然界中周期性现象的重要数学工具，它在测量、物理等方面都有着广泛应用.
          </p>
          <p>
            本单元主要将角的概念推广到任意情形，引入弧度制、任意角的三角函数，学习三角函数基本公式及任意角的三角函数的图像和性质.本单元将借助图像理解任意角的三角函数的概念，利用直观想象发现三角函数中数与形之间的联系，会表达其特征与关系；感受用直观想象从具体问题中抽象出数学问题的过程，认识数学中的通性、通法；通过对三角函数具体问题的分析，利用逻辑推理进行三角函数基本公式的推导；初步感知三角函数模型所刻画的简单的周期性函数；提升数学运算、直观想象、逻辑推理和数学抽象等核心素养.
          </p>
        </div>
      </div>
    </div>
    <!-- 目标 -->
    <div class="page-box" page="161">
      <div v-if="showPageList.indexOf(161) > -1">

        <div class="padding-116">目标</div>
        <div class="padding-116">
          <p class="left"><img class="inline2" alt="" src="../../assets/images/xxmb.jpg" /></p>
          <div class="fieldset">
            <p>1.角的概念推广.</p>
            <p>知道推广角的意义和任意角所在的象限，能识别终边相同的角.</p>
            <p>2.弧度制.</p>
            <p>知道引入弧度制的意义，会进行角度与弧度的换算.</p>
            <p>3.任意角的正弦函数、余弦函数和正切函数.</p>
            <p>
              能根据任意角的三角函数（正弦函数、余弦函数和正切函数）定义，判断三角函数值的符号.
            </p>
            <p>4.同角三角函数的基本关系.</p>
            <p>
              会根据三角函数的定义或借助单位圆，推导同角三角函数的平方关系和商数关系，能进行有关化简和计算.
            </p>
            <p>5.诱导公式.</p>
            <p>知道诱导公式在三角函数求值与化简中的作用.</p>
            <p>6.正弦函数、余弦函数的图像和性质.</p>
            <p>会借助代数运算与几何直观，认识正弦函数、余弦函数的图像和性质；</p>
            <p>知道运用“五点法”可以画出正弦函数、余弦函数在一个周期上的简图.</p>
            <p>7.已知三角函数值求指定范围的角.</p>
            <p>知道特殊的三角函数值与[0，2<i>π</i>]范围内角的对应关系；</p>
            <p>会用计算工具进行有关的三角计算.</p>
          </div>
        </div>
      </div>
    </div>



    <!-- 155 -->
    <div class="page-box" page="162">
      <div v-if="showPageList.indexOf(162) > -1">
@@ -32,7 +59,31 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b030">
            5.1 角的概念推广<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <h3 id="c048">5.1.1 角的概念的推广<span class="fontsz2">＞＞＞</span></h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            （1）
            中国跳水队享有奥运“梦之队”的美誉.自1984年到2016年，奥运会跳水项目一共产生了56枚奥运金牌，中国跳水队一共夺得了40枚，约占其中的71.4%.如图5-1（1）
            所示，跳水比赛中有“向前翻腾一周半”和“向后翻腾两周半”的动作，你知道这两个动作分别表示的旋转的角度是多少吗？
          </p>
          <p>
            （2）
            环青海湖国际公路自行车赛是我国规模最大、参赛队伍最多的竞赛，也是世界上海拔最高的国际性竞赛，“绿色、人文、和谐”的竞赛主题倡导体育运动应低碳环保，促进文化交流、人与自然和谐共生.如图5-1（2）
            所示，选手在骑自行车时，自行车车轮在前进和后退的过程中旋转形成的角一样吗？
          </p>
          <p class="center"><img class="img-b" alt="" src="../../assets/images/0166-1.jpg" /></p>
          <p class="img">图5-1</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            生活中随处可见超出0°～360°范围的角.问题（1）
            中“向前翻腾一周半”和“向后翻腾两周半”的跳水动作，不仅有超出360°的“一周半”和“两周半”的角，而且旋转的方向也不同，产生的效果也不一样；问题（2）
            中自行车前进时车轮若是逆时针方向旋转，可以旋转几百圈甚至上万圈，后退时车轮则是顺时针方向旋转，其形成的角是不一样的.因此，要准确描述这些现象，就应知道旋转度数和旋转方向，这就需要对角的概念进行推广.
          </p>
        </div>
      </div>
    </div>
    <!-- 156 -->
@@ -44,7 +95,47 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            我们规定，一条射线绕其端点按逆时针方向旋转形成的角叫作<b>正角</b>，如图5-2（1）
            所示.按顺时针方向旋转形成的角叫作<b>负角</b>，如图5-2（2）
            所示.如果一条射线没有做任何旋转，就称它形成了一个<b>零角</b>，如图5-2（3）
            所示.
          </p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0167-1.jpg" /></p>
          <p class="img">图5-2</p>
          <p>这样我们就把角的概念推广到了<b>任意角</b>，包括正角、负角和零角.</p>
          <div class="bk-hzjl">
            <div class="bj1-hzjl">
              <p class="left">
                <img class="img-gn2" alt="" src="../../assets/images/hzjl.jpg" />
              </p>
            </div>
            <p class="block">
              类比实数<i>a</i>与-<i>a</i>互为相反数，角<i>α</i>与角-<i>α</i>是什么关系呢？类比实数减法的“减去一个数等于加上这个数的相反数”，角的减法可以转化为角的加法吗？即<i>α</i>-<i>β</i>=<i>α</i>+（-<i>β</i>）
              成立吗？不妨画图试试.
            </p>
          </div>
          <p>
            为了简便起见，在不引起混淆的前提下，我们把“角<i>α</i>”或“∠<i>α</i>”简记为“<i>α</i>”.今后我们可以用小写希腊字母<i>α</i>，<i>β</i>，<i>γ</i>，…来表示角.
          </p>
          <p>
            在问题（1）
            中，若“向前翻腾一周半”记为<i>α</i>=540°，那么“向后翻腾两周半”则记为<i>α</i>=-900°.在问题（2）
            中，自行车前进或后退，车轮按逆时针方向旋转形成正角，按顺时针方向旋转形成负角.
          </p>
          <p>
            为了方便研究，通常在平面直角坐标系内讨论角.我们将角的顶点与原点重合，角的始边与<i>x</i>轴的非负半轴重合.这样，角的终边在第几象限，就说这个角是第几象限角.
          </p>
          <p>例如，图5-3中的690°角、-210°角分别是第四象限角和第二象限角.</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0167-2.jpg" /></p>
          <p class="img">图5-3</p>
          <p>
            如果角的终边在坐标轴上，那么就认为这个角不属于任何一个象限（也称界限角）.例如，0°，90°，180°，270°，360°角.

          </p>
        </div>
      </div>
    </div>
    <!-- 157 -->
@@ -59,7 +150,37 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例1</b></span>　在平面直角坐标系中，分别画出下列各角，并指出它们是第几象限角.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0168-1.jpg" /></p>
          <p class="img">图5-4</p>
          <p>（1） 225°；（2） -300°.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1）
            以<i>x</i>轴的非负半轴为始边，逆时针方向旋转225°，即形成225°角，如图5-4（1）
            所示.因为225°角的终边在第三象限内，所以225°角是第三象限角.
          </p>
          <p>
            （2）
            以<i>x</i>轴的非负半轴为始边，顺时针方向旋转300°，即形成-300°角，如图5-4（2）
            所示.因为-300°角的终边在第一象限内，所以-300°角是第一象限角.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.如图所示，已知锐角∠<i>AOB</i>=45°，写出下图中箭头所示角的度数.</p>
            <p class="center">
              <img class="img-b" alt="" src="../../assets/images/0168-2.jpg" />
            </p>
            <p class="img">第1题图</p>
            <p>2.在平面直角坐标系中，分别画出下列各角，并指出它们各是第几象限角.</p>
            <p class="center">
              <img class="img-b" alt="" src="../../assets/images/0168-3.jpg" />
            </p>
            <p class="img">第2题图</p>
          </div>
        </div>
      </div>
    </div>
    <!-- 158 -->
@@ -71,7 +192,39 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>3.判断下列说法是否正确，正确的画“√”，错误的画“×”.</p>
            <p>（1） 锐角是第一象限角，钝角是第二象限角.（　）</p>
            <p>（2） 小于90°的角一定是锐角.（　）</p>
            <p>（3） 直角是第一象限角或第二象限角.（　）</p>
            <p>（4） 第二象限角一定比第一象限角大.（　）</p>
            <p>
              4.（1）
              若0°＜<i>α</i>＜90°，则<i>α</i>是第___象限角；若90°＜<i>α</i>＜180°，则<i>α</i>是第___象限角；若180°＜<i>α</i>＜270°，则<i>α</i>是第___象限角；若270°＜<i>α</i>＜360°，则<i>α</i>是第___象限角.
            </p>
            <p>
              （2）
              若-90°＜<i>α</i>＜0°，则<i>α</i>是第___象限角；若-180°＜<i>α</i>＜-90°，则<i>α</i>是第___象限角；若-270°＜<i>α</i>＜-180°，则<i>α</i>是第___象限角；若-360°＜<i>α</i>＜-270°，则<i>α</i>是第___象限角.
            </p>
          </div>
          <h3 id="c049">5.1.2 终边相同的角<span class="fontsz2">＞＞＞</span></h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            如图5-5所示，在平面直角坐标系中，分别画出了-330°，30°，390°角，观察其终边有何联系？-330°，390°与30°在数值上有什么关系？
          </p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0169-1.jpg" /></p>
          <p class="img">图5-5</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            观察发现，图5-5中-330°，390°与30°角终边相同，并且与30°角终边相同的这些角都可以表示成30°角与<i>k</i>个（<i>k</i>∈<b>Z</b>）周角的和，如
          </p>
          <p class="center">-330°=30°-360°（这里<i>k</i>=-1），</p>
          <p class="center">390°=30°+360°（这里<i>k</i>=1）.</p>
          <p>
            进一步分析可知，与30°角终边相同的所有角都可以表示成30°角与<i>k</i>（<i>k</i>∈<b>Z</b>）个周角的和，因此可用集合<i>S</i>={<i>β</i>|<i>β</i>=30°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}表示与30°角终边相同的角.显然，-330°，390°角都是集合<i>S</i>中的元素，30°角也是<i>S</i>中的元素（此时<i>k</i>=0）.反之，集
          </p>
        </div>
      </div>
    </div>
    <!-- 159 -->
@@ -85,7 +238,53 @@
            <p><span>159</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            合<i>S</i>中的任何一个元素都与30°角终边相同.
          </p>
          <p>与45°，60°，70°，100°，…角终边相同的角构成的集合又应该如何表达呢？</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            一般地，所有与<i>α</i>终边相同的角，连同<i>α</i>在内，可以组成一个集合
          </p>
          <p class="center">
            <i>S</i>={<i>β</i>|<i>β</i>=<i>α</i>+<i>k</i>·360°，<i>k</i>∈<i>Z</i>}.
          </p>
          <p>
            <b>任意的与<i>α</i>终边相同的角都可以表示成<i>α</i>与整数个周角（360°的整数倍）的和.</b>例如，与100°角终边相同的角组成的集合为<i>S</i>={<i>β</i>|<i>β</i>=100°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}，当<i>k</i>=0时，<i>β</i>=100°；<i>k</i>=1时，<i>β</i>=460°；<i>k</i>=-1时，<i>β</i>=-260°.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　在0°～360°内，找出与下列各角终边相同的角，并分别判断它们是第几象限角.
          </p>
          <p>（1） 600°；（2） -230°；（3） -890°.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 因为600°=240°+360°，所以600°角与240°角终边相同，是第三象限角.
          </p>
          <p>
            （2） 因为-230°=130°-360°，所以-230°角与130°角终边相同，是第二象限角.
          </p>
          <p>
            （3） 因为-890°=190°-3×360°，所以-890°角与190°角终边相同，是第三象限角.
          </p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　写出下列角的集合.
          </p>
          <p>（1） 终边在<i>y</i>轴正半轴上的角的集合；</p>
          <p>（2） 终边在<i>y</i>轴负半轴上的角的集合；</p>
          <p>（3） 终边在<i>y</i>轴上的角的集合.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 在0°～360°内，终边在<i>y</i>轴正半轴上的角是90°角，
          </p>
          <p>所以，终边在<i>y</i>轴正半轴上的角的集合是</p>
          <p class="center">
            <i>S</i>1={<i>β</i>|<i>β</i>=90°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}.
          </p>
          <p>（2） 在0°～360°内，终边在<i>y</i>轴负半轴上的角是270°角，</p>
          <p>所以，终边在<i>y</i>轴负半轴上的角的集合是</p>
          <p class="center">
            <i>S</i>2={<i>β</i>|<i>β</i>=270°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}.
          </p>
        </div>
      </div>
    </div>

@@ -97,7 +296,58 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>（3） 终边在<i>y</i>轴上的角的集合是</p>
          <p><i>S</i>=<i>S</i><sub>1</sub>∪<i>S</i><sub>2</sub></p>
          <p>
            ={<i>β</i>|<i>β</i>=90°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}∪{<i>β</i>|<i>β</i>=270°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}
          </p>
          <p>
            ={<i>β</i>|<i>β</i>=90°+2<i>k</i>·180°，<i>k</i>∈<b>Z</b>}∪{<i>β</i>|<i>β</i>=90°+（2<i>k</i>+1）·180°，<i>k</i>∈<b>Z</b>}
          </p>
          <p>={<i>β</i>|<i>β</i>=90°+<i>m</i>·180°，<i>m</i>∈<b>Z</b>}.</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.选择题.</p>
            <p>（1） 与60°角终边相同的角的集合表示正确的是（　）.</p>
            <p>
              <i>A</i>.{<i>β</i>|<i>β</i>=60°+<i>k</i>·360°}
              <i>B</i>.{<i>β</i>|<i>β</i>=60°+<i>k</i>·180°，<i>k</i>∈<b>Z</b>}
            </p>
            <p><i>C</i>.{<i>β</i>|<i>β</i>=60°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}</p>
            <p>
              <i>D</i>.{<i>β</i>|<i>β</i>=-60°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}
            </p>
            <p>（2） 与-70°角终边相同的角的集合表示正确的是（　）.</p>
            <p><i>A</i>.{<i>α</i>|<i>α</i>=-70°+<i>k</i>·360°}</p>
            <p><i>B</i>.{<i>α</i>|<i>α</i>=70°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}</p>
            <p>
              <i>C</i>.{<i>α</i>|<i>α</i>=-70°+<i>k</i>·180°，<i>k</i>∈<b>Z</b>}
            </p>
            <p>
              <i>D</i>.{<i>α</i>|<i>α</i>=-70°+<i>k</i>·360°，<i>k</i>∈<b>Z</b>}
            </p>
            <p>2.填空题.</p>
            <p>
              （1）
              在0°～360°内，与-50°角终边相同的角是____，则-50°角是第___象限角；
            </p>
            <p>
              （2） 在0°～360°内，与390°角终边相同的角是____，则390°是第___象限角；
            </p>
            <p>
              （3）
              在0°～360°内，与-480°角终边相同的角是____，则-480°角是第___象限角；
            </p>
            <p>
              （4） 在0°～360°内，与800°角终边相同的角是____，则800°角是第___象限角.
            </p>
            <p>3.写出与下列角终边相同的角的集合.</p>
            <p>（1） 与0°角终边相同的角的集合是__________________；</p>
            <p>（2） 与180°角终边相同的角的集合是__________________；</p>
            <p>（3） 终边在<i>x</i>轴上的角的集合是__________________.</p>
          </div>
        </div>
      </div>
    </div>
    <!-- 161 -->
@@ -112,7 +362,56 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <h3 id="c050">习题5.1<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.230° 角是第（　）象限角.</p>
            <p>A.一</p>
            <p>B.二</p>
            <p>C.三</p>
            <p>D.四</p>
            <p>2.与-420°角终边相同的角是（　）.</p>
            <p>A.420°</p>
            <p>B.-120°</p>
            <p>C.280°</p>
            <p>D.-60°</p>
            <p>3.（1） 与70°角终边相同的角的集合表示为__________________；</p>
            <p>（2） 与-120°角终边相同的角的集合表示为__________________.</p>
            <p>
              4.在0°～360°内，找出与下列各角终边相同的角，并判断它们分别是第几象限角.
            </p>
            <p>（1）-285°；（2） 570°.</p>
            <p>
              5.把下列各角化成<i>α</i>+<i>k</i>·360°（0°≤<i>α</i>＜360°，<i>k</i>∈<b>Z</b>）的形式，并指出它们是第几象限角.
            </p>
            <p>（1） 675°；（2） -520°.</p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>1.找出与1 200°角终边相同且绝对值最小的负角.</p>
            <p>
              2.设<i>α</i>为第二象限角，指出<math display="0">
                <mfrac>
                  <mi>α</mi>
                  <mn>2</mn>
                </mfrac>
              </math>是第几象限角.
            </p>
            <p>
              3.分别写出与下列各角终边相同的角组成的集合，并把满足不等式-360°＜<i>β</i>＜360°的<i>β</i>写出来.
            </p>
            <p>（1） 125°；（2） -380°；（3） 485°.</p>
          </div>
          <h2 id="b031">5.2 弧度制<span class="fontsz1">＞＞＞＞＞＞＞＞</span></h2>
          <h3 id="c051">5.2.1 弧度制的定义<span class="fontsz2">＞＞＞</span></h3>
          <p class="block">
            2016年9月25日，具有我国自主知识产权的世界最大单口径、最灵敏的球面射电望远镜“中国天眼”在贵州平塘落成启用.这个500
            m口径球面射电望远镜
          </p>
        </div>
      </div>
    </div>
    <!-- 162 -->
@@ -124,7 +423,97 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            主要用于实现巡视宇宙中的中性氢、观测脉冲星等科学目标和空间飞行器测量与通信等应用目标.
          </p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0172-2.jpg" /></p>
          <p>
            在衡量天体之间的距离时，我们可以用光年、米的单位制来度量；对于面积，我们可以用平方米、公顷等不同的单位制来度量；质量可以用千克、吨等不同的单位制来度量.角的大小，我们是否也能用不同的单位制来度量？
          </p>
          <div class="bk">
            <div class="bj1">
              <p class="left"><img class="img-gn1" alt="" src="../../assets/images/gn.jpg" /></p>
            </div>
            <p class="block">角度制</p>
            <p class="block">弧度制</p>
            <p class="block">弧度</p>
          </div>
          <p>
            我们知道，角可以以度为单位进行度量，把周角的<math display="0">
              <mfrac>
                <mn>1</mn>
                <mn>360</mn>
              </mfrac>
            </math>所对应的圆心角规定为1度的角，记为1°.这种以度为单位来度量角的单位制，叫作<b>角度制</b>.
          </p>
          <p>
            在数学和其他科学研究中，经常使用另一种度量角的单位制——<b>弧度制</b>.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            我们规定，长度等于半径的圆弧所对的圆心角叫作1弧度的角，弧度单位用符号rad表示，读作弧度.1弧度的角就记作1
            rad，读作“1弧度”，如图5-6所示.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0173-2.jpg" /></p>
          <p class="img">图5-6</p>
          <p>
            根据上述规定可知，在半径为<i>r</i>的圆中，若弧长为<i>l</i>的弧所对的圆心角为<i>α</i>
            rad，则<i>α</i>的大小为
          </p>
          <math display="block">
            <mo stretchy="false">|</mo>
            <mi>α</mi>
            <mrow>
              <mo stretchy="false">|</mo>
            </mrow>
            <mo>=</mo>
            <mfrac>
              <mi>l</mi>
              <mi>r</mi>
            </mfrac>
            <mtext>.&nbsp;</mtext>
          </math>
          <p>
            <i>α</i>的正负由<i>α</i>的始边到终边的旋转方向决定，逆时针方向旋转为正，顺时针方向旋转为负.
          </p>
          <p>
            当一个圆的半径为<i>r</i>时，若圆心角∠<i>AOB</i>所对的圆弧长为2<i>r</i>，则∠<i>AOB</i>的弧度数就为<math display="0">
              <mfrac>
                <mrow>
                  <mn>2</mn>
                  <mi>r</mi>
                </mrow>
                <mi>r</mi>
              </mfrac>
              <mo>=</mo>
              <mn>2</mn>
              <mrow>
                <mi mathvariant="normal">r</mi>
                <mi mathvariant="normal">a</mi>
                <mi mathvariant="normal">d</mi>
              </mrow>
            </math>=2
            rad（如图5-7（1））；若圆心角∠<i>AOB</i>所对的圆弧长为整个圆周长2<i>πr</i>，则∠<i>AOB</i>的弧度数就为<math display="0">
              <mfrac>
                <mrow>
                  <mn>2</mn>
                  <mi>π</mi>
                  <mi>r</mi>
                </mrow>
                <mi>r</mi>
              </mfrac>
              <mo>=</mo>
              <mn>2</mn>
              <mi>π</mi>
              <mrow>
                <mi mathvariant="normal">r</mi>
                <mi mathvariant="normal">a</mi>
                <mi mathvariant="normal">d</mi>
              </mrow>
            </math>（如图5-7（2）），即一个周角的弧度数是2<i>π rad</i>.
          </p>
        </div>
      </div>
    </div>
    <!-- 163 -->
@@ -138,7 +527,275 @@
            <p><span>163</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bk">
            <p>360°=2<i>π</i> rad； 180°=<i>π</i> rad；</p>
            <p>
              <math display="0">
                <msup>
                  <mn>1</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo>=</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>180</mn>
                </mfrac>
                <mi>rad</mi>
                <mo>≈</mo>
                <mn>0.01745</mn>
                <mrow>
                  <mi mathvariant="normal">r</mi>
                  <mi mathvariant="normal">a</mi>
                  <mi mathvariant="normal">d</mi>
                </mrow>
              </math>；
            </p>
            <p>
              <math display="0">
                <mn>1</mn>
                <mrow>
                  <mi mathvariant="normal">r</mi>
                  <mi mathvariant="normal">a</mi>
                  <mi mathvariant="normal">d</mi>
                </mrow>
                <mo>=</mo>
                <mfrac>
                  <msup>
                    <mn>180</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <mi>π</mi>
                </mfrac>
                <mo>≈</mo>
                <msup>
                  <mn>57.30</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo>=</mo>
                <msup>
                  <mn>57</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <msup>
                  <mn>18</mn>
                  <mrow>
                    <mi data-mjx-alternate="1" mathvariant="normal">′</mi>
                  </mrow>
                </msup>
              </math>.
            </p>
          </div>
          <p class="center"><img class="img-b" alt="" src="../../assets/images/0174-3.jpg" /></p>
          <p class="img">图5-7</p>
          <p>
            为了简便起见，以弧度为单位表示角的大小时，单位“弧度”或“rad”一般省略不写.例如，1
            rad，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mrow>
                <mi mathvariant="normal">r</mi>
                <mi mathvariant="normal">a</mi>
                <mi mathvariant="normal">d</mi>
              </mrow>
            </math>，0 rad 可简写成1，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>，0.
          </p>
          <p>一般地，正角的弧度数为正数，负角的弧度数为负数，零角的弧度数为0.</p>
          <p>
            当形成角的射线旋转一周后继续旋转，就可以得到弧度数大于2<i>π</i>或小于-2<i>π</i>的角.这样就可以得到任意弧度数的角.
          </p>
          <p>
            因此，每一个确定的角都有唯一确定的实数与它对应；反之，每一个确定的实数也都有唯一确定的角与它对应，如图5-8所示.这样，角与实数之间就建立了一一对应的关系.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0174-6.jpg" /></p>
          <p class="img">图5-8</p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　把下列各角化为弧度.
          </p>
          <p>（1） 30°；（2） -225°；（3） 0°.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（1）</mo>
              <msup>
                <mn>30</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mn>30</mn>
              <mo>×</mo>
              <mfrac>
                <mi>π</mi>
                <mn>180</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>.</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（2）</mo>
              <mo>−</mo>
              <msup>
                <mn>225</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mo>−</mo>
              <mn>225</mn>
              <mo>×</mo>
              <mfrac>
                <mi>π</mi>
                <mn>180</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
              <mo>.</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（3）</mo>
              <msup>
                <mn>0</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mn>0</mn>
              <mo>×</mo>
              <mfrac>
                <mi>π</mi>
                <mn>180</mn>
              </mfrac>
              <mo>=</mo>
              <mn>0</mn>
              <mo>.</mo>
            </math>
          </p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　把下列各角化为角度.
          </p>
          <p>
            （1）
            <math display="0">
              <mo>−</mo>
              <mfrac>
                <mrow>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>；（2） 5rad（结果精确到0.01）.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（1）</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msup>
                  <mn>180</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <msup>
                <mn>60</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>.</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（2）</mo>
              <mn>5</mn>
              <mrow>
                <mi mathvariant="normal">r</mi>
                <mi mathvariant="normal">a</mi>
                <mi mathvariant="normal">d</mi>
              </mrow>
              <mo>=</mo>
              <mn>5</mn>
              <mo>×</mo>
              <mfrac>
                <msup>
                  <mn>180</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mi>π</mi>
              </mfrac>
              <mo>≈</mo>
              <msup>
                <mn>286.44</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>.</mo>
            </math>
          </p>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              弧度化角度时,如果式子里有<i>π</i> ,直接把<i>π</i>转化成180°即可.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -150,7 +807,68 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例3</b></span>　利用科学计算器，把下列各角进行弧度与角度的互化.（结果精确到0.01）
          </p>
          <p>（1） -5.6；（2） 154°13′.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1）
            先将科学计算器的精确度设置为0.01，再将科学计算器设置为角度计算模式，科学计算器Ⅰ按<img class="inline" alt=""
              src="../../assets/images/0175-1.jpg" />，科学计算器Ⅱ按<img class="inline" alt=""
              src="../../assets/images/0175-2.jpg" />.之后依次按下列各键.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0175-3.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0175-4.jpg" /></p>
          <p>所以　-5.6 <i>rad</i> ≈-320.86°.</p>
          <p>
            （2）
            先将科学计算器的精确度设置为0.01，再将科学计算器设置为弧度计算模式，科学计算器Ⅰ按<img class="inline" alt=""
              src="../../assets/images/0175-5.jpg" />，科学计算器Ⅱ按<img class="inline" alt=""
              src="../../assets/images/0175-6.jpg" />.之后依次按下列各键.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0175-7.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0175-8.jpg" /></p>
          <p>所以　154°13′≈2.69 rad.</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.补充下列表格.</p>
            <p class="img">表5-1</p>
            <p class="center">
              <img class="img-a" alt="" src="../../assets/images/0175-9.jpg" />
            </p>
            <p>2.角度与弧度互化.</p>
            <p>（1） 225°=____；（2） -330°=____；</p>
            <p>
              （3）
              <math display="0">
                <mfrac>
                  <mrow>
                    <mn>9</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>4</mn>
                </mfrac>
                <mo>=</mo>
              </math>____；（4）
              <math display="0">
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>6</mn>
                </mfrac>
                <mo>=</mo>
              </math>____.
            </p>
            <p>3.利用科学计算器，进行弧度与角度的互化.（结果精确到0.01）</p>
            <p>（1） -3 <i>rad</i> =____；（2） 12°=____.</p>
          </div>
        </div>
      </div>
    </div>
    <!-- 165 -->
@@ -164,7 +882,255 @@
            <p><span>165</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h3 id="c052">
            5.2.2 弧长公式、扇形的面积公式<span class="fontsz2">＞＞＞</span>
          </h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>学习了弧度制后，你能推导出弧度制下的弧长和扇形的面积公式吗？</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0176-1.jpg" /></p>
          <p class="img">图5-9</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            如图5-9所示，已知半径为<i>r</i>的圆，设圆心角<i>α</i>=<i>n</i>°，且0°＜<i>α</i>＜360°，<i>α</i>所对的<math display="0">
              <mover>
                <mrow>
                  <mi>A</mi>
                  <mi>B</mi>
                </mrow>
                <mo>⏜</mo>
              </mover>
            </math>长为<i>l</i>，<i>α</i>所对应的扇形面积为<i>S</i>，则
          </p>
          <p class="center">
            <math display="">
              <mfrac>
                <mi>l</mi>
                <mrow>
                  <mn>2</mn>
                  <mi>π</mi>
                  <mi>r</mi>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mi>n</mi>
                <mn>360</mn>
              </mfrac>
            </math>，即<math display="0">
              <mi>l</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mi>n</mi>
                  <mi>π</mi>
                  <mi>r</mi>
                </mrow>
                <mn>180</mn>
              </mfrac>
            </math>(<i>n</i>°的圆心角所对的弧长为<math display="0">
              <mfrac>
                <mrow>
                  <mi>n</mi>
                  <mi>π</mi>
                  <mi>r</mi>
                </mrow>
                <mn>180</mn>
              </mfrac>
            </math>).
          </p>
          <p class="center">
            <math display="">
              <mfrac>
                <mi>S</mi>
                <mrow>
                  <mi>π</mi>
                  <msup>
                    <mi>r</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mi>n</mi>
                <mn>360</mn>
              </mfrac>
            </math>，即<math display="0">
              <mi>S</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mi>n</mi>
                  <mi>π</mi>
                  <msup>
                    <mi>r</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                </mrow>
                <mn>360</mn>
              </mfrac>
            </math>（<i>n</i>°的圆心角所对应的扇形面积为<math display="0">
              <mfrac>
                <mrow>
                  <mi>n</mi>
                  <mi>π</mi>
                  <msup>
                    <mi>r</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                </mrow>
                <mn>360</mn>
              </mfrac>
            </math>）.
          </p>
          <p>
            我们知道，弧长<i>l</i>与半径<i>r</i>的比值等于所对圆心角的弧度数，即<i>α</i>，<i>r</i>，<i>l</i>三者之间满足关系式
          </p>
          <math display="block">
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mi>l</mi>
              <mi>r</mi>
            </mfrac>
            <mo>.</mo>
          </math>
          <p>所以，弧长公式为<i>l</i>=<i>αr</i>.</p>
          <p>
            扇形的圆心角为<i>α</i>（0＜<i>α</i>＜2<i>π</i>），圆周角为2<i>π</i>，圆面积为<i>πr</i><sup>2</sup>，所以圆心角为<i>α</i>的扇形面积为
          </p>
          <math display="block">
            <mi>S</mi>
            <mo>=</mo>
            <mfrac>
              <mi>α</mi>
              <mrow>
                <mn>2</mn>
                <mi>π</mi>
              </mrow>
            </mfrac>
            <mo>⋅</mo>
            <mi>π</mi>
            <msup>
              <mi>r</mi>
              <mrow>
                <mn>2</mn>
              </mrow>
            </msup>
            <mo>=</mo>
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
            <mi>α</mi>
            <msup>
              <mi>r</mi>
              <mrow>
                <mn>2</mn>
              </mrow>
            </msup>
            <mo>=</mo>
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
            <mi>r</mi>
            <mo>⋅</mo>
            <mi>α</mi>
            <mi>r</mi>
            <mo>=</mo>
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
            <mi>r</mi>
            <mi>l</mi>
            <mo>.</mo>
          </math>
          <p>
            将采用角度制表示的和弧度制表示的弧长公式与扇形的面积公式进行对比可知，采用弧度制后弧长公式和扇形的面积公式就更简洁了.
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0176-11.jpg" /></p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　截至2021年4月，中国高速公路总里程约为16万千米，位居全球第一.某高速公路转弯处为一弧形高架桥，测得此处公路中线的总长为1
            200 m，该弧形高架桥所对应的圆心角为<math display="0">
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>5</mn>
              </mfrac>
            </math>，求该弧形高架桥的转弯半径（结果精确到1 m）.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 由题意可知，<i>l</i>=1 200，<math display="0">
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>5</mn>
              </mfrac>
            </math>，由<i>l</i>=<i>αr</i>可得
          </p>
          <math display="block">
            <mi>r</mi>
            <mo>=</mo>
            <mfrac>
              <mi>l</mi>
              <mi>α</mi>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mn>1200</mn>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>5</mn>
              </mfrac>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mn>1200</mn>
                <mo>×</mo>
                <mn>5</mn>
              </mrow>
              <mrow>
                <mn>3</mn>
                <mi>π</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mn>2000</mn>
              <mi>π</mi>
            </mfrac>
            <mo>≈</mo>
            <mn>645</mn>
            <mo stretchy="false">(</mo>
            <mrow>
              <mtext>&nbsp;</mtext>
              <mi mathvariant="normal">m</mi>
            </mrow>
            <mo stretchy="false">)</mo>
            <mo>.</mo>
          </math>
          <p>所以，该弧形高架桥的转弯半径约为645 m.</p>

        </div>
      </div>
    </div>

@@ -176,8 +1142,128 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0177-1.jpg" /></p>
          <p class="img">图5-10</p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　如图5-10所示，要在一块废铁皮上剪出一个扇形，用于制作一个圆锥筒，要求这个扇形的圆心角为60°，半径为90
            cm .请求出这个扇形的弧长与面积.（结果分别精确到0.01 cm和0.01
            cm<sup>2</sup>）
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 由于<math display="0">
              <msup>
                <mn>60</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>， 所以
          </p>
          <math display="block">
            <mtable columnalign="left" columnspacing="1em" rowspacing="4pt">
              <mtr>
                <mtd>
                  <mi>l</mi>
                  <mo>=</mo>
                  <mi>α</mi>
                  <mi>r</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>3</mn>
                  </mfrac>
                  <mo>×</mo>
                  <mn>90</mn>
                  <mo>=</mo>
                  <mn>30</mn>
                  <mi>π</mi>
                  <mo>≈</mo>
                  <mn>94.26</mn>
                  <mo stretchy="false">(</mo>
                  <mrow>
                    <mtext>&nbsp;</mtext>
                    <mi mathvariant="normal">c</mi>
                    <mi mathvariant="normal">m</mi>
                  </mrow>
                  <mo stretchy="false">)</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mi>S</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mn>1</mn>
                    <mn>2</mn>
                  </mfrac>
                  <mi>r</mi>
                  <mi>l</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mn>1</mn>
                    <mn>2</mn>
                  </mfrac>
                  <mo>×</mo>
                  <mn>90</mn>
                  <mo>×</mo>
                  <mn>30</mn>
                  <mi>π</mi>
                  <mo>≈</mo>
                  <mn>4241.70</mn>
                  <mrow data-mjx-texclass="INNER">
                    <mo data-mjx-texclass="OPEN">(</mo>
                    <msup>
                      <mrow>
                        <mtext>&nbsp;</mtext>
                        <mi mathvariant="normal">c</mi>
                        <mi mathvariant="normal">m</mi>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="CLOSE">)</mo>
                  </mrow>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <p>所以，这个扇形的弧长约为94.26 cm，面积约为4 241.70 cm<sup>2</sup>.</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.要在一个半径为120 mm的圆形塑料上切割一片弧长为144
              mm的扇形物料，切割时，该弧所对应的圆心角（正角）为____弧度.2.若弧形花台的弧长为20
              m，该弧所对应的圆心角为1.6 rad，则该弧形花台对应的转弯半径是____m.
            </p>
            <p>
              3.若扇形的圆心角为<math display="0">
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
              </math>，半径为5 cm，则此扇形的弧长为____cm，面积为____cm<sup>2</sup>.
            </p>
          </div>
          <h3 id="c053">习题5.2<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.在下图中填入适当的值.</p>
            <p class="center">
              <img class="img-f" alt="" src="../../assets/images/0177-5.jpg" />
            </p>
            <p class="img">第1题图</p>
          </div>
        </div>
      </div>
    </div>

@@ -193,7 +1279,179 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>2.角度与弧度互化.</p>
            <p>（1） 135°=____；（2） -225°=____；</p>
            <p>
              （3） -300°=___；（4）
              <math display="0">
                <mfrac>
                  <mn>2</mn>
                  <mn>3</mn>
                </mfrac>
                <mi>π</mi>
                <mo>=</mo>
              </math>
            </p>
            <p>
              （5）
              <math display="0">
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>4</mn>
                </mfrac>
                <mo>=</mo>
              </math>；（6）-3<i>π</i>=____.
            </p>
            <p>
              3.（1） 若<math display="0">
                <mi>α</mi>
                <mo>∈</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo>,</mo>
                  <mi>π</mi>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>，则<i>α</i>是第____象限角，<math display="0">
                <mfrac>
                  <mi>α</mi>
                  <mn>2</mn>
                </mfrac>
              </math>是第____象限角；
            </p>
            <p>
              （2） 若<math display="0">
                <mi>α</mi>
                <mo>∈</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mi>π</mi>
                  <mo>,</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>，则<i>α</i>是第____象限角，<math display="0">
                <mfrac>
                  <mi>α</mi>
                  <mn>2</mn>
                </mfrac>
              </math>是第____象限角.
            </p>
            <p>4.三角形的三个内角的度数之比为1∶2∶3，求最小内角的弧度数.</p>
            <p>5.经过1 <i>h</i>，钟表的时针和分针各转了多少度？分别是多少弧度？</p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>1.已知扇形的面积为2，扇形的圆心角的弧度数为4，求该扇形的周长.</p>
            <p>
              2.要在半径为100 cm的圆金属板上截取一块扇形板，使它的弧长为112
              cm，求该弧所对的圆心角的弧度数与角度数.（结果精确到1°）
            </p>
            <p>
              3.已知长50 cm 的弧所对的圆心角为200°，求该弧所在圆的半径.（结果精确到1
              cm）
            </p>
          </div>
          <h2 id="b032">
            5.3 任意角的正弦函数、余弦函数和正切函数<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/zshg.jpg" />
              </p>
            </div>
            <p class="block">
              初中我们在Rt△<i>ABC</i>中定义了锐角<i>α</i>的正弦、余弦和正切，如图5-11所示.
            </p>
            <p class="center">
              <img class="img-c" alt="" src="../../assets/images/0178-7.jpg" />
            </p>
            <p class="img">图5-11</p>
            <p class="block">
              正弦：<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mi>a</mi>
                  <mi>c</mi>
                </mfrac>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mi mathvariant="normal">∠</mi>
                    <mi>α</mi>
                    <mtext>&nbsp;的对边&nbsp;</mtext>
                  </mrow>
                  <mtext>&nbsp;斜边&nbsp;</mtext>
                </mfrac>
              </math>.
            </p>
            <p class="block">
              余弦：<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mi>b</mi>
                  <mi>c</mi>
                </mfrac>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mi mathvariant="normal">∠</mi>
                    <mi>α</mi>
                    <mtext>&nbsp;的邻边&nbsp;</mtext>
                  </mrow>
                  <mtext>&nbsp;斜边&nbsp;</mtext>
                </mfrac>
              </math>.
            </p>
            <p class="block">
              正切：<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mi>a</mi>
                  <mi>b</mi>
                </mfrac>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mi mathvariant="normal">∠</mi>
                    <mi>α</mi>
                    <mtext>&nbsp;的对边&nbsp;</mtext>
                  </mrow>
                  <mrow>
                    <mi mathvariant="normal">∠</mi>
                    <mi>α</mi>
                    <mtext>&nbsp;的邻边&nbsp;</mtext>
                  </mrow>
                </mfrac>
              </math>.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -206,7 +1464,220 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0179-1.jpg" /></p>
          <p class="img">图5-12</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            现在我们将一个锐角<i>α</i>放入平面直角坐标系中，使得顶点与原点重合，始边与<i>x</i>轴的非负半轴重合，如图5-12所示.已知点<i>P</i>（<i>x</i>，<i>y</i>）是锐角<i>α</i>终边上的任意一点，点
            <i>P</i>与原点<i>O</i>的距离<i>OP</i>=<i>r</i>（<i>r</i>＞0），你能利用锐角三角函数的定义计算出锐角<i>α</i>所对应的三角函数值吗？
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            过点<i>P</i>作<i>x</i>轴的垂线，垂足为<i>M</i>，则线段<i>OM</i>的长度为<i>x</i>，线段<i>MP</i>的长度为<i>y</i>.
          </p>
          <p>
            在<i>Rt</i> △<i>OMP</i>中，根据勾股定理可得，<math display="0">
              <mi>r</mi>
              <mo>=</mo>
              <msqrt>
                <msup>
                  <mi>x</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mo>+</mo>
                <msup>
                  <mi>y</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
              </msqrt>
              <mo>&gt;</mo>
              <mn>0</mn>
            </math>.
          </p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>M</mi>
                <mi>P</mi>
              </mrow>
              <mrow>
                <mi>O</mi>
                <mi>P</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>r</mi>
            </mfrac>
            <mo>,</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>O</mi>
                <mi>M</mi>
              </mrow>
              <mrow>
                <mi>O</mi>
                <mi>P</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mi>x</mi>
              <mi>r</mi>
            </mfrac>
            <mo>,</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>M</mi>
                <mi>P</mi>
              </mrow>
              <mrow>
                <mi>O</mi>
                <mi>M</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>x</mi>
            </mfrac>
            <mo>.</mo>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>在弧度制下，我们已将<i>α</i>的范围扩展到了全体实数.</p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0179-4.jpg" /></p>
          <p class="img">图5-13</p>
          <p>
            一般地，如图5-13所示，当<i>α</i>为任意角时，点<i>P</i>（<i>x</i>，<i>y</i>）是<i>α</i>的终边上异于原点的任意一点，点<i>P</i>到原点的距离为<math
              display="0">
              <mi>r</mi>
              <mo>=</mo>
              <msqrt>
                <msup>
                  <mi>x</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mo>+</mo>
                <msup>
                  <mi>y</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
              </msqrt>
            </math>.我们仍然将<i>α</i>的正弦、余弦、正切分别定义如下.
          </p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>r</mi>
            </mfrac>
            <mo>,</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mi>x</mi>
              <mi>r</mi>
            </mfrac>
            <mo>,</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>x</mi>
            </mfrac>
            <mo stretchy="false">(</mo>
            <mi>x</mi>
            <mo>≠</mo>
            <mn>0</mn>
            <mo stretchy="false">)</mo>
            <mo>.</mo>
          </math>
          <p>注意：当<i>α</i>的终边不在<i>y</i>轴上时，tan<i>α</i>才有意义.</p>
          <p>
            对于每一个确定的<i>α</i>，其正弦、余弦及正切都分别对应一个确定的比值，因此，正弦、余弦及正切都是以<i>α</i>为自变量的函数，分别叫作正弦函数、余弦函数及正切函数.
          </p>
          <p>
            当点<i>P</i>的横坐标<i>x</i>=0时，<i>α</i>的终边在<i>y</i>轴上，此时<math display="0">
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>+</mo>
              <mi>k</mi>
              <mi>π</mi>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>，<math display="0">
              <mfrac>
                <mi>y</mi>
                <mi>x</mi>
              </mfrac>
              <mo>=</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
            </math>无意义.除此之外，对于确定的<i>α</i>，三个函数都有意义.
          </p>
          <p>我们将正弦函数、余弦函数和正切函数统称为三角函数，通常记为：</p>
          <p><b>正弦函数</b> <i>y</i>=sin <i>x</i>，<i>x</i>∈<b>R</b>；</p>
          <p><b>余弦函数</b> <i>y</i>=cos <i>x</i>，<i>x</i>∈<b>R</b>；</p>
          <p>
            <b>正切函数</b> <i>y</i>=tan <i>x</i>，<math display="0">
              <mi>x</mi>
              <mo>≠</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>+</mo>
              <mi>k</mi>
              <mi>π</mi>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>.
          </p>
        </div>
      </div>
    </div>

@@ -221,7 +1692,131 @@
            <p><span>169</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例1</b></span>　如图5-14所示，已知<i>α</i>的终边经过点 <i>P</i>（3，-4），
            求sin<i>α</i>，cos<i>α</i>，tan<i>α</i>的值.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0180-3.jpg" /></p>
          <p class="img">图5-14</p>
          <p>
            <span class="zt-ls"><b>解</b></span> 由已知有<i>x</i>=3，<i>y</i>=-4，
          </p>
          <p>则</p>
          <math display="block">
            <mi>r</mi>
            <mo>=</mo>
            <msqrt>
              <msup>
                <mn>3</mn>
                <mrow>
                  <mn>2</mn>
                </mrow>
              </msup>
              <mo>+</mo>
              <mo stretchy="false">(</mo>
              <mo>−</mo>
              <mn>4</mn>
              <msup>
                <mo stretchy="false">)</mo>
                <mrow>
                  <mn>2</mn>
                </mrow>
              </msup>
            </msqrt>
            <mo>=</mo>
            <mn>5</mn>
            <mo>.</mo>
          </math>
          <p>于是</p>
          <math display="block">
            <mtable columnalign="left" columnspacing="1em" rowspacing="4pt">
              <mtr>
                <mtd>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mi>y</mi>
                    <mi>r</mi>
                  </mfrac>
                  <mo>=</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mn>4</mn>
                    <mn>5</mn>
                  </mfrac>
                  <mo>,</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mi>x</mi>
                    <mi>r</mi>
                  </mfrac>
                  <mo>=</mo>
                  <mfrac>
                    <mn>3</mn>
                    <mn>5</mn>
                  </mfrac>
                  <mo>,</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mi>y</mi>
                    <mi>x</mi>
                  </mfrac>
                  <mo>=</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mn>4</mn>
                    <mn>3</mn>
                  </mfrac>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.（1） 正弦函数表示为<i>y</i>=____，<i>x</i>∈____；</p>
            <p>（2） 余弦函数表示为<i>y</i>=____，<i>x</i>∈____；</p>
            <p>（3） 正切函数表示为<i>y</i>=____，<i>x</i>≠____.</p>
            <p>
              2.若<i>α</i>的终边过点（-8，6），则sin<i>α</i>=____，cos<i>α</i>=____，tan<i>α</i>=
              ____.
            </p>
            <p>
              3.若<i>α</i>的终边过点（5，12），则sin<i>α</i>=____，cos<i>α</i>=____，tan<i>α</i>=
              ____.
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            从<i>α</i>的正弦、余弦和正切的定义与实例可知，任意角的正弦值、余弦值和正切值在不同的象限有不同的符号.下面我们来研究各个象限内，任意角的正弦值、余弦值和正切值的符号的规律.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            以第二象限角为例，根据任意角的正弦、余弦和正切的定义，试分析它们在第二象限的符号情况.
          </p>
          <p>
            因为<i>α</i>的终边在第二象限，任取终边上异于原点的一点<i>P</i>（<i>x</i>，<i>y</i>），有
          </p>
          <p class="center"><i>x</i>＜0， <i>y</i>＞0， <i>OP</i>= <i>r</i>＞0.</p>
          <p>根据任意角的正弦、余弦和正切的定义可知，</p>
        </div>
      </div>
    </div>

@@ -233,7 +1828,76 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            （1）<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>y</mi>
                <mi>r</mi>
              </mfrac>
              <mo>&gt;</mo>
              <mn>0</mn>
            </math>；
          </p>
          <p>
            （2）<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>x</mi>
                <mi>r</mi>
              </mfrac>
              <mo>&lt;</mo>
              <mn>0</mn>
            </math>；
          </p>
          <p>
            （3）<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>y</mi>
                <mi>x</mi>
              </mfrac>
              <mo>&lt;</mo>
              <mn>0</mn>
            </math>.
          </p>
          <p>所以，可以得出第二象限各值的符号，见表5-2.</p>
          <p class="img">表5-2</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0181-4.jpg" /></p>
          <p>同理，可得出其他象限内各值的符号.</p>
          <p>
            一般地，<i>α</i>为任意角，<i>P</i>（<i>x</i>，<i>y</i>）为<i>α</i>终边上异于原点的任意一点，点
            <i>P</i>与原点<i>O</i>的距离<i>OP</i>=<i>r</i>.因为<i>r</i>＞0，由定义可知，
          </p>
          <p><b>正弦值的符号与点<i>P</i>的纵坐标<i>y</i>的符号相同；</b></p>
          <p><b>余弦值的符号与点<i>P</i>的横坐标<i>x</i>的符号相同；</b></p>
          <p>
            <b>正切值的符号与点<i>P</i>的纵坐标与横坐标的比值</b><math display="0">
              <mfrac>
                <mi>y</mi>
                <mi>x</mi>
              </mfrac>
            </math><b>的符号相同.</b>
          </p>
          <p>
            将点<i>P</i>（<i>x</i>，<i>y</i>）的坐标与各象限角的正弦值、余弦值和正切值的符号列表，如表5-3所示.
          </p>
          <p class="img">表5-3</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0181-6.jpg" /></p>
          <p>
            为了便于记忆，我们将sin<i>α</i>，cos<i>α</i>，tan<i>α</i>的符号标在各象限内，如图5-15所示.
          </p>
        </div>
      </div>
    </div>
    <!-- 171 -->
@@ -248,7 +1912,129 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0182-1.jpg" /></p>
          <p class="img">图5-15</p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　确定下列各三角函数值的符号.
          </p>
          <p>
            （1） sin（-210°）；（2） tan760°；（3）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>17</mn>
                  <mi>π</mi>
                </mrow>
                <mn>12</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 因为-210°是第二象限角，所以
          </p>
          <p class="center">sin（-210°）＞0.</p>
          <p>
            （2）
            因为760°=40°+2×360°，可知760°角与40°角的终边相同，是第一象限角，所以
          </p>
          <p class="center">tan 760°＞0.</p>
          <p>
            （3） 由<math display="0">
              <mfrac>
                <mrow>
                  <mn>17</mn>
                  <mi>π</mi>
                </mrow>
                <mn>12</mn>
              </mfrac>
              <mo>=</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mn>5</mn>
                <mn>12</mn>
              </mfrac>
              <mi>π</mi>
            </math>，可看出<math display="0">
              <mi>π</mi>
              <mo>&lt;</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>12</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mrow>
                  <mn>6</mn>
                  <mi>π</mi>
                </mrow>
                <mn>12</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
            </math>，是第三象限角，
          </p>
          <p>所以</p>
          <math display="block">
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>17</mn>
                <mi>π</mi>
              </mrow>
              <mn>12</mn>
            </mfrac>
            <mo>&lt;</mo>
            <mn>0</mn>
          </math>
          <p>
            <span class="zt-ls"><b>例3</b></span>　根据sin <i>α</i>＞0，且cos <i>α</i>＜0，确定<i>α</i>是第几象限角.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为sin
            <i>α</i>＞0，所以<i>α</i>的终边在第一或第二象限或<i>y</i>轴的正半轴上；又因为cos<i>α</i>＜0，所以<i>α</i>的终边在第二或第三象限或<i>x</i>轴的负半轴上.因此，<i>α</i>为第二象限角.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.用“＜”“＞”或“=”填空.</p>
            <p>
              （1） sin160°___0；（2）<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>18</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
              </math>
              ___0；
            </p>
            <p>（3） tan590°___0.</p>
            <p>2.（1） 若sin<i>α</i>＞0，则<i>α</i>的终边在______；</p>
            <p>（2） 若cos<i>α</i>＜0，则<i>α</i>的终边在______；</p>
            <p>（3） 若tan<i>α</i>＞0，则<i>α</i>的终边在第___或第___象限.</p>
            <p>3.若sin<i>α</i>＜0，且tan<i>α</i>＜0，则<i>α</i>是第___象限角.</p>
          </div>
        </div>
      </div>
    </div>

@@ -261,7 +2047,80 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img style="width: 24%;" alt="" src="../../assets/images/0183-1.jpg" /></p>
          <p class="img">图5-16</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            如图5-16所示，两个三角板上有几个特殊的锐角：30°，45°，60°.初中已研究了它们对应的正弦值、余弦值和正切值.现将角的范围进行了推广，已经在平面直角坐标系中研究了各象限角的正弦值、余弦值和正切值的符号分布规律.对于在平面直角坐标系中不属于任何象限的特殊角，如0°，90°，180°，270°等，它们的正弦值、余弦值和正切值又是多少？以180°为例，试求出它的正弦值、余弦值和正切值.
          </p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0183-2.jpg" /></p>
          <p class="img">图5-17</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            在平面直角坐标系中，180°角的终边正好与<i>x</i>轴的负半轴重合，如图5-17所示.以坐标原点为圆心、半径为单位长度的圆（简称单位圆）与<i>x</i>轴交于点<i>P</i>（-1，0），于是有
          </p>
          <p class="center"><i>x</i>=-1，<i>y</i>=0，<i>r</i>=1.</p>
          <p>根据任意角的正弦、余弦和正切的定义可知，</p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <msup>
              <mn>180</mn>
              <mrow>
                <mo>∘</mo>
              </mrow>
            </msup>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>r</mi>
            </mfrac>
            <mo>=</mo>
            <mn>0</mn>
            <mo>;</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <msup>
              <mn>180</mn>
              <mrow>
                <mo>∘</mo>
              </mrow>
            </msup>
            <mo>=</mo>
            <mfrac>
              <mi>x</mi>
              <mi>r</mi>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mn>1</mn>
            <mo>;</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <msup>
              <mn>180</mn>
              <mrow>
                <mo>∘</mo>
              </mrow>
            </msup>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>x</mi>
            </mfrac>
            <mo>=</mo>
            <mn>0</mn>
            <mo>.</mo>
          </math>
          <p><b>类比归纳</b></p>
          <p>
            一般地，取单位圆与坐标轴的交点就可以得到0°，90°，180°和270°等特殊角的正弦值、余弦值和正切值，如表5-4所示.
          </p>
          <p class="img">表5-4</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0183-4.jpg" /></p>
          <p>表中360°角与0°角的终边相同，对应的三角函数值也相同.</p>
        </div>
      </div>
    </div>
    <!-- 173 -->
@@ -275,8 +2134,264 @@
            <p><span>173</span></p>
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例4</b></span>　求5sin180°-4sin90°+2tan180°-7sin270°的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 5sin 180°-4sin 90°+2 tan 180°-7sin
            270°
          </p>
          <p>=5×0-4×1+2×0-7×（-1）</p>
          <p>=3.</p>
          <p>
            <span class="zt-ls"><b>例5</b></span>　求<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>−</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>+</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>π</mi>
              <mo>−</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
            </math>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mtable displaystyle="true"
                columnalign="right left right left right left right left right left right left"
                columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mi>π</mi>
                      <mn>6</mn>
                    </mfrac>
                    <mo>−</mo>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mi>π</mi>
                      <mn>3</mn>
                    </mfrac>
                    <mo>+</mo>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                        <mi>π</mi>
                      </mrow>
                      <mn>2</mn>
                    </mfrac>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd>
                    <mo>=</mo>
                  </mtd>
                  <mtd>
                    <mfrac>
                      <mn>1</mn>
                      <mn>2</mn>
                    </mfrac>
                    <mo>−</mo>
                    <mfrac>
                      <mn>1</mn>
                      <mn>2</mn>
                    </mfrac>
                    <mo>+</mo>
                    <mo stretchy="false">(</mo>
                    <mo>−</mo>
                    <mn>1</mn>
                    <mo stretchy="false">)</mo>
                    <mo>−</mo>
                    <mo stretchy="false">(</mo>
                    <mo>−</mo>
                    <mn>1</mn>
                    <mo stretchy="false">)</mo>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd>
                    <mo>=</mo>
                  </mtd>
                  <mtd>
                    <mn>0</mn>
                    <mo>.</mo>
                  </mtd>
                </mtr>
              </mtable>
            </math>
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.<math display="0">
                <mfrac>
                  <mrow>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <msup>
                      <mn>60</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                  </mrow>
                  <mrow>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <msup>
                      <mn>45</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                  </mrow>
                </mfrac>
                <mo>=</mo>
                <mo stretchy="false">(</mo>
                <mstyle scriptlevel="0">
                  <mspace width="1em"></mspace>
                </mstyle>
                <mo stretchy="false">)</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>6</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>2</mn>
                  </msqrt>
                  <mn>4</mn>
                </mfrac>
              </math>
            </p>
            <p>2.2cos 270°+5sin 0°+2cos 180°=____.</p>
            <p>
              3.<math display="0">
                <mn>5</mn>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>+</mo>
                <mn>2</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo>−</mo>
                <mn>3</mn>
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>π</mi>
                <mo>+</mo>
                <mn>6</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mn>0</mn>
                <mo>=</mo>
              </math>____.
            </p>
          </div>
          <h3 id="c054">习题5.3<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.若<i>α</i>的终边经过点<i>P</i>（-3，-4），则tan<i>α</i>=（　）.</p>
            <p>A.-3</p>
            <p>B.-4</p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>4</mn>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <mn>3</mn>
                  <mn>4</mn>
                </mfrac>
              </math>
            </p>
            <p>2.（1） 若sin<i>α</i>＜0，则<i>α</i>的终边在______；</p>
            <p>（2） 若cos<i>α</i>＞0，则<i>α</i>的终边在______；</p>
            <p>（3） 若tan<i>α</i>＜0，则<i>α</i>的终边在第___或第___象限.</p>
            <p>3.用“＜”“＞”或“=”填空.</p>
            <p>（1） sin 210°___0；（2） cos（-30°）___0；</p>
            <p>（3） tan 240°___0；（4） sin 150°___0.</p>
          </div>
        </div>
      </div>
    </div>

@@ -289,7 +2404,142 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>
              4.若<i>α</i>的终边经过点<i>P</i>，求sin<i>α</i>，cos<i>α</i>，tan<i>α</i>的值.
            </p>
            <p>（1） <i>P</i>（3，4）；（2） <i>P</i>（12，-5）.</p>
            <p>5.计算.</p>
            <p>
              （1）<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo>+</mo>
                <mn>2</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>3</mn>
                </mfrac>
                <mo>−</mo>
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>4</mn>
                </mfrac>
              </math>；
            </p>
            <p>（2） 2sin 0°-3sin 90°+4sin 180°-5sin 270°-6sin 360°；</p>
            <p>
              （3）<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mn>0</mn>
                <mo>−</mo>
                <mn>2</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>+</mo>
                <mn>3</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mn>4</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo>+</mo>
                <mn>5</mn>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mn>2</mn>
                <mi>π</mi>
              </math>.
            </p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>
              1.若tan<i>α</i>·cos<i>α</i>＞0，且cos<i>α</i>·sin<i>α</i>＜0，求<i>α</i>所在的象限.
            </p>
            <p>
              2.若<i>α</i>的终边经过点<i>Ρ</i>（3，<i>y</i>），且满足<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>5</mn>
                </mfrac>
              </math>，求sin<i>α</i>，tan<i>α</i>的值.
            </p>
            <p>
              3.已知<i>α</i>的终边经过点<i>P</i>（3<i>a</i>，-4<i>a</i>）（<i>a</i>≠0），求sin<i>α</i>，cos<i>α</i>，tan<i>α</i>的值.
            </p>
          </div>
          <h2 id="b033">
            5.4 同角三角函数的基本关系<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <p>
            在上一节，我们学习了三角函数的定义以及在各个象限的符号，那么同一个角的三角函数值之间是否存在某种关系呢？
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0185-4.jpg" /></p>
          <p class="img">图5-18</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            我们知道，在平面直角坐标系中，单位圆是以原点为圆心、单位长度为半径的圆.下面我们利用单位圆来研究同角三角函数的基本关系.如图5-18所示，已知点<i>P</i>（<i>x</i>，<i>y</i>）是角<i>α</i>的终边与单位圆的交点.过点<i>P</i>作<i>x</i>轴的垂线，垂足为<i>M</i>，则△<i>OMP</i>是直角三角形，且<i>OM</i>=｜<i>x</i>｜，<i>PM</i>=｜<i>y</i>｜，<i>OP</i>=<i>r</i>=1.
          </p>
          <p>根据正弦、余弦和正切的定义可知，在单位圆上，</p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mi>y</mi>
            <mo>;</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mi>x</mi>
            <mo>;</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mi>y</mi>
              <mi>x</mi>
            </mfrac>
            <mo>,</mo>
            <mi>x</mi>
            <mo>≠</mo>
            <mn>0</mn>
            <mo>.</mo>
          </math>
          <p>
            在Rt △<i>OPM</i>中，由勾股定理有<i>OM</i><sup>2</sup>+<i>PM</i><sup>2</sup>=<i>OP</i><sup>2</sup>，
          </p>
        </div>
      </div>
    </div>

@@ -304,7 +2554,397 @@
            <p><span>175</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>即<i>x</i><sup>2</sup>+<i>y</i><sup>2</sup>=1，</p>
          <p>所以sin<sup>2</sup><i>α</i>+cos<sup>2</sup><i>α</i>=1.</p>
          <p>显然，当<i>α</i>的终边与坐标轴重合时，这个公式也成立.</p>
          <p>
            根据正切的定义，当<math display="0">
              <mi>α</mi>
              <mo>≠</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>+</mo>
              <mi>k</mi>
              <mi>π</mi>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>时，
          </p>
          <math display="block">
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
            </mfrac>
            <mo>.</mo>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>一般地，可以得到同角三角函数的基本关系式.</p>
          <p>
            <b>（1） 平方关系：</b>sin<sup>2</sup><i>α</i>+cos<sup>2</sup><i>α</i>=1.
          </p>
          <p>
            <b>（2） 商数关系：</b><math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </mrow>
                <mrow>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </mrow>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　已知<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mn>3</mn>
                <mn>5</mn>
              </mfrac>
            </math>， 且<i>α</i>是第四象限角，求sin<i>α</i>，tan<i>α</i>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为
            <i>α</i>是第四象限角，所以sin<i>α</i>＜0 .
          </p>
          <math display="block">
            <mtable columnspacing="1em" rowspacing="4pt">
              <mtr>
                <mtd>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mo>−</mo>
                  <msqrt>
                    <mn>1</mn>
                    <mo>−</mo>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </msqrt>
                  <mo>=</mo>
                  <mo>−</mo>
                  <msqrt>
                    <mn>1</mn>
                    <mo>−</mo>
                    <msup>
                      <mrow data-mjx-texclass="INNER">
                        <mo data-mjx-texclass="OPEN">(</mo>
                        <mfrac>
                          <mn>3</mn>
                          <mn>5</mn>
                        </mfrac>
                        <mo data-mjx-texclass="CLOSE">)</mo>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                  </msqrt>
                  <mo>=</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mn>4</mn>
                    <mn>5</mn>
                  </mfrac>
                  <mo>,</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mi>sin</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>α</mi>
                    </mrow>
                    <mrow>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>α</mi>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mo>−</mo>
                      <mfrac>
                        <mn>4</mn>
                        <mn>5</mn>
                      </mfrac>
                    </mrow>
                    <mfrac>
                      <mn>3</mn>
                      <mn>5</mn>
                    </mfrac>
                  </mfrac>
                  <mo>=</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mn>4</mn>
                    <mn>3</mn>
                  </mfrac>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              根据sin<sup>2</sup><i>α</i>+cos<sup>2</sup><i>α</i>=1，可得<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <msqrt>
                  <mn>1</mn>
                  <mo>−</mo>
                  <msup>
                    <mi>cos</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </msqrt>
              </math>或<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mo>−</mo>
                <msqrt>
                  <mn>1</mn>
                  <mo>−</mo>
                  <msup>
                    <mi>cos</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </msqrt>
              </math>.其开方后的符号是由正弦值的象限符号来确定的.同理，开方后余弦值的符号也一样.
            </p>
          </div>
          <p>
            <span class="zt-ls"><b>例2</b></span>　已知<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mn>12</mn>
                <mn>5</mn>
              </mfrac>
            </math>，且<i>α</i>是第三象限角，求sin <i>α</i>，cos <i>α</i>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <math display="block">
            <mtext>&nbsp;由&nbsp;</mtext>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mn>12</mn>
              <mn>5</mn>
            </mfrac>
            <mtext>&nbsp;得,&nbsp;</mtext>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mn>12</mn>
              <mn>5</mn>
            </mfrac>
            <mtext>, 即&nbsp;</mtext>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mn>12</mn>
              <mn>5</mn>
            </mfrac>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mtext>.&nbsp;</mtext>
          </math>
          <p>把①代入</p>
          <math display="block">
            <msup>
              <mi>sin</mi>
              <mrow>
                <mn>2</mn>
              </mrow>
            </msup>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>+</mo>
            <msup>
              <mi>cos</mi>
              <mrow>
                <mn>2</mn>
              </mrow>
            </msup>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mn>1</mn>
            <mo>,</mo>
          </math>
          <p class="right">①</p>
          <p>得</p>
          <math display="block">
            <mtable columnspacing="1em" rowspacing="4pt">
              <mtr>
                <mtd>
                  <msup>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <mfrac>
                        <mn>12</mn>
                        <mn>5</mn>
                      </mfrac>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>α</mi>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo>+</mo>
                  <msup>
                    <mi>cos</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mn>1</mn>
                  <mo>,</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mfrac>
                    <mn>169</mn>
                    <mn>25</mn>
                  </mfrac>
                  <msup>
                    <mi>cos</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mn>1</mn>
                  <mo>,</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <msup>
                    <mi>cos</mi>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </msup>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mn>25</mn>
                    <mn>169</mn>
                  </mfrac>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <p>因为<i>α</i>是第三象限角，所以cos<i>α</i>＜0.</p>
          <p>
            所以<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mn>5</mn>
                <mn>13</mn>
              </mfrac>
            </math>.
          </p>
        </div>
      </div>
    </div>

@@ -316,7 +2956,538 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            把<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mn>5</mn>
                <mn>13</mn>
              </mfrac>
            </math>代入①式，得
          </p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mn>12</mn>
              <mn>5</mn>
            </mfrac>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>=</mo>
            <mfrac>
              <mn>12</mn>
              <mn>5</mn>
            </mfrac>
            <mo>×</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mo>−</mo>
              <mfrac>
                <mn>5</mn>
                <mn>13</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>=</mo>
            <mo>−</mo>
            <mfrac>
              <mn>12</mn>
              <mn>13</mn>
            </mfrac>
            <mo>.</mo>
          </math>
          <p>
            <span class="zt-ls"><b>例3</b></span>　求证sin<sup>4</sup><i>α</i>-cos<sup>4</sup><i>α</i>=2sin
            <sup>2</sup><i>α</i>-1.
          </p>
          <p>
            <b>证明</b> sin<sup>4</sup><i>α</i>-cos<sup>4</sup><i>α</i>=（sin
            <sup>2</sup><i>α</i>+cos<sup>2</sup><i>α</i>）（sin<sup>2</sup><i>α</i>-cos<sup>2</sup><i>α</i>）
          </p>
          <p>=sin<sup>2</sup><i>α</i>-cos<sup>2</sup><i>α</i></p>
          <p>=sin<sup>2</sup><i>α</i>-（1-sin<sup>2</sup><i>α</i>）</p>
          <p>=2sin<sup>2</sup><i>α</i>-1.</p>
          <p>
            <span class="zt-ls"><b>例4</b></span>　化简<math display="0">
              <mfrac>
                <mrow>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>−</mo>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
                <mrow>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>−</mo>
                  <mn>1</mn>
                </mrow>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <math display="block">
            <mo>由</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>θ</mi>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>θ</mi>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>θ</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mn>3</mn>
            <mo>,</mo>
            <mo>得</mo>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>θ</mi>
            <mo>=</mo>
            <mo>−</mo>
            <mn>3</mn>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>θ</mi>
            <mo>.</mo>
            <mtable displaystyle="true" columnalign="right left right left right left right left right left right left"
              columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
              <mtr>
                <mtd>
                  <mfrac>
                    <mrow>
                      <mn>4</mn>
                      <mi>sin</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                      <mo>−</mo>
                      <mn>2</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mn>5</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                      <mo>+</mo>
                      <mn>3</mn>
                      <mi>sin</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                </mtd>
                <mtd>
                  <mi></mi>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mn>4</mn>
                      <mo stretchy="false">(</mo>
                      <mo>−</mo>
                      <mn>3</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                      <mo stretchy="false">)</mo>
                      <mo>−</mo>
                      <mn>2</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mn>5</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                      <mo>+</mo>
                      <mn>3</mn>
                      <mo stretchy="false">(</mo>
                      <mo>−</mo>
                      <mn>3</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                      <mo stretchy="false">)</mo>
                    </mrow>
                  </mfrac>
                </mtd>
              </mtr>
              <mtr>
                <mtd></mtd>
                <mtd>
                  <mi></mi>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mo>−</mo>
                      <mn>14</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mo>−</mo>
                      <mn>4</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mfrac>
                    <mn>7</mn>
                    <mn>2</mn>
                  </mfrac>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              方法一的运算思路是由正弦函数、余弦函数变化为正切函数求出结果，我们简称为“弦化切”；方法二的运算思路是由正切函数变化为正弦函数和余弦函数的关系后求出结果，我们简称为“切化弦”.
            </p>
          </div>
          <p>
            <span class="zt-ls"><b>例5</b></span>　已知tan<i>θ</i>=-3，求<math display="0">
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>−</mo>
                  <mn>2</mn>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
                <mrow>
                  <mn>5</mn>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>+</mo>
                  <mn>3</mn>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
              </mfrac>
            </math>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 方法一：显然cos <i>θ</i>≠0，
          </p>
          <p class="left1">
            <math display="">
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>−</mo>
                  <mn>2</mn>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
                <mrow>
                  <mn>5</mn>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>+</mo>
                  <mn>3</mn>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mfrac>
                    <mrow>
                      <mn>4</mn>
                      <mi>sin</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>2</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                </mrow>
                <mrow>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                  <mo>+</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>sin</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                    <mrow>
                      <mi>cos</mi>
                      <mo data-mjx-texclass="NONE">⁡</mo>
                      <mi>θ</mi>
                    </mrow>
                  </mfrac>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                  <mo>−</mo>
                  <mn>2</mn>
                </mrow>
                <mrow>
                  <mn>5</mn>
                  <mo>+</mo>
                  <mn>3</mn>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mo>×</mo>
                  <mo stretchy="false">(</mo>
                  <mo>−</mo>
                  <mn>3</mn>
                  <mo stretchy="false">)</mo>
                  <mo>−</mo>
                  <mn>2</mn>
                </mrow>
                <mrow>
                  <mn>5</mn>
                  <mo>+</mo>
                  <mn>3</mn>
                  <mo>×</mo>
                  <mo stretchy="false">(</mo>
                  <mo>−</mo>
                  <mn>3</mn>
                  <mo stretchy="false">)</mo>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>7</mn>
                <mn>2</mn>
              </mfrac>
              <mo>.</mo>
            </math>
          </p>
          <p>方法二：</p>
          <p class="left1">
            <math display="">
              <mo>由</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>θ</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
                <mrow>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>θ</mi>
                </mrow>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mn>3</mn>
              <mo>,</mo>
              <mo>得</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>θ</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mn>3</mn>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>θ</mi>
              <mo>.</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mtable displaystyle="true"
                columnalign="right left right left right left right left right left right left"
                columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
                <mtr>
                  <mtd>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                        <mo>−</mo>
                        <mn>2</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                      </mrow>
                      <mrow>
                        <mn>5</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                        <mo>+</mo>
                        <mn>3</mn>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                      </mrow>
                    </mfrac>
                  </mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                        <mo stretchy="false">(</mo>
                        <mo>−</mo>
                        <mn>3</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                        <mo stretchy="false">)</mo>
                        <mo>−</mo>
                        <mn>2</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                      </mrow>
                      <mrow>
                        <mn>5</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                        <mo>+</mo>
                        <mn>3</mn>
                        <mo stretchy="false">(</mo>
                        <mo>−</mo>
                        <mn>3</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                        <mo stretchy="false">)</mo>
                      </mrow>
                    </mfrac>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mo>−</mo>
                        <mn>14</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                      </mrow>
                      <mrow>
                        <mo>−</mo>
                        <mn>4</mn>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>θ</mi>
                      </mrow>
                    </mfrac>
                    <mo>=</mo>
                    <mfrac>
                      <mn>7</mn>
                      <mn>2</mn>
                    </mfrac>
                    <mo>.</mo>
                  </mtd>
                </mtr>
              </mtable>
            </math>
          </p>
        </div>
      </div>
    </div>

@@ -332,7 +3503,210 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>5</mn>
                  <mn>13</mn>
                </mfrac>
              </math>，且<i>α</i>是第二象限角，则cos<i>α</i>=____，tan<i>α</i>=____.
            </p>
            <p>
              2.已知<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mo>−</mo>
                <msqrt>
                  <mo>−</mo>
                  <mn>3</mn>
                </msqrt>
              </math>，且<i>α</i>是第四象限角，则sin<i>α</i>=____，cos <i>α</i>=____.
            </p>
            <p>3.化简.</p>
            <p>
              （1）
              cos<i>α</i>·tan<i>α</i>=____；（2）（1-sin<i>x</i>）（1+sin<i>x</i>）=____.
            </p>
          </div>
          <h3 id="c055">习题5.4<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>5</mn>
                </mfrac>
              </math>，且<i>α</i>是第四象限角，则cos<i>α</i>=（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>4</mn>
                  <mn>5</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>4</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>3</mn>
                  <mn>5</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <mn>4</mn>
                  <mn>5</mn>
                </mfrac>
              </math>
            </p>
            <p>2.sin<sup>2</sup>25°+cos<sup>2</sup>25°=____.</p>
            <p>
              3.已知<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>，且<i>α</i>是第三象限角，则sin<i>α</i>=____，tan<i>α</i>=____.
            </p>
            <p>
              4.已知tan <i>α</i>=-1，且<i>α</i>是第四象限角，求sin <i>α</i>，cos
              <i>α</i>的值.
            </p>
            <p>
              5.化简<math display="0">
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>1</mn>
                  </mrow>
                  <mrow>
                    <mn>1</mn>
                    <mo>−</mo>
                    <mn>2</mn>
                    <msup>
                      <mi>sin</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                </mfrac>
              </math>.
            </p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>3</mn>
                </mfrac>
              </math>，且<math display="0">
                <mi>α</mi>
                <mo>∈</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>2</mn>
                  </mfrac>
                  <mo>,</mo>
                  <mn>2</mn>
                  <mi>π</mi>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>，求cos<i>α</i>，tan<i>α</i>的值.
            </p>
            <p>
              2.设tan<i>α</i>=3，求<math display="0">
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>+</mo>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                  <mrow>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>2</mn>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                </mfrac>
              </math>的值.
            </p>
            <p>3.求证.</p>
            <p>
              （1） sin<sup>4</sup><i>α</i>+sin<sup>2</sup><i>α</i>·cos<sup>2</sup><i>α</i>+cos<sup>2</sup><i>α</i>=1；
            </p>
            <p>
              （2） tan<sup>2</sup><i>α</i>-sin<sup>2</sup><i>α</i>=tan<sup>2</sup><i>α</i>·sin<sup>2</sup><i>α</i>.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -344,7 +3718,160 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b034">
            5.5 诱导公式<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <p>
            我们知道，图像的对称性是函数性质（如奇偶性）的重要几何特征.在上一节，我们借助单位圆推导了同角三角函数的基本关系式.下面，我们继续利用在平面直角坐标系中关于原点中心对称的单位圆，推导三角函数的诱导公式.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            我们知道，<math display="0">
              <mfrac>
                <mi>α</mi>
                <mn>3</mn>
              </mfrac>
            </math>和<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>（<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>可写为<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>+</mo>
              <mn>2</mn>
              <mi>π</mi>
            </math>）所对应的角是终边相同的角.想一想，<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>之间有什么关系？
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0189-9.jpg" /></p>
          <p class="img">图5-19</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            在平面直角坐标系中，由于<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>和<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>所对应的角的终边相同，所以由三角函数的定义可知，<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            如图5-19所示，角<i>α</i>的终边与单位圆的交点为<i>P</i>（cos<i>α</i>，sin<i>α</i>），终边继续旋转2<i>πk</i>（<i>k</i>∈<b>Z</b>）后，点<i>P</i>（cos<i>α</i>，sin<i>α</i>）又回到原来的位置，所以各三角函数值并不发生变化.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            我们知道，所有与<i>α</i>终边相同的角，连同<i>α</i>在内，可以组成一个集合
          </p>
          <p class="center">
            <i>S</i>={<i>β</i>|<i>β</i>=<i>α</i>+2<i>kπ</i>，<i>k</i>∈<b>Z</b>}.
          </p>
          <p>
            由三角函数的定义可知，角<i>α</i>+2<i>kπ</i>（<i>k</i>∈<b>Z</b>）与角<i>α</i>的同名三角函数的值相等（“同名”指同为正弦、余弦或正切，下同）.于是，当<i>k</i>∈<b>Z</b>时，
          </p>
          <div class="bj">
            <p class="center">
              有sin（<i>α</i>+2<i>kπ</i>）=sin <i>α</i>（<i>k</i>∈<b>Z</b>）；
            </p>
            <p class="center">
              　　　　cos（<i>α</i>+2<i>kπ</i>）=cos<i>α</i>（<i>k</i>∈<i>Z</i>）；
              公式一
            </p>
            <p class="center">
              tan（<i>α</i>+2<i>kπ</i>）=tan<i>α</i>（<i>k</i>∈<b>Z</b>）.
            </p>
          </div>
          <p>即终边相同的角同名三角函数值相等.</p>
        </div>
      </div>
    </div>

@@ -360,7 +3887,326 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例1</b></span>　求下列三角函数的值.
          </p>
          <p>
            （1）
            <math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>13</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>；（2）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>；（3） tan 405°.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1）<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>13</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo>+</mo>
                <mn>2</mn>
                <mi>π</mi>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>；
          </p>
          <p>
            （2）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>3</mn>
                </mfrac>
                <mo>−</mo>
                <mn>2</mn>
                <mi>π</mi>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>；
          </p>
          <p>（3） tan405°=tan（45°+360°）=tan45°=1.</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.sin 750°=（　）.</p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              2.<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>25</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>6</mn>
                </mfrac>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              3.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>7</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>4</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>=（　）.
            </p>
            <p>A.-1</p>
            <p>B.1</p>
            <p>
              C.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <msqrt>
                  <mn>3</mn>
                </msqrt>
              </math>
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/gcsk.jpg" /></p>
          <p>
            如图5-20所示，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>和<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>（<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>可写为<math display="0">
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>）所对应的角的终边关于坐标原点对称.想一想，<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>之间有什么关系？
          </p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0190-25.jpg" /></p>
          <p class="img">图5-20</p>
        </div>
      </div>
    </div>
    <!-- 180 -->
@@ -371,7 +4217,162 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            如图5-20所示，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>和<math display="0">
              <mfrac>
                <mrow>
                  <mn>7</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>所对应的角的终边与单位圆的交点分别是点<i>P</i>与点<i>P</i>′.根据对称性可知，它们的横坐标与纵坐标都互为相反数.
          </p>
          <p>由此得到</p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mi>π</mi>
              <mn>6</mn>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>7</mn>
                <mi>π</mi>
              </mrow>
              <mn>6</mn>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>,</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mi>π</mi>
              <mn>6</mn>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>7</mn>
                <mi>π</mi>
              </mrow>
              <mn>6</mn>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>.</mo>
          </math>
          <p>
            如图5-21所示，设单位圆与任意角<i>α</i>，π+<i>α</i>的终边分别相交于点<i>P</i>和<i>P</i>′，则点<i>P</i>和<i>P</i>′关于原点中心对称.如果点<i>P</i>的坐标是（cos
            <i>α</i>，sin
            <i>α</i>），那么点<i>P</i>′的坐标应该是（-cos<i>α</i>，-sin<i>α</i>）.又由于点<i>P</i>′作为角π+<i>α</i>的终边与单位圆的交点，其坐标应该是（cos（π+<i>α</i>），sin（π+<i>α</i>）），由此得到
          </p>
          <p class="center">
            cos（π+<i>α</i>）=-cos<i>α</i>，sin（π+<i>α</i>）=-sin<i>α</i>.
          </p>
          <p>由同角三角函数的关系式知</p>
          <math display="block">
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mo stretchy="false">(</mo>
            <mi>π</mi>
            <mo>+</mo>
            <mi>α</mi>
            <mo stretchy="false">)</mo>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mi>π</mi>
                <mo>+</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mi>π</mi>
                <mo>+</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mo>−</mo>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
              <mrow>
                <mo>−</mo>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>.</mo>
          </math>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0191-5.jpg" /></p>
          <p class="img">图5-21</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            与任意角<i>α</i>的终边关于原点中心对称的角π+<i>α</i>的正弦函数、余弦函数和正切函数的计算公式如下.
          </p>
          <div class="bj">
            <p class="center">　sin（π+<i>α</i>）=-sin<i>α</i>；</p>
            <p class="center">　　　　cos（π+<i>α</i>）=-cos<i>α</i>；公式二</p>
            <p class="center">tan（π+<i>α</i>）=tan<i>α</i>.</p>
          </div>
        </div>
      </div>
    </div>
    <!-- 181 -->
@@ -385,7 +4386,391 @@
            <p><span>181</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>例2</b></span>　求下列三角函数的值.
          </p>
          <p>
            （1） sin 225°；（2）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>；（3） tan 570°.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1）<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>225</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <msup>
                  <mn>180</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo>+</mo>
                <msup>
                  <mn>45</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>45</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
            </math>；
          </p>
          <p>
            　（2）<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>π</mi>
                <mo>+</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>3</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>；
          </p>
          <p>
            　（3）<math display="0">
              <mtable displaystyle="true"
                columnalign="right left right left right left right left right left right left"
                columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
                <mtr>
                  <mtd>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <msup>
                      <mn>570</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                  </mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <msup>
                        <mn>210</mn>
                        <mrow>
                          <mo>∘</mo>
                        </mrow>
                      </msup>
                      <mo>+</mo>
                      <msup>
                        <mn>360</mn>
                        <mrow>
                          <mo>∘</mo>
                        </mrow>
                      </msup>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                    <mo>=</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <msup>
                      <mn>210</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                    <mo>=</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <msup>
                        <mn>180</mn>
                        <mrow>
                          <mo>∘</mo>
                        </mrow>
                      </msup>
                      <mo>+</mo>
                      <msup>
                        <mn>30</mn>
                        <mrow>
                          <mo>∘</mo>
                        </mrow>
                      </msup>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                    <mo>=</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <msup>
                      <mn>30</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mfrac>
                      <msqrt>
                        <mn>3</mn>
                      </msqrt>
                      <mn>3</mn>
                    </mfrac>
                  </mtd>
                </mtr>
              </mtable>
            </math>
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.sin 240°（　）.</p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              2.<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>10</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              3.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>21</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>4</mn>
                </mfrac>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>A.-1</p>
            <p>
              B.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>C.1</p>
            <p>
              D.<math display="0">
                <msqrt>
                  <mn>3</mn>
                </msqrt>
              </math>
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/gcsk.jpg" /></p>
          <p>
            如图5-22所示，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>和<math display="0">
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>所对应的角的终边关于<i>x</i>轴对称.想一想，<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>之间有什么关系？
          </p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0192-23.jpg" /></p>
          <p class="img">图5-22</p>
        </div>
      </div>
    </div>
    <!-- 182 -->
@@ -396,7 +4781,368 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">

          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            如图5-22所示，<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>和<math display="0">
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>所对应的角的终边与单位圆的交点分别是点<i>P</i>与点<i>P</i>′.根据对称性可知，点<i>P</i>与点<i>P</i>′的横坐标相同、纵坐标互为相反数.
          </p>
          <p>
            由此得到<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>，<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0193-5.jpg" /></p>
          <p class="img">图5-23</p>
          <p>
            如图5-23所示，设单位圆与任意角<i>α</i>，-<i>α</i>的终边分别相交于点<i>P</i>和点<i>P</i>′，则点<i>P</i>与点<i>P</i>′关于<i>x</i>轴对称.如果点<i>P</i>的坐标是（cos<i>α</i>，sin<i>α</i>），那么点<i>P</i>′的坐标是（cos<i>α</i>，-sin<i>α</i>）.由于点<i>P</i>′作为角-<i>α</i>的终边与单位圆的交点，其坐标应该是（cos（-<i>α</i>），sin（-<i>α</i>）），于是得到
          </p>
          <p class="center">
            cos（-<i>α</i>）=cos<i>α</i>，sin（-<i>α</i>）=-sin<i>α</i>.
          </p>
          <p>由同角三角函数的关系式知</p>
          <math display="block">
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mo stretchy="false">(</mo>
            <mo>−</mo>
            <mi>α</mi>
            <mo stretchy="false">)</mo>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mo>−</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mo>−</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mo>−</mo>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>.</mo>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            与任意角<i>α</i>的终边关于<i>x</i>轴对称的角-<i>α</i>的正弦函数、余弦函数和正切函数的计算公式如下.
          </p>
          <div class="bj">
            <p class="center">sin（-<i>α</i>）=-sin<i>α</i>；</p>
            <p class="center">cos（-<i>α</i>）=cos<i>α</i>；公式三</p>
            <p class="center">tan（-<i>α</i>）=-tan<i>α</i>.</p>
          </div>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              利用公式三，可以把负角的三角函数转化为正角的三角函数.
            </p>
          </div>
          <p>
            <span class="zt-ls"><b>例3</b></span>　求下列三角函数的值.
          </p>
          <p>
            （1） sin（-45°）；（2） cos（-390°）；（3）
            <math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mo stretchy="false">(</mo>
              <mo>−</mo>
              <mfrac>
                <mrow>
                  <mn>16</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo stretchy="false">)</mo>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（1）</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <msup>
                  <mn>45</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>45</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
              <mo>;</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（2）</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <msup>
                  <mn>390</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>390</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <msup>
                  <mn>30</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo>+</mo>
                <msup>
                  <mn>360</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>30</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mfrac>
                <msqrt>
                  <mn>3</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
              <mo>;</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（3）</mo>
              <mtable displaystyle="true"
                columnalign="right left right left right left right left right left right left"
                columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
                <mtr>
                  <mtd>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <mo>−</mo>
                      <mfrac>
                        <mrow>
                          <mn>16</mn>
                          <mi>π</mi>
                        </mrow>
                        <mn>3</mn>
                      </mfrac>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                  </mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mrow>
                        <mn>16</mn>
                        <mi>π</mi>
                      </mrow>
                      <mn>3</mn>
                    </mfrac>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <mfrac>
                        <mrow>
                          <mn>4</mn>
                          <mi>π</mi>
                        </mrow>
                        <mn>3</mn>
                      </mfrac>
                      <mo>+</mo>
                      <mn>4</mn>
                      <mi>π</mi>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                        <mi>π</mi>
                      </mrow>
                      <mn>3</mn>
                    </mfrac>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mrow data-mjx-texclass="INNER">
                      <mo data-mjx-texclass="OPEN">(</mo>
                      <mi>π</mi>
                      <mo>+</mo>
                      <mfrac>
                        <mi>π</mi>
                        <mn>3</mn>
                      </mfrac>
                      <mo data-mjx-texclass="CLOSE">)</mo>
                    </mrow>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mfrac>
                      <mi>π</mi>
                      <mn>3</mn>
                    </mfrac>
                    <mo>=</mo>
                    <mo>−</mo>
                    <msqrt>
                      <mn>3</mn>
                    </msqrt>
                  </mtd>
                </mtr>
              </mtable>
            </math>
          </p>
        </div>
      </div>
    </div>
    <!-- 183  -->
@@ -410,7 +5156,223 @@
            <p><span>183</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>6</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              2.<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>4</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>3</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              3.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>9</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>4</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>A.1</p>
            <p>
              B.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>C.-1</p>
            <p>
              D.<math display="0">
                <msqrt>
                  <mn>3</mn>
                </msqrt>
              </math>
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/gcsk.jpg" /></p>
          <p>
            如图5-24所示，<i>α</i>和π-<i>α</i>所对应的角的终边关于<i>y</i>轴对称.想一想，sin<i>α</i>与sin（π-<i>α</i>），cos<i>α</i>与cos（π-<i>α</i>）之间有什么关系？
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0194-14.jpg" /></p>
          <p class="img">图5-24</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            如图5-24所示，设单位圆与角<i>α</i>，π-<i>α</i>的终边分别相交于点<i>P</i>和点<i>P</i>′，则点<i>P</i>与点<i>P</i>′关于<i>y</i>轴对称.如果点<i>P</i>的坐标是（cos<i>α</i>，sin<i>α</i>），那么点<i>P</i>′的坐标是（-cos<i>α</i>，sin<i>α</i>）.由于点<i>P</i>′作为角π-<i>α</i>的终边与单位圆的交点，其坐标应该是（cos（π-<i>α</i>），sin（π-<i>α</i>）），
          </p>
          <p class="center">
            cos（π-<i>α</i>）=-cos<i>α</i>， sin（π-<i>α</i>）=sin<i>α</i>.
          </p>
          <p>由同角三角函数的关系式知</p>
          <math display="block">
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mo stretchy="false">(</mo>
            <mi>π</mi>
            <mo>−</mo>
            <mi>α</mi>
            <mo stretchy="false">)</mo>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
              <mrow>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mo stretchy="false">(</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mi>α</mi>
                <mo stretchy="false">)</mo>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mfrac>
              <mrow>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
              <mrow>
                <mo>−</mo>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
              </mrow>
            </mfrac>
            <mo>=</mo>
            <mo>−</mo>
            <mi>tan</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mi>α</mi>
            <mo>.</mo>
          </math>
        </div>
      </div>
    </div>
    <!-- 184 -->
@@ -421,7 +5383,420 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            与任意角<i>α</i>的终边关于<i>y</i>轴对称的角π-<i>α</i>的正弦函数、余弦函数和正切函数的计算公式如下.
          </p>
          <div class="bj">
            <p class="center">sin（π-<i>α</i>）=sin<i>α</i>；</p>
            <p class="center">cos（π-<i>α</i>）=-cos<i>α</i>； 公式四</p>
            <p class="center">tan（π-<i>α</i>）=-tan<i>α</i>.</p>
          </div>
          <p>
            公式一至公式四统称为三角函数的诱导公式.利用这些公式可以把任意角的三角函数转化为锐角三角函数.
          </p>
          <p>
            <span class="zt-ls"><b>例4</b></span>　求下列三角函数的值.
          </p>
          <p>
            （1） cos 135°；（2）
            <math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>8</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>；（3）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>11</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（1）</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>135</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <msup>
                  <mn>180</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo>−</mo>
                <msup>
                  <mn>45</mn>
                  <mrow>
                    <mo>∘</mo>
                  </mrow>
                </msup>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <msup>
                <mn>45</mn>
                <mrow>
                  <mo>∘</mo>
                </mrow>
              </msup>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
              <mo>;</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（2）</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>8</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>+</mo>
                <mn>2</mn>
                <mi>π</mi>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>2</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>3</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <msqrt>
                <mn>3</mn>
              </msqrt>
              <mo>;</mo>
            </math>
          </p>
          <p class="left1">
            <math display="">
              <mo stretchy="false">（3）</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>11</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>4</mn>
                </mfrac>
                <mo>+</mo>
                <mn>2</mn>
                <mi>π</mi>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>4</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mo>−</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>4</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
              <mo>.</mo>
            </math>
          </p>
          <p>
            <span class="zt-ls"><b>例5</b></span>　化简：<math display="0">
              <mfrac>
                <mrow>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mo stretchy="false">(</mo>
                  <mn>2</mn>
                  <mi>π</mi>
                  <mo>−</mo>
                  <mi>α</mi>
                  <mo stretchy="false">)</mo>
                  <mo>⋅</mo>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mo stretchy="false">(</mo>
                  <mn>3</mn>
                  <mi>π</mi>
                  <mo>+</mo>
                  <mi>α</mi>
                  <mo stretchy="false">)</mo>
                </mrow>
                <mrow>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mo stretchy="false">(</mo>
                  <mo>−</mo>
                  <mi>π</mi>
                  <mo>+</mo>
                  <mi>α</mi>
                  <mo stretchy="false">)</mo>
                  <mo>⋅</mo>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mo stretchy="false">(</mo>
                  <mn>3</mn>
                  <mi>π</mi>
                  <mo>−</mo>
                  <mi>α</mi>
                  <mo stretchy="false">)</mo>
                  <mo>⋅</mo>
                  <mi>tan</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mo stretchy="false">(</mo>
                  <mo>−</mo>
                  <mi>α</mi>
                  <mo>−</mo>
                  <mi>π</mi>
                  <mo stretchy="false">)</mo>
                </mrow>
              </mfrac>
            </math>
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
          </p>
          <p class="left1">
            <math display="">
              <mtable displaystyle="true"
                columnalign="right left right left right left right left right left right left"
                columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" rowspacing="3pt">
                <mtr>
                  <mtd>
                    <mtext>&nbsp;原式&nbsp;</mtext>
                  </mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mo>−</mo>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo>⋅</mo>
                        <mo stretchy="false">(</mo>
                        <mo>−</mo>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo stretchy="false">)</mo>
                      </mrow>
                      <mrow>
                        <mo>−</mo>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo stretchy="false">(</mo>
                        <mo>−</mo>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo stretchy="false">)</mo>
                        <mo stretchy="false">(</mo>
                        <mo>−</mo>
                        <mi>tan</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo stretchy="false">)</mo>
                      </mrow>
                    </mfrac>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mfrac>
                      <mrow>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo>⋅</mo>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                      </mrow>
                      <mrow>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo>⋅</mo>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                        <mo>⋅</mo>
                        <mfrac>
                          <mrow>
                            <mi>sin</mi>
                            <mo data-mjx-texclass="NONE">⁡</mo>
                            <mi>α</mi>
                          </mrow>
                          <mrow>
                            <mi>cos</mi>
                            <mo data-mjx-texclass="NONE">⁡</mo>
                            <mi>α</mi>
                          </mrow>
                        </mfrac>
                      </mrow>
                    </mfrac>
                  </mtd>
                </mtr>
                <mtr>
                  <mtd></mtd>
                  <mtd>
                    <mi></mi>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mfrac>
                      <mrow>
                        <mi>sin</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                      </mrow>
                      <mrow>
                        <mi>cos</mi>
                        <mo data-mjx-texclass="NONE">⁡</mo>
                        <mi>α</mi>
                      </mrow>
                    </mfrac>
                    <mo>=</mo>
                    <mo>−</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>.</mo>
                  </mtd>
                </mtr>
              </mtable>
            </math>
          </p>
          <p><b>归纳总结</b></p>
          <p>
            利用诱导公式，把任意角的三角函数值转化为锐角的三角函数值的一般步骤为：
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0195-6.jpg" /></p>
        </div>
      </div>
    </div>

@@ -436,7 +5811,189 @@
            <p><span>185</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            事实上，以上步骤体现了将未知转化为已知的化归思想.利用公式一至公式四，按上述步骤解决了求三角函数值这个重要而困难的问题.现在，由于计算工具的便捷使用，对于三角函数的“求值”已不是问题，但其中的思想方法在解决三角函数的各种问题中却依然有重要的作用.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.sin 150°=（　）.</p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              2.<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>14</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>3</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              3.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>29</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>6</mn>
                </mfrac>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>A.-1</p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>C.1</p>
            <p>
              D.<math display="0">
                <msqrt>
                  <mn>3</mn>
                </msqrt>
              </math>
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            前面我们探究了求特殊角的三角函数值的方法，而对于不是特殊角的三角函数值又该如何求值呢？使用计算工具就能很容易地解决这个问题.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            利用科学计算器的<img class="inline" alt="" src="../../assets/images/0196-13.jpg" />键，可以方便地计算任意角的三角函数值.
          </p>
          <p>
            主要步骤如下：设置精确度→设置模式（角度制或弧度制）→按键<img class="inline" alt="" src="../../assets/images/0196-14.jpg" />（或键<img
              class="inline" alt="" src="../../assets/images/0196-15.jpg" />）→输入角的大小→按键<img class="inline" alt=""
              src="../../assets/images/0196-16.jpg" />显示结果.
          </p>
          <p>
            <span class="zt-ls"><b>例6</b></span>　利用科学计算器计算.（结果精确到0.01）
          </p>
          <p>
            （1） sin 63°52′41″；（2）<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>；（3）
            <math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>6</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>.
          </p>
        </div>
      </div>
    </div>
    <!-- 186 -->
@@ -447,7 +6004,90 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            <span class="zt-ls"><b>解</b></span>（1）
            先将精确度设置为0.01，再将科学计算器设置为角度计算模式，然后依次按下列各键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-1.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-2.jpg" /></p>
          <p>所以 sin 63°52′41″≈0.90.</p>
          <p>
            （2）
            先将精确度设置为0.01，再将科学计算器设置为弧度计算模式，然后依次按下列各键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-3.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-4.jpg" /></p>
          <p>
            所以
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>4</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mo>−</mo>
              <mn>0.50</mn>
              <mo>.</mo>
            </math>
          </p>
          <p>
            （3）
            先将精确度设置为0.01，再将科学计算器设置为弧度计算模式，然后依次按下列各键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-6.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0197-7.jpg" /></p>
          <p>
            所以<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>6</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>≈</mo>
              <mo>−</mo>
              <mn>0.73</mn>
              <mo>.</mo>
            </math>
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>1.利用科学计算器求值.（结果精确到0.01）</p>
            <p>
              （1） sin 1 480°10′12″____；（2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>9</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
                <mo>≈</mo>
              </math>____；
            </p>
            <p>（3） tan（-3.6）≈____.</p>
          </div>
        </div>
      </div>
    </div>

@@ -462,7 +6102,344 @@
            <p><span>187</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>2.先填“＜”“＞”或“=”，再用科学计算器加以验证.</p>
            <p>
              （1） sin 516°____0；（2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>16</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
              </math>；
            </p>
            <p>（3） tan（-1 050°）____0.</p>
          </div>
          <h3 id="c056">习题5.5<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>
              1.<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>14</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>=</mo>
              </math>（　）.
            </p>
            <p>
              A.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>2.tan 315°=（　）.</p>
            <p>A.1</p>
            <p>B.-1</p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>
            </p>
            <p>3.填空题.（（7）～（9）小题用科学计算器完成，结果精确到0.001）</p>
            <p>（1） sin 240°=____；（2） cos330°=____；</p>
            <p>
              （3） tan 225°=____；（4）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>13</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>=</mo>
              </math>；
            </p>
            <p>
              （5）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>7</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>6</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>；（6）
              <math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>17</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>4</mn>
                </mfrac>
                <mo>=</mo>
              </math>____；
            </p>
            <p>
              （7）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>12</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
                <mo>≈</mo>
              </math>____；（8）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>7</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>≈</mo>
              </math>____；
            </p>
            <p>（9） tan236°7′≈____.</p>
            <p>
              4.计算<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>6</mn>
                </mfrac>
                <mo>−</mo>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>+</mo>
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>4</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>.
            </p>
            <p>
              5.化简<math display="0">
                <mfrac>
                  <mrow>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                    <mo>⋅</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>2</mn>
                    <mi>π</mi>
                    <mo stretchy="false">)</mo>
                    <mo>⋅</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mn>2</mn>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                  </mrow>
                  <mrow>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mi>π</mi>
                    <mo>+</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                  </mrow>
                </mfrac>
              </math>.
            </p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mi>α</mi>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>3</mn>
                </mfrac>
              </math>，求sin（π-<i>α</i>）的值.
            </p>
            <p>2.求值sin（-1 200°）·cos 1 290°+cos（-1 020°）·sin（-1 050°）.</p>
            <p>
              3.化简<math display="0">
                <mfrac>
                  <mrow>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                    <mo>⋅</mo>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mn>3</mn>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                    <mo>⋅</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mi>π</mi>
                    <mo stretchy="false">)</mo>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mn>2</mn>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                  </mrow>
                  <mrow>
                    <mi>tan</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mn>4</mn>
                    <mi>π</mi>
                    <mo>−</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                    <mo>⋅</mo>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mo stretchy="false">(</mo>
                    <mn>5</mn>
                    <mi>π</mi>
                    <mo>+</mo>
                    <mi>α</mi>
                    <mo stretchy="false">)</mo>
                  </mrow>
                </mfrac>
              </math>.
            </p>
          </div>
        </div>
      </div>
    </div>
    <!-- 188 -->
@@ -473,7 +6450,50 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b035">
            5.6 正弦函数的图像和性质<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <h3 id="c057">5.6.1 正弦函数的图像<span class="fontsz2">＞＞＞</span></h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/gcsk.jpg" /></p>
          <p>
            如果今天是2021年3月17日星期三，那么往前推7天是周几？往后推7天是周几？再过7天又是周几？
          </p>
          <p>显然，前面所有问题都是同一个答案：周三.</p>
          <p>
            生活中，像这样每隔7天，“周三”又会重复出现，这个“7天”就是我们常说的一周（一个周期），这种每隔一段时间便会重复出现的现象称为周期现象.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0199-1.jpg" /></p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            我们知道，单位圆上任意一点在圆周上旋转一周就回到原来的位置，这说明，
            在函数<i>y</i>=sin<i>x</i>中，当自变量每间隔2π个单位长度时，对应的函数值都会重复出现，即sin（<i>x</i>+2π）=sin<i>x</i>.
          </p>
          <div class="bk">
            <div class="bj1">
              <p class="left"><img class="img-gn1" alt="" src="../../assets/images/gn.jpg" /></p>
            </div>
            <p class="block">周期函数</p>
            <p class="block">周期</p>
            <p class="block">最小正周期</p>
          </div>
          <p>
            一般地，对于函数<i>y</i>=<i>f</i>（<i>x</i>），如果存在一个非零常数<i>T</i>，当<i>x</i>取定义域<i>D</i>内的每一个值时，都有<i>x</i>+<i>T</i>∈<i>D</i>，并且都满足
          </p>
          <p class="center"><i>f</i>（<i>x</i>+<i>T</i>）=<i>f</i>（<i>x</i>），</p>
          <p>
            则称函数<i>y</i>=<i>f</i>（<i>x</i>）为<b>周期函数</b>，非零常数<i>T</i>叫作这个函数的一个<b>周期</b>.
          </p>
          <p>
            例如，函数<i>y</i>=sin<i>x</i>中，对于任意<i>x</i>∈<b>R</b>，都有<i>x</i>+2π∈<b>R</b>，且满足<i>f</i>（<i>x</i>+2π）=<i>f</i>（<i>x</i>）.可见，正弦函数是周期函数，且2π是它的一个周期.
          </p>
          <p>
            又由sin（<i>x</i>+2π<i>k</i>）=sin<i>x</i>（<i>k</i>∈<b>Z</b>），可知2π，4π，6π，…及-2π，-4π，-6π，…都是正弦函数<i>y</i>=sin<i>x</i>的周期.
          </p>
          <p>
            对于一个周期函数<i>y</i>=<i>f</i>（<i>x</i>），如果在它的所有的周期中存在一个最小的正数，就称这个最小的正数为<i>y</i>=<i>f</i>（<i>x</i>）的<b>最小正周期</b>.
          </p>
        </div>
      </div>
    </div>

@@ -488,7 +6508,82 @@
            <p><span>189</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            由此可见，2π就是正弦函数<i>y</i>=sin<i>x</i>的最小正周期.为了简便起见，本书所指的三角函数的周期一般指函数的最小正周期.因此，我们说正弦函数的周期是2π.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            <i>y</i>=sin<i>x</i>是以2π为周期的函数，所以只要画出它在一个完整周期内的图像，再利用周期性就可以得到正弦函数的图像.
          </p>
          <p>
            首先，列表.自变量<i>x</i>的取值如表5-5所示，利用科学计算器求出<i>y</i>=sin<i>x</i>的各个值并填入表中.
          </p>
          <p class="img">表5-5</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0200-1.jpg" /></p>
          <p>
            其次，描点连线.根据表中数值描点，然后用光滑的曲线把各点连接起来，绘制出在［0，2π］上的图像，如图5-25所示.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0200-2.jpg" /></p>
          <p class="img">图5-25</p>
          <p>
            由图5-25可以看出，决定函数<i>y</i>=sin<i>x</i>（<i>x</i>∈0，2π）
            图像形状的有五个关键点，即
          </p>
          <math display="block">
            <mo stretchy="false">(</mo>
            <mn>0</mn>
            <mo>,</mo>
            <mn>0</mn>
            <mo stretchy="false">)</mo>
            <mo>,</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>,</mo>
              <mn>1</mn>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>,</mo>
            <mo stretchy="false">(</mo>
            <mi>π</mi>
            <mo>,</mo>
            <mn>0</mn>
            <mo stretchy="false">)</mo>
            <mo>,</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
              <mo>,</mo>
              <mo>−</mo>
              <mn>1</mn>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>,</mo>
            <mo stretchy="false">(</mo>
            <mn>2</mn>
            <mi>π</mi>
            <mo>,</mo>
            <mn>0</mn>
            <mo stretchy="false">)</mo>
            <mo>.</mo>
          </math>
          <p>
            因此，在精确度要求不高时，经常先找出这五个关键点，再用光滑的曲线将它们连接起来，得到函数<i>y</i>=sin<i>x</i>（<i>x</i>∈0，2π）的简图，我们称这种画图方法为“五点（画图）法”.
          </p>
          <p>
            最后，利用正弦函数的周期性，我们将函数<i>y</i>=sin<i>x</i>（<i>x</i>∈0，2π）的图像向左或向右平移2π，4π，…，即可画出<i>y</i>=sin<i>x</i>在<b>R</b>的图像，如图5-26所示.
          </p>
        </div>
      </div>
    </div>

@@ -500,7 +6595,33 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0201-1.jpg" /></p>
          <p class="img">图5-26</p>
          <p>正弦函数<i>y</i>=sin<i>x</i>，<i>x</i>∈<b>R</b>的图像叫作正弦曲线.</p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　用“五点法”画出下列函数在区间［0，2π］内的简图.
          </p>
          <p>（1） <i>y</i>=-sin<i>x</i>；（2） <i>y</i>=1+sin<i>x</i>.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 列表（表5-6）.
          </p>
          <p class="img">表5-6</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0201-2.jpg" /></p>
          <p>
            描点连线得<i>y</i>=-sin<i>x</i>在区间［0，2π］内的简图，如图5-27所示.
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0201-3.jpg" /></p>
          <p class="img">图5-27</p>
          <p>（2） 列表（表5-7）.</p>
          <p class="img">表5-7</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0201-4.jpg" /></p>
          <p>
            描点连线得<i>y</i>=1+sin<i>x</i>在区间［0，2π］内的简图，如图5-28所示.
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0201-5.jpg" /></p>
          <p class="img">图5-28</p>
        </div>
      </div>
    </div>

@@ -516,7 +6637,89 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bk-hzjl">
            <div class="bj1-hzjl">
              <p class="left">
                <img class="img-gn2" alt="" src="../../assets/images/hzjl.jpg" />
              </p>
            </div>
            <p class="block">
              <i>y</i>=-sin<i>x</i>与<i>y</i>=sin<i>x</i>的图像有什么关系？
              <i>y</i>=1+sin<i>x</i>与<i>y</i>=sin<i>x</i>的图像有什么关系？
            </p>
          </div>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              完成下表（表5-8），并利用“五点法”画出<i>y</i>=3sin
              <i>x</i>在区间［0，2π］内的简图，并说明<i>y</i>=3sin
              <i>x</i>的图像与正弦函数<i>y</i>=sin <i>x</i>的图像的区别和联系.
            </p>
            <p class="img">表5-8</p>
            <p class="center">
              <img class="img-a" alt="" src="../../assets/images/0202-1.jpg" />
            </p>
            <p class="center">
              <img class="img-d" alt="" src="../../assets/images/0202-2.jpg" />
            </p>
          </div>
          <h3 id="c058">
            5.6.2 正弦函数的性质（一）<span class="fontsz2">＞＞＞</span>
          </h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            通过观察<i>y</i>=sin<i>x</i>的图像可知正弦函数<i>y</i>=sin<i>x</i>的性质.本节主要研究正弦函数的定义域、值域、周期性和奇偶性.
          </p>
          <p>1.定义域.</p>
          <p><i>y</i>=sin<i>x</i>的定义域是<b>R</b>.</p>
          <p>2.值域.</p>
          <p>
            曲线夹在两条直线<i>y</i>=1和<i>y</i>=-1之间，因此-1≤sin<i>x</i>≤1，即<i>y</i>=sin<i>x</i>的值域是［-1，1］.
          </p>
          <p>
            当<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mn>2</mn>
              <mi>k</mi>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>时，<i>y</i>=sin <i>x</i>取得最大值1；
          </p>
          <p>
            当<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mn>2</mn>
              <mi>k</mi>
              <mi>π</mi>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>时，<i>y</i>=sin <i>x</i>取得最小值-1.
          </p>
        </div>
      </div>
    </div>

@@ -528,8 +6731,207 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>3.周期性.</p>
          <p><i>y</i>=sin<i>x</i>是周期函数，周期是2π.</p>
          <p>4.奇偶性.</p>
          <p>
            因为sin（-<i>x</i>）=-sin<i>x</i>，所以<i>y</i>=sin<i>x</i>是奇函数，其图像关于原点对称.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　已知<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mo>−</mo>
                  <mi>a</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
            </math>，求<i>a</i>的取值范围.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为　-1≤sin<i>x</i>≤1，
          </p>
          <p>
            所以　<math display="0">
              <mo>−</mo>
              <mn>1</mn>
              <mo>⩽</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mo>−</mo>
                  <mi>a</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
              <mo>⩽</mo>
              <mn>1</mn>
            </math>，
          </p>
          <p>解得　1≤<i>a</i>≤5.</p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　求使下列函数取得最大值、最小值的<i>x</i>的集合，并求出这些函数的最大值、最小值.
          </p>
          <p>（1） <i>y</i>=3+sin<i>x</i>；（2） <i>y</i>=-2sin<i>x</i>.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1）
            使函数<i>y</i>=3+sin<i>x</i>取得最大值的<i>x</i>的集合，就是使函数<i>y</i>=sin<i>x</i>取得最大值的<i>x</i>的集合<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">{</mo>
                <mi>x</mi>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">|</mo>
                  <mstyle scriptlevel="0">
                    <mspace width="thinmathspace"></mspace>
                  </mstyle>
                  <mi>x</mi>
                  <mo>=</mo>
                  <mn>2</mn>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>+</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo>
                </mrow>
                <mo>,</mo>
                <mi>k</mi>
                <mo>∈</mo>
                <mrow>
                  <mi mathvariant="bold">Z</mi>
                </mrow>
                <mo data-mjx-texclass="CLOSE">}</mo>
              </mrow>
            </math>.这时函数<i>y</i>=3+sin<i>x</i>的最大值为<i>y</i>=3+1=4.
          </p>
          <p>
            使函数<i>y</i>=3+sin<i>x</i>取得最小值的<i>x</i>的集合，就是使函数<i>y</i>=sin<i>x</i>取得最小值的<i>x</i>的集合<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">{</mo>
                <mi>x</mi>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">|</mo>
                  <mstyle scriptlevel="0">
                    <mspace width="thinmathspace"></mspace>
                  </mstyle>
                  <mi>x</mi>
                  <mo>=</mo>
                  <mn>2</mn>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo>
                </mrow>
                <mo>,</mo>
                <mi>k</mi>
                <mo>∈</mo>
                <mrow>
                  <mi mathvariant="bold">Z</mi>
                </mrow>
                <mo data-mjx-texclass="CLOSE">}</mo>
              </mrow>
            </math>.这时函数<i>y</i>=3+sin<i>x</i>的最小值为<i>y</i>=3+（-1）=2.
          </p>
          <p>
            （2）
            使函数<i>y</i>=-2sin<i>x</i>取得最大值的<i>x</i>的集合，就是使函数<i>y</i>=sin<i>x</i>取得最小值的<i>x</i>的集合<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">{</mo>
                <mi>x</mi>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">|</mo>
                  <mstyle scriptlevel="0">
                    <mspace width="thinmathspace"></mspace>
                  </mstyle>
                  <mi>x</mi>
                  <mo>=</mo>
                  <mn>2</mn>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo>
                </mrow>
                <mo>,</mo>
                <mi>k</mi>
                <mo>∈</mo>
                <mrow>
                  <mi mathvariant="bold">Z</mi>
                </mrow>
                <mo data-mjx-texclass="CLOSE">}</mo>
              </mrow>
            </math>.这时函数<i>y</i>=-2sin<i>x</i>的最大值为<i>y</i>=-2×（-1）=2.
          </p>
          <p>
            使函数<i>y</i>=-2sin<i>x</i>取得最小值的<i>x</i>的集合，就是使函数<i>y</i>=sin<i>x</i>取得最大值的<i>x</i>的集合<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">{</mo>
                <mi>x</mi>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">|</mo>
                  <mstyle scriptlevel="0">
                    <mspace width="thinmathspace"></mspace>
                  </mstyle>
                  <mi>x</mi>
                  <mo>=</mo>
                  <mn>2</mn>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>+</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo>
                </mrow>
                <mo>,</mo>
                <mi>k</mi>
                <mo>∈</mo>
                <mrow>
                  <mi mathvariant="bold">Z</mi>
                </mrow>
                <mo data-mjx-texclass="CLOSE">}</mo>
              </mrow>
            </math>.这时函数<i>y</i>=-2sin<i>x</i>的最小值为<i>y</i>=-2×1=-2.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mi>a</mi>
                    <mo>−</mo>
                    <mn>1</mn>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
              </math>，则<i>a</i>的取值范围为____.
            </p>
            <p>
              2.（1）函数<i>y</i>=1+0.6sin <i>x</i>的最大值为____， 最小值为____；
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -544,7 +6946,257 @@
            <p><span>193</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">

            <p>
              （2）函数<math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>2</mn>
                  <mn>3</mn>
                </mfrac>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>的最大值为____，最小值为____.
            </p>
          </div>
          <h3 id="c059">
            5.6.3 正弦函数的性质（二）<span class="fontsz2">＞＞＞</span>
          </h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>5.单调性.</p>
          <p>
            如图5-29所示，选取正弦曲线在长度为2π的区间<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>内的图像进行考查.
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0204-3.jpg" /></p>
          <p class="img">图5-29</p>
          <p>
            <i>y</i>=sin<i>x</i> 在区间<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是增函数，在区间<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是减函数.由正弦函数的周期性可知：<i>y</i>=sin<i>x</i>在每一个区间<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mn>2</mn>
                <mi>k</mi>
                <mi>π</mi>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mn>2</mn>
                <mi>k</mi>
                <mi>π</mi>
                <mo>+</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上都是增函数，函数值由-1增大到1；在每一个区间<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mn>2</mn>
                <mi>k</mi>
                <mi>π</mi>
                <mo>+</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mn>2</mn>
                <mi>k</mi>
                <mi>π</mi>
                <mo>+</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上都是减函数，函数值由1减小到-1.
          </p>
          <p><b>例</b> 不求值，利用正弦函数的单调性，比较下列各对正弦值的大小.</p>
          <p>
            （1）<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>2</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
            </math>；（2）<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>9</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>与<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>10</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 因为　<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>2</mn>
                  <mi>π</mi>
                </mrow>
                <mn>3</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
            </math>，
          </p>
          <p>
            而<i>y</i>=sin <i>x</i> 在<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是减函数，所以
          </p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>3</mn>
                <mi>π</mi>
              </mrow>
              <mn>4</mn>
            </mfrac>
            <mo>&lt;</mo>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>2</mn>
                <mi>π</mi>
              </mrow>
              <mn>3</mn>
            </mfrac>
            <mtext>.&nbsp;</mtext>
          </math>
        </div>
      </div>
    </div>

@@ -556,7 +7208,211 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">

          <p>
            （2） 因为　<math display="0">
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>9</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>10</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mn>0</mn>
            </math>，
          </p>
          <p>
            而<i>y</i>=sin <i>x</i>在<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mn>0</mn>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是增函数，所以
          </p>
          <math display="block">
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>9</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>&lt;</mo>
            <mi>sin</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>10</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>不求值，利用正弦函数的单调性，比较下列各对正弦函数值的大小.</p>
            <p>
              （1）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>12</mn>
                </mfrac>
              </math>____<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>11</mn>
                </mfrac>
              </math>；（2） sin 250°____sin 260°.
            </p>
          </div>
          <h3 id="c060">习题5.6<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.比较大小.</p>
            <p>
              （1） sin 53°____sin 78°；（2）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
              </math>____<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>4</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
              </math>.
            </p>
            <p>2.函数<i>y</i>=2sin <i>x</i>的最大值为____，最小值为____.</p>
            <p>
              3.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>a</mi>
                    <mo>−</mo>
                    <mn>1</mn>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
              </math>，则<i>a</i>的取值范围为____.
            </p>
            <p>
              4.求函数<math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>，<i>x</i>∈0，2π的单调区间.
            </p>
            <p>
              5.利用“五点法”画出函数<math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>2</mn>
                </mfrac>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>在一个周期内的图像.
            </p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>1.求函数<i>y</i>=-1-1.5sin <i>x</i>的最大值与最小值.</p>
            <p>2.求函数<i>y</i>=3-2sin <i>x</i>，<i>x</i>∈<b>R</b>的单调区间.</p>
            <p>3.不求值，利用函数的单调性，比较下列各对正弦值的大小.</p>
            <p>（1） sin 500°与sin 140°；</p>
            <p>
              （2）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>6</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>与<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>6</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>7</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -571,7 +7427,41 @@
            <p><span>195</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b036">
            5.7 余弦函数的图像和性质<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <h3 id="c061">5.7.1 余弦函数的图像<span class="fontsz2">＞＞＞</span></h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            我们学习了正弦函数的图像和性质，你能用类似的方法绘制出余弦函数的图像，并根据图像研究它的性质吗？
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>根据诱导公式可知，</p>
          <p class="center">cos（<i>x</i>+2π）=cos <i>x</i>.</p>
          <p>
            由周期函数的定义可知，余弦函数<i>y</i>=cos<i>x</i>是以2π为周期的周期函数.为画出函数<i>y</i>=cos<i>x</i>的图像，可仿照正弦曲线的画法，先用描点法画出它在一个周期［0，2π］内的图像，然后利用周期性画出其完整图像.
          </p>
          <p>
            首先，列表.自变量<i>x</i>取值如表5-9所示，利用科学计算器求出cos
            <i>x</i>的各个值并填入表中.
          </p>
          <p class="img">表5-9</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0206-1.jpg" /></p>
          <p>
            其次，描点连线.根据表中数值描点，用光滑的曲线把各点连接起来，得出图像如图5-30所示.
          </p>
          <p class="center"><img class="img-d" alt="" src="../../assets/images/0206-2.jpg" /></p>
          <p class="img">图5-30</p>
          <p>
            最后，利用余弦函数的周期性，把<i>y</i>=cos
            <i>x</i>在［0，2π］内的图像向左或向右平移2π，4π，…就可以画出<i>y</i>=cos
            <i>x</i>在<b>R</b>上的图像，如图5-31所示.
          </p>
          <p>
            余弦函数<i>y</i>=cos<i>x</i>，<i>x</i>∈<b>R</b>的图像叫作<b>余弦曲线</b>.
          </p>
        </div>
      </div>
    </div>
    <!-- 196 -->
@@ -582,7 +7472,90 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0207-1.jpg" /></p>
          <p class="img">图5-31</p>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              1.与画正弦函数的图像一样，也可以用“五点法”画出<i>y</i>=cos<i>x</i>在0，2π内的简图，这五个关键点是：
            </p>
            <math display="block">
              <mo stretchy="false">(</mo>
              <mn>0</mn>
              <mo>,</mo>
              <mn>1</mn>
              <mo stretchy="false">)</mo>
              <mo>,</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mn>0</mn>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>,</mo>
              <mo stretchy="false">(</mo>
              <mi>π</mi>
              <mo>,</mo>
              <mo>−</mo>
              <mn>1</mn>
              <mo stretchy="false">)</mo>
              <mo>,</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mn>0</mn>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>,</mo>
              <mo stretchy="false">(</mo>
              <mn>2</mn>
              <mi>π</mi>
              <mo>,</mo>
              <mn>1</mn>
              <mo stretchy="false">)</mo>
              <mtext>.&nbsp;</mtext>
            </math>
            <p class="block">
              2.正弦函数的图像向左平移<math display="0">
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
              </math>个单位长度即可得到余弦函数的图像，如图5-32所示.
            </p>
            <p class="center">
              <img class="img-c" alt="" src="../../assets/images/0207-4.jpg" />
            </p>
            <p class="img">图5-32</p>
          </div>
          <p><b>例</b> 用“五点法”画出下列函数在区间［0，2π］内的简图.</p>
          <p>（1） <i>y</i>=2cos <i>x</i>；（2） <i>y</i>=-1+cos <i>x</i>.</p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 列表（表5-10）.
          </p>
          <p class="img">表5-10</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0207-5.jpg" /></p>
          <p class="center"><img class="img-f" alt="" src="../../assets/images/0207-6.jpg" /></p>
          <p class="img">图5-33</p>
          <p>描点连线得<i>y</i>=2cos <i>x</i>在区间［0，2π］</p>
          <p>内的简图，如图5-33所示.</p>
        </div>
      </div>
    </div>

@@ -597,7 +7570,35 @@
            <p><span>197</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>（2） 列表（表5-11）.</p>
          <p class="img">表5-11</p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0208-1.jpg" /></p>
          <p>
            描点连线得<i>y</i>=-1+cos <i>x</i>在区间［0，2π］内的简图，如图5-34所示.
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0208-2.jpg" /></p>
          <p class="img">图5-34</p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              完成下表（表5-12），利用“五点法”画出<i>y</i>=1-cos
              <i>x</i>在区间［0，2π］内的简图，并说明<i>y</i>=1-cos
              <i>x</i>的图像与<i>y</i>=cos <i>x</i>的图像的区别和联系.
            </p>
            <p class="img">表5-12</p>
            <p class="center">
              <img class="img-a" alt="" src="../../assets/images/0208-3.jpg" />
            </p>
            <p>
              对比<i>y</i>=cos <i>x</i>的图像，<i>y</i>=1-cos
              <i>x</i>图像是将<i>y</i>=cos <i>x</i>的图像通过____变化而得到的.
            </p>
            <p class="center">
              <img class="img-d" alt="" src="../../assets/images/0208-4.jpg" />
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -609,7 +7610,60 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h3 id="c062">5.7.2 余弦函数的性质<span class="fontsz2">＞＞＞</span></h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/cxgk.jpg" /></p>
          <p>
            通过观察<i>y</i>=cos <i>x</i>的图像可知，余弦函数<i>y</i>=cos
            <i>x</i>的性质有：
          </p>
          <p>1.定义域.</p>
          <p><i>y</i>=cos <i>x</i>的定义域是<b>R</b>.</p>
          <p>2.值域.</p>
          <p>
            由余弦函数的图像可以看出，曲线夹在两条直线<i>y</i>=1和<i>y</i>=-1之间，因此-1≤cos
            <i>x</i>≤1，即<i>y</i>=cos <i>x</i>的值域是［-1，1］.
          </p>
          <p>
            当<i>x</i>=2<i>k</i>π（<i>k</i>∈<b>Z</b>）时，<i>y</i>=cos
            <i>x</i>取得最大值1；
          </p>
          <p>
            当<i>x</i>=2<i>k</i>π+π（<i>k</i>∈<b>Z</b>）时，<i>y</i>=cos
            <i>x</i>取得最小值-1.
          </p>
          <p>3.周期性.</p>
          <p><i>y</i>=cos <i>x</i>是周期函数，周期是2π.</p>
          <p>4.奇偶性.</p>
          <p>
            因为cos（-<i>x</i>）=cos <i>x</i>，所以<i>y</i>=cos
            <i>x</i>是偶函数，其图像关于<i>y</i>轴对称.
          </p>
          <p>5.单调性.</p>
          <p>
            <i>y</i>=cos <i>x</i>在区间［0，π］上是减函数，在［π，2π］上是增函数.
          </p>
          <p>
            余弦函数<i>y</i>=cos
            <i>x</i>在每一个区间［2<i>k</i>π，2<i>k</i>π+π］（<i>k</i>∈<b>Z</b>）上都是减函数，其值由1减小到-1；在每一个区间［2<i>k</i>π+π，2<i>k</i>π+2π］（<i>k</i>∈<b>Z</b>）上都是增函数，其值由-1增大到1.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　求函数<i>y</i>=-1+cos <i>x</i>的最大值、最小值、最小正周期及值域.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>
            当<i>x</i>=2<i>k</i>π（<i>k</i>∈<b>Z</b>）时，函数<i>y</i>=-1+cos
            <i>x</i>的最大值为<i>y</i>=1-1=0；
          </p>
          <p>
            当<i>x</i>=2<i>k</i>π+π（<i>k</i>∈<i>Z</i>）时，函数<i>y</i>=-1+cos
            <i>x</i>的最小值为<i>y</i>=-1-1=-2；
          </p>
          <p>
            函数<i>y</i>=-1+cos <i>x</i>的最小正周期为2π；函数<i>y</i>=-1+cos
            <i>x</i>的值域为[-2，0].
          </p>
        </div>
      </div>
    </div>
    <!-- 199 -->
@@ -624,7 +7678,319 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bk-hzjl">
            <div class="bj1-hzjl">
              <p class="left">
                <img class="img-gn2" alt="" src="../../assets/images/hzjl.jpg" />
              </p>
            </div>
            <p class="block">
              1.<i>y</i>=2cos <i>x</i>与<i>y</i>=cos <i>x</i>的图像有什么关系？
            </p>
            <p class="block">
              2.<i>y</i>=-1+cos <i>x</i>与<i>y</i>=cos <i>x</i>的图像有什么关系？
            </p>
          </div>
          <p>
            <span class="zt-ls"><b>例2</b></span>　不求值，利用余弦函数的单调性，比较下列各对余弦值的大小.
          </p>
          <p>
            （1）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>6</mn>
                  <mi>π</mi>
                </mrow>
                <mn>5</mn>
              </mfrac>
            </math>与<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>；（2）
            <math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>7</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>与<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>8</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>（1） 因为<math display="0">
              <mi>π</mi>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>6</mn>
                  <mi>π</mi>
                </mrow>
                <mn>5</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>2</mn>
              </mfrac>
            </math>，而函数<i>y</i>=cos <i>x</i>在<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mi>π</mi>
                <mo>,</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是增函数，所以
          </p>
          <math display="block">
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>6</mn>
                <mi>π</mi>
              </mrow>
              <mn>5</mn>
            </mfrac>
            <mo>&lt;</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mfrac>
              <mrow>
                <mn>5</mn>
                <mi>π</mi>
              </mrow>
              <mn>4</mn>
            </mfrac>
            <mo>.</mo>
          </math>
          <p>
            （2）<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>7</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>7</mn>
              </mfrac>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>8</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>8</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            因为<math display="0">
              <mn>0</mn>
              <mo>&lt;</mo>
              <mfrac>
                <mi>π</mi>
                <mn>8</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mi>π</mi>
                <mn>7</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
            </math>，而函数<i>y</i>=cos <i>x</i>在0，<math display="0">
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mn>0</mn>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>上是减函数，所以<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>7</mn>
              </mfrac>
              <mo>&lt;</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>8</mn>
              </mfrac>
            </math>，即
          </p>
          <math display="block">
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>7</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>&lt;</mo>
            <mi>cos</mi>
            <mo data-mjx-texclass="NONE">⁡</mo>
            <mrow data-mjx-texclass="INNER">
              <mo data-mjx-texclass="OPEN">(</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>8</mn>
              </mfrac>
              <mo data-mjx-texclass="CLOSE">)</mo>
            </mrow>
            <mo>.</mo>
          </math>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.（1） 函数<math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mn>1</mn>
                <mo>+</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>的最大值为____，最小值为____，最小正周期为____；
            </p>
            <p>（2） 函数<i>y</i>=1+4cos <i>x</i>的最大值为____，最小值为____.</p>
            <p>2.比较大小.</p>
            <p>
              （1） cos 157°____cos 160°；（2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>5</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>______<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>6</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>.
            </p>
            <p>3.下列等式是否成立？并说明理由.</p>
            <p>（1） cos<sup>2</sup><i>x</i>=1；（2） 2cos <i>x</i>=3.</p>
            <p>
              4.求使下列函数取得最大值、最小值的<i>x</i>的集合，并求出这个函数的最大值、最小值.
            </p>
            <p>
              （1） <i>y</i>=-3cos <i>x</i>；（2）
              <math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mn>4</mn>
                <mo>+</mo>
                <mfrac>
                  <mn>5</mn>
                  <mn>6</mn>
                </mfrac>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -636,7 +8002,174 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h3 id="c063">习题5.7<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.比较大小.</p>
            <p>
              （1） cos 153°____cos 173°；（2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>8</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
              </math>____<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>9</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
              </math>.
            </p>
            <p>2.（1） 函数<i>y</i>=3cos <i>x</i>的最大值为____，最小值为____；</p>
            <p>（2） 函数<i>y</i>=-0.5cos <i>x</i>的最大值为____，最小值为____.</p>
            <p>
              3.函数<i>y</i>=1+3cos
              <i>x</i>，<i>x</i>∈0，2π，当<i>x</i>=____时，<i>y</i>取最大值；当<i>x</i>=____时，<i>y</i>取最小值.
            </p>
            <p>4.求函数<i>y</i>=2+cos <i>x</i>，<i>x</i>∈0，2π的单调区间.</p>
            <p>5.利用“五点法”画出函数<i>y</i>=-4cos <i>x</i>在区间0，2π内的图像.</p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>1.求函数<i>y</i>=|cos <i>x</i>的最小正周期.</p>
            <p>
              2.求函数<math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>2</mn>
                  <mrow>
                    <mo>−</mo>
                    <mn>1</mn>
                    <mo>+</mo>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>x</mi>
                  </mrow>
                </mfrac>
              </math>的定义域.
            </p>
            <p>3.不求值，利用函数的单调性，比较下列各对函数值的大小.</p>
            <p>
              （1）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>23</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>5</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>与<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>17</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>4</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>；
            </p>
            <p>
              （2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
              </math>与<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>5</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>.
            </p>
          </div>
          <h2 id="b037">
            5.8 已知三角函数值，求指定范围的角<span class="fontsz1">＞＞＞＞＞＞＞＞</span>
          </h2>
          <h3 id="c064">
            5.8.1 已知特殊三角函数值求角<span class="fontsz2">＞＞＞</span>
          </h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            如果<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>，那么<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>；反之，如果<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>，那么<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>吗？
          </p>
        </div>
      </div>
    </div>
    <!-- 201 -->
@@ -650,7 +8183,247 @@
            <p><span>201</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            由<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>可知，<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>是满足<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>的一个角，还有没有更多的角也能满足<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>呢？我们借助正弦曲线来探究问题.
          </p>
          <p>
            如图5-35所示，条件中的<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>，在图像中就可以表示为<math display="0">
              <mi>y</mi>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>，问题就转化为求当<math display="0">
              <mi>y</mi>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>的值，即直线<math display="0">
              <mi>y</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>与正弦曲线<i>y</i>=sin <i>x</i>交点所对应的<i>x</i>的值.
          </p>
          <p class="center"><img class="img-a" alt="" src="../../assets/images/0212-9.jpg" /></p>
          <p class="img">图5-35</p>
          <p>
            观察图像可知，直线<math display="0">
              <mi>y</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>与正弦曲线<i>y</i>=sin <i>x</i>的交点有无数个.
          </p>
          <p>
            现将问题的范围限定为<i>x</i>∈［0，2π］，由图像可知，满足条件的交点共有两个.
          </p>
          <p>
            因为<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
              <mo>＞</mo>
              <mn>0</mn>
            </math>，所以<i>x</i>是第一或第二象限角.
          </p>
          <p>
            满足<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>的锐角是<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>，所以符合条件的第一象限的角是<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            由诱导公式<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>π</mi>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>6</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>可知，<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mi>π</mi>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>，所以符合条件的第二象限角是<math display="0">
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            所以
            <math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>6</mn>
              </mfrac>
            </math>或<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>6</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　已知<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
            </math>，且<i>x</i>∈［0，2π］ ，求<i>x</i>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
              <mo>＜</mo>
              <mn>0</mn>
            </math>，所以<i>x</i>是第二或第三象限角.
          </p>
        </div>
      </div>
    </div>

@@ -662,7 +8435,335 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            满足<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mn>2</mn>
              </mfrac>
            </math>的锐角是<math display="0">
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>4</mn>
              </mfrac>
            </math>，所以
          </p>
          <p>
            符合条件的第二象限角是<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mi>π</mi>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>4</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>；
          </p>
          <p>
            符合条件的第三象限角是<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mi>π</mi>
              <mo>+</mo>
              <mfrac>
                <mi>π</mi>
                <mn>4</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            所以<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>3</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>或<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mn>5</mn>
                  <mi>π</mi>
                </mrow>
                <mn>4</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　已知<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <msqrt>
                <mn>3</mn>
              </msqrt>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>x</mi>
                <mo>≠</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>+</mo>
                <mi>k</mi>
                <mi>π</mi>
                <mo>,</mo>
                <mi>k</mi>
                <mo>∈</mo>
                <mrow>
                  <mi mathvariant="bold">Z</mi>
                </mrow>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>，且0°≤<i>x</i>≤360°，求<i>x</i>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <msqrt>
                <mn>3</mn>
              </msqrt>
              <mo>＞</mo>
              <mn>0</mn>
            </math>，所以<i>x</i>是第一或第三象限角.
          </p>
          <p>
            由<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mn>60</mn>
              <mrow>
                <mo>°</mo>
              </mrow>
              <mo>=</mo>
              <msqrt>
                <mn>3</mn>
              </msqrt>
            </math>可知，符合条件的第一象限角是<i>x</i>=60°.
          </p>
          <p>
            又因为<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mn>180</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
                <mo>+</mo>
                <mn>60</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mn>60</mn>
              <mrow>
                <mo>°</mo>
              </mrow>
              <mo>=</mo>
              <msqrt>
                <mn>3</mn>
              </msqrt>
            </math>，
          </p>
          <p>所以符合条件的第三象限角是<i>x</i>=180°+60°=240°.</p>
          <p>所以<i>x</i>=60°或<i>x</i>=240°.</p>
          <div class="bk">
            <div class="bj1">
              <p class="left">
                <img class="img-gn1" alt="" src="../../assets/images/tbts.jpg" />
              </p>
            </div>
            <p class="block">
              已知三角函数值，求给定范围的角<i>x</i>的值，其基本步骤如下.
            </p>
            <p class="block">
              （1） 根据已知三角函数值的符号，判定角<i>x</i>所在的象限；
            </p>
            <p class="block">（2） 求出满足三角函数值的锐角<i>x</i>′；</p>
            <p class="block">
              （3）
              根据<i>x</i>所在的象限和诱导公式，写出满足题目给定范围的<i>x</i>的值.
            </p>
          </div>
          <p>
            <span class="zt-ls"><b>例3</b></span>　已知<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>，且<i>x</i>∈［-π，π］，求<i>x</i>的值.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
              <mo>＞</mo>
              <mn>0</mn>
            </math>，所以<i>x</i>是第一或第四象限角.
          </p>
          <p>
            满足<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>的锐角是<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>，所以符合条件的第一象限角是<math display="0">
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            因为<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>3</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
              <mo>=</mo>
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
            </math>，
          </p>
          <p>
            所以符合条件的第四象限角是<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>.
          </p>
          <p>
            所以<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>或<math display="0">
              <mi>x</mi>
              <mo>=</mo>
              <mo>−</mo>
              <mfrac>
                <mi>π</mi>
                <mn>3</mn>
              </mfrac>
            </math>.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <msqrt>
                    <mn>2</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>，且<i>x</i>∈［0，2π］，则<i>x</i>的值为____.
            </p>
            <p>
              2.已知<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>，且<i>x</i>∈［0，2π］，则<i>x</i>的值为____.
            </p>

          </div>
        </div>
      </div>
    </div>

@@ -677,7 +8778,113 @@
            <p><span>203</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>
              3.已知<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mi>x</mi>
                  <mo>≠</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo>+</mo>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>,</mo>
                  <mi>k</mi>
                  <mo>∈</mo>
                  <mrow>
                    <mi mathvariant="bold">Z</mi>
                  </mrow>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>，且<i>x</i>∈［0，2π］，则<i>x</i>的值为____.
            </p>
          </div>
          <h3 id="c065">
            5.8.2 已知任意三角函数值求角<span class="fontsz2">＞＞＞</span>
          </h3>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/wttc.jpg" /></p>
          <p>
            我们已经探究了已知特殊的三角函数值求角的方法，而对于不是特殊的三角函数值，又该如何求角呢？
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/fxlj.jpg" /></p>
          <p>
            根据已知特殊的三角函数值求角的方法，借助计算工具，可以解决已知任意三角函数值求角的问题.
          </p>
          <p>
            <span class="zt-ls"><b>例1</b></span>　已知<math display="0">
              <mi>α</mi>
              <mo>∈</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>，求<i>α</i>的值.（结果精确到0.000 1）
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为<math display="0">
              <mi>α</mi>
              <mo>∈</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">[</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">]</mo>
              </mrow>
            </math>，所以<i>α</i>在<i>y</i>=sin <i>α</i>的一个单调区间内，这时使sin
            <i>α</i>=0.943 7的角<i>α</i>的值是唯一的.
          </p>
          <p>
            先将科学计算器的精确度设置为0.000
            1，再将科学计算器设置为弧度计算模式，然后依次按键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0214-4.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0214-5.jpg" /></p>
          <p>所以　<i>α</i>≈1.233 6.</p>
          <p>
            <span class="zt-ls"><b>例2</b></span>　已知cos <i>α</i>=0.694
            3，0°≤<i>α</i>≤180°，求<i>α</i>的值.（结果精确到0.000 1）
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span>　因为0°≤<i>α</i>≤180°，所以<i>α</i>在<i>y</i>=cos
            <i>α</i>的一个单调区间内，这时使cos <i>α</i>=0.694
            3的角<i>α</i>的值是唯一的.
          </p>
        </div>
      </div>
    </div>
    <!-- 204 -->
@@ -689,7 +8896,85 @@
          <li>上册</li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>
            先将科学计算器的精确度设置为0.000
            1，再将科学计算器设置为角度计算模式，然后依次按键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-1.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-2.jpg" /></p>
          <p>所以<i>α</i>≈46.028 5°.</p>
          <p>
            注意：应当区分所给条件中角的单位是角度还是弧度.如果是角度，计算时应用角度计算模式；
            如果是弧度，计算时应用弧度计算模式.
          </p>
          <p>
            <span class="zt-ls"><b>例3</b></span>　已知tan <i>α</i>=-2.747 0，<math display="0">
              <mi>α</mi>
              <mo>∈</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>，求<i>α</i>的值.（结果精确到0.000 1）
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 因为<math display="0">
              <mi>α</mi>
              <mo>∈</mo>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mo>−</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>,</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>，所以<i>α</i>在<i>y</i>=tan <i>α</i>的一个单调区间内，这时使tan
            <i>α</i>=-2.747 0的角<i>α</i>的值是唯一的.
          </p>
          <p>
            先将科学计算器的精确度设置为0.000
            1，再将科学计算器设置为弧度计算模式，然后依次按键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-5.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-6.jpg" /></p>
          <p>所以　<i>α</i>≈-1.221 7.</p>
          <p>
            <span class="zt-ls"><b>例4</b></span>　已知sin <i>α</i>=-0.857
            2，<i>α</i>∈［0，2π］，求<i>α</i>的值.（结果精确到0.000 1）
          </p>
          <p class="block">
            <span class="zt-ls2"><b>分析</b></span>　因为sin <i>α</i>=-0.857
            2＜0，在［0，2π］范围内有两个<i>α</i>值满足条件，它们分别位于第三象限和第四象限，即<i>α</i>在［π，2π］范围内.可用科学计算器先求出sin
            <i>α</i>=0.857 2所对应的锐角，再利用诱导公式求出所求的角.
          </p>
          <p>
            <span class="zt-ls"><b>解</b></span> 先将科学计算器的精确度设置为0.000
            1，再将科学计算器设置为弧度计算模式，然后依次按键：
          </p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-7.jpg" /></p>
          <p>结果显示：</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0215-8.jpg" /></p>
        </div>
      </div>
    </div>

@@ -705,7 +8990,132 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">
          <p>即</p>
          <p class="center">sin 1.029 8≈0.857 2.</p>
          <p>因为</p>
          <p class="center">sin（π+1.029 8）=-sin 1.029 8≈-0.857 2，</p>
          <p>所以符合条件的第三象限角是π+1.029 8≈4.171 4.</p>
          <p>因为</p>
          <p class="center">sin（2π-1.029 8）=-sin 1.029 8≈-0.857 2，</p>
          <p>所以符合条件的第四象限角是2π-1.029 8≈5.253 4.</p>
          <p>
            所以满足sin <i>α</i>=-0.857
            2，<i>α</i>∈［0，2π］的角<i>α</i>的集合为{4.171 4，5.253 4}.
          </p>
          <p class="left"><img class="img-gn" alt="" src="../../assets/images/stlx.jpg" /></p>
          <div class="bj">
            <p>借助科学计算器，求出下列指定范围内的角.（结果精确到0.000 1）</p>
            <p>
              1.已知<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>β</mi>
                <mo>−</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>7</mn>
                </mfrac>
                <mo>=</mo>
                <mn>0</mn>
              </math>，<i>β</i>∈［0，π］，则<i>β</i>的值为____.
            </p>
            <p>
              2.已知tan<i>γ</i>=-0.234 5，-90°＜<i>γ</i>＜90°，则<i>γ</i>的值为____.
            </p>
            <p>
              3.已知sin <i>α</i>=0.973 4，0°≤<i>α</i>≤360°，则<i>α</i>的值为____.
            </p>
            <p>
              4.已知cos <i>β</i>=-0.202 8，<i>β</i>∈［-π，π］，则<i>β</i>的值为____.
            </p>
          </div>
          <h3 id="c066">习题5.8<span class="fontsz2">＞＞＞</span></h3>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>
              1.已知<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>，<math display="0">
                <mi>x</mi>
                <mo>∈</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">[</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo>,</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">]</mo>
                </mrow>
              </math>，则<i>x</i>的值为____.
            </p>
            <p>
              2.已知<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>2</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>，0°≤<i>x</i>≤180°，则<i>x</i>的值为____.
            </p>
            <p>
              3.已知<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <msqrt>
                  <mn>3</mn>
                </msqrt>
              </math>，-90°＜<i>x</i>＜90°，则<i>x</i>的值为____.
            </p>
            <p>
              4.借助科学计算器，求适合下列各式中的<i>x</i>（0≤<i>x</i>＜2π）的值的集合.（结果精确到0.000
              1）
            </p>
            <p>（1） sin <i>x</i>=0.318 5；（2） cos <i>x</i>=-0.789 0.</p>
            <p>5.求适合下列各式中的<i>x</i>（-π≤<i>x</i>≤π）的值的集合.</p>
            <p>
              （1）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>；（2） cos <i>x</i>=-0.5；（3） tan <i>x</i>=-1.
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -717,7 +9127,109 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">

            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>
              1.借助科学计算器，求适合下列各式中<i>x</i>的值.（结果精确到0.000 1）
            </p>
            <p>
              （1）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>5</mn>
                </mfrac>
              </math>，<math display="0">
                <mfrac>
                  <mi>π</mi>
                  <mn>2</mn>
                </mfrac>
                <mo>＜</mo>
                <mi>x</mi>
                <mo>＜</mo>
                <mi>π</mi>
              </math>；（2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>4</mn>
                </mfrac>
              </math>，<math display="0">
                <mi>π</mi>
                <mo>＜</mo>
                <mi>x</mi>
                <mo>＜</mo>
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
              </math>；
            </p>
            <p>
              （3）
              <math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>3</mn>
                </mfrac>
              </math>，<math display="0">
                <mfrac>
                  <mrow>
                    <mn>3</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>2</mn>
                </mfrac>
                <mo>＜</mo>
                <mi>x</mi>
                <mo>＜</mo>
                <mn>2</mn>
                <mi>π</mi>
              </math>.
            </p>
            <p>
              2.求满足<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mn>2</mn>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <mn>1</mn>
                  <mn>2</mn>
                </mfrac>
              </math>的角<i>x</i>（0°≤<i>x</i>≤180°）的值的集合.
            </p>
            <p>
              3.借助科学计算器，求出下面指定范围内的角<i>β</i>的值的集合：cos
              2<i>β</i>=-0.690 9，0°≤<i>β</i>≤180°.（结果精确到0.000 1）
            </p>
          </div>
        </div>
      </div>
    </div>

@@ -733,7 +9245,33 @@
          </li>
        </ul>

        <div class="padding-116"></div>
        <div class="padding-116">

          <h2 id="b038">数学园地<span class="fontsz1">＞＞＞＞＞＞＞＞</span></h2>
          <p class="center">三角学在我国的发展</p>
          <p>
            我国很早就开始了对三角知识的研究.我国古老的数学书籍《周髀算经》一书中，记载了古时候人们计算地面上一点到太阳距离的方法.魏晋时期的著名数学家刘徽在古人“重差术”的基础上，编撰了《海岛算经》一书.
          </p>
          <p>
            春秋时期的《考工记》一书，对“角”已有初步认识，并用“倨句”表示角度的多少，其中直角叫作“矩”.
          </p>
          <p>
            唐朝开元六年至十四年（718—726），唐代文学家翟昙悉达修撰《开元占经》一百二十卷，将印度数学家编制的三角函数表载于其中，这是传入我国的最早的三角函数表.
          </p>
          <p>
            由我国著名数学家徐光启（1562—1633）等人共同编译的《大测》二卷序言中说：“大测者，测三角之法也.”我国“三角学”一词即由此而来.该书介绍了三角函数值的造表方法和正弦定理、余弦定理等.
          </p>
          <p>
            明末清初数学家薛凤祚著有《三角算法》一书，这是我国数学家自己撰写的第一部三角学著作.书中所介绍的三角学知识，要比《大测》《测量全义》中的内容更详细与完备.
          </p>
          <p>
            清初著名数学家梅文鼎研究三角学数年，对所传入的三角学知识进行了通俗的解释，并著有《平三角举要》五卷.其内容由浅入深，循序渐进，条理清楚，是当时以及后人学习三角学的主要教科书.
          </p>
          <p>
            如果想知道更多的关于三角学在我国发展历程中所经历的人和事，你可以通过不同的途径（如上网搜索）查找资料，整理出更为丰富的史料来.对此，你不妨与同学合作，试一试.
          </p>
          <p>——摘录自沈文选、杨清桃编著的《数学史话览胜》一书，引用时有改动</p>
        </div>
      </div>
    </div>

@@ -745,7 +9283,178 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b039">单元小结<span class="fontsz1">＞＞＞＞＞＞＞＞</span></h2>
          <p class="bj2"><b>学习导图</b></p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0219-1.jpg" /></p>
          <p class="bj2"><b>学习指导</b></p>
          <p>
            1.与角<i>α</i>终边相同的角的集合：<i>S</i>={<i>β</i>|<i>β</i>=<i>α</i>+<i>k</i>·2π，<i>k</i>∈<b>Z</b>}.
          </p>
          <p>2.弧度与角度的换算.</p>
          <math display="block">
            <mtable columnspacing="1em" rowspacing="4pt">
              <mtr>
                <mtd>
                  <mi>π</mi>
                  <mo>=</mo>
                  <msup>
                    <mn>180</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <mo>;</mo>
                  <mn>2</mn>
                  <mi>π</mi>
                  <mo>=</mo>
                  <msup>
                    <mn>360</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <mo>.</mo>
                </mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mn>1</mn>
                  <mrow>
                    <mi mathvariant="normal">r</mi>
                    <mi mathvariant="normal">a</mi>
                    <mi mathvariant="normal">d</mi>
                  </mrow>
                  <mo>=</mo>
                  <mfrac>
                    <msup>
                      <mn>180</mn>
                      <mrow>
                        <mo>∘</mo>
                      </mrow>
                    </msup>
                    <mi>π</mi>
                  </mfrac>
                  <mo>≈</mo>
                  <msup>
                    <mn>57.30</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <mo>=</mo>
                  <msup>
                    <mn>57</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <msup>
                    <mn>18</mn>
                    <mrow>
                      <mi data-mjx-alternate="1" mathvariant="normal">′</mi>
                    </mrow>
                  </msup>
                  <mo>;</mo>
                  <msup>
                    <mn>1</mn>
                    <mrow>
                      <mo>∘</mo>
                    </mrow>
                  </msup>
                  <mo>=</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>180</mn>
                  </mfrac>
                  <mrow>
                    <mi mathvariant="normal">r</mi>
                    <mi mathvariant="normal">a</mi>
                    <mi mathvariant="normal">d</mi>
                  </mrow>
                  <mo>≈</mo>
                  <mn>0.01745</mn>
                  <mrow>
                    <mi mathvariant="normal">r</mi>
                    <mi mathvariant="normal">a</mi>
                    <mi mathvariant="normal">d</mi>
                  </mrow>
                  <mo>.</mo>
                </mtd>
              </mtr>
            </mtable>
          </math>
          <p>
            3.弧长公式为<i>l</i>=<i>αr</i>，扇形的面积公式为<math display="0">
              <mi>S</mi>
              <mo>=</mo>
              <mfrac>
                <mn>1</mn>
                <mn>2</mn>
              </mfrac>
              <mi>r</mi>
              <mi>l</mi>
            </math>.
          </p>
          <p>4.任意角的正弦、余弦和正切.</p>
          <p>
            点<i>P</i>（<i>x</i>，<i>y</i>）是角<i>α</i>的终边上异于原点的任意一点，点<i>P</i>到原点的距离为<math display="0">
              <mi>r</mi>
              <mo>=</mo>
              <msqrt>
                <msup>
                  <mi>x</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mo>+</mo>
                <msup>
                  <mi>y</mi>
                  <mrow>
                    <mn>2</mn>
                  </mrow>
                </msup>
              </msqrt>
              <mo>&gt;</mo>
              <mn>0</mn>
            </math>，则<math display="0">
              <mi>sin</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>y</mi>
                <mi>r</mi>
              </mfrac>
            </math>，<math display="0">
              <mi>cos</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>x</mi>
                <mi>r</mi>
              </mfrac>
            </math>，<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mi>y</mi>
                <mi>x</mi>
              </mfrac>
              <mrow data-mjx-texclass="INNER">
                <mo data-mjx-texclass="OPEN">(</mo>
                <mi>x</mi>
                <mo>≠</mo>
                <mn>0</mn>
                <mo data-mjx-texclass="CLOSE">)</mo>
              </mrow>
            </math>.
          </p>
        </div>
      </div>
    </div>
    <!-- 209 -->
@@ -759,7 +9468,59 @@
            <p><span>209</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <p>5.任意角的正弦函数、余弦函数和正切函数.</p>
          <p>正弦函数 <i>y</i>=sin <i>x</i>，<i>x</i>∈<b>R</b>；</p>
          <p>余弦函数 <i>y</i>=cos <i>x</i>，<i>x</i>∈<b>R</b>；</p>
          <p>
            正切函数 <i>y</i>=tan <i>x</i>，<math display="0">
              <mi>x</mi>
              <mo>≠</mo>
              <mfrac>
                <mi>π</mi>
                <mn>2</mn>
              </mfrac>
              <mo>+</mo>
              <mi>k</mi>
              <mi>π</mi>
              <mo stretchy="false">(</mo>
              <mi>k</mi>
              <mo>∈</mo>
              <mrow>
                <mi mathvariant="bold">Z</mi>
              </mrow>
              <mo stretchy="false">)</mo>
            </math>.
          </p>
          <p>6.同角三角函数基本关系式.</p>
          <p>
            （1） 平方关系：sin <sup>2</sup> <i>α</i>+cos <sup>2</sup> <i>α</i>=1；
          </p>
          <p>
            （2） 商数关系：<math display="0">
              <mi>tan</mi>
              <mo data-mjx-texclass="NONE">⁡</mo>
              <mi>α</mi>
              <mo>=</mo>
              <mfrac>
                <mrow>
                  <mi>sin</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </mrow>
                <mrow>
                  <mi>cos</mi>
                  <mo data-mjx-texclass="NONE">⁡</mo>
                  <mi>α</mi>
                </mrow>
              </mfrac>
            </math>.
          </p>
          <p>7.诱导公式表（<i>k</i>∈<b>Z</b>）.</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0220-3.jpg" /></p>
          <p>8.正弦函数、余弦函数的图像和性质.</p>
          <p class="center"><img class="img-c" alt="" src="../../assets/images/0220-4.jpg" /></p>
        </div>
      </div>
    </div>
    <!-- 210 -->
@@ -771,7 +9532,267 @@
          <li>数学.基础模块</li>
          <li>上册</li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <h2 id="b040">单元检测<span class="fontsz1">＞＞＞＞＞＞＞＞</span></h2>
          <div class="bj">
            <p>
              <span class="bj-sp"><b>水平一</b></span>
            </p>
            <p>1.选择题.</p>
            <p>（1） 下列正确的是（　）.</p>
            <p>
              A.<math display="0">
                <mn>15</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
                <mo>=</mo>
                <mfrac>
                  <mi>π</mi>
                  <mn>11</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mn>1200</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
                <mo>=</mo>
                <mfrac>
                  <mrow>
                    <mn>21</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>6</mn>
                </mfrac>
                <mo>=</mo>
                <mn>150</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
              </math>
            </p>
            <p>
              D.<math display="0">
                <mo>−</mo>
                <mfrac>
                  <mrow>
                    <mn>7</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>8</mn>
                </mfrac>
                <mo>=</mo>
                <mo>−</mo>
                <mn>220</mn>
                <mrow>
                  <mo>°</mo>
                </mrow>
              </math>
            </p>
            <p>（2） 下列正确的是（　）.</p>
            <p>A.cos（-60°）＜0</p>
            <p>B.tan 320°＞0</p>
            <p>
              C.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>4</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>5</mn>
                </mfrac>
                <mo>＞</mo>
                <mn>0</mn>
              </math>
            </p>
            <p>D.cos 330°＞0</p>
            <p>（3） 下列正确的是（　）.</p>
            <p>
              A.<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>4</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
                <mo>&lt;</mo>
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>5</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>7</mn>
                </mfrac>
              </math>
            </p>
            <p>
              B.<math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>3</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>5</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>&gt;</mo>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>4</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>5</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>
            </p>
            <p>
              C.<math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>9</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>7</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>&gt;</mo>
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>9</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>8</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>
            </p>
            <p>D.sin 6＞sin 7</p>
            <p>
              2.（1） 在0°～360°范围内，与1
              458°角终边相同的角是____，它是第____象限角；
            </p>
            <p>
              （2） 在0°～360°范围内，与-330°角终边相同的角是____，它是第____象限角.
            </p>
            <p>3.计算.</p>
            <p>
              （1）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mfrac>
                  <mrow>
                    <mn>25</mn>
                    <mi>π</mi>
                  </mrow>
                  <mn>3</mn>
                </mfrac>
                <mo>+</mo>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>17</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>4</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>+</mo>
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mo>−</mo>
                  <mfrac>
                    <mrow>
                      <mn>23</mn>
                      <mi>π</mi>
                    </mrow>
                    <mn>6</mn>
                  </mfrac>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
                <mo>=</mo>
              </math>______；
            </p>
            <p>
              （2） sin <sup>2</sup>150°+2sin 390°+cos
              <sup>2</sup>（-120°）+tan（-60°）= ；
            </p>
            <p>（3） tan1+cos 2+sin 3=______.（使用科学计算器）</p>
            <p>
              4.已知角<i>α</i>的终边上有一点 <i>P</i>（5，-12），求 sin
              <i>α</i>，cos <i>α</i>，tan <i>α</i>的值.
            </p>
            <p>
              5.已知<i>α</i>为锐角，且<math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>α</mi>
                <mo>=</mo>
                <mfrac>
                  <mn>3</mn>
                  <mn>5</mn>
                </mfrac>
              </math>，求cos <i>α</i>，tan <i>α</i>的值.
            </p>
            <p>
              6.已知 <i>tanθ</i>=3，且<i>θ</i>为第三象限角，求sin <i>θ</i>，cos
              <i>θ</i>的值.
            </p>
          </div>
        </div>
      </div>
    </div>
    <!-- 211 -->
@@ -785,7 +9806,244 @@
            <p><span>211</span></p>
          </li>
        </ul>
        <div class="padding-116"></div>
        <div class="padding-116">
          <div class="bj">
            <p>7.已知tan <i>α</i>=2，求下列各式的值.</p>
            <p>
              （1）
              <math display="0">
                <mfrac>
                  <mrow>
                    <mn>2</mn>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>+</mo>
                    <mn>3</mn>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                  <mrow>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>+</mo>
                    <mn>2</mn>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                </mfrac>
              </math>；
            </p>
            <p>
              （2）
              <math display="0">
                <mfrac>
                  <mrow>
                    <msup>
                      <mi>sin</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>+</mo>
                    <mn>3</mn>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                  <mrow>
                    <mn>3</mn>
                    <msup>
                      <mi>sin</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>2</mn>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                </mfrac>
              </math>.
            </p>
            <p>8.求使下列函数取得最大值、最小值时<i>x</i>的集合.</p>
            <p>（1） <i>y</i>=-1+2sin <i>x</i>；</p>
            <p>
              （2）
              <math display="0">
                <mi>y</mi>
                <mo>=</mo>
                <mn>1</mn>
                <mo>+</mo>
                <msqrt>
                  <mn>2</mn>
                </msqrt>
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
              </math>.
            </p>
            <p>
              <span class="bj-sp"><b>水平二</b></span>
            </p>
            <p>
              1.在半径为10 cm
              的圆中，60°的圆心角所对的弧长是____，对应的扇形面积是____.
            </p>
            <p>2.求下列指定范围内的角<i>x</i>的集合.</p>
            <p>
              （1）
              <math display="0">
                <mi>sin</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mo>−</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>，<i>x</i>∈-π，π，则<i>x</i>=____；
            </p>
            <p>
              （2）
              <math display="0">
                <mi>cos</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <msqrt>
                    <mn>2</mn>
                  </msqrt>
                  <mn>2</mn>
                </mfrac>
              </math>， <i>x</i>∈0，2π，则<i>x</i>=____；
            </p>
            <p>
              （3）
              <math display="0">
                <mi>tan</mi>
                <mo data-mjx-texclass="NONE">⁡</mo>
                <mi>x</mi>
                <mo>=</mo>
                <mfrac>
                  <msqrt>
                    <mn>3</mn>
                  </msqrt>
                  <mn>3</mn>
                </mfrac>
                <mrow data-mjx-texclass="INNER">
                  <mo data-mjx-texclass="OPEN">(</mo>
                  <mi>x</mi>
                  <mo>≠</mo>
                  <mfrac>
                    <mi>π</mi>
                    <mn>2</mn>
                  </mfrac>
                  <mo>+</mo>
                  <mi>k</mi>
                  <mi>π</mi>
                  <mo>,</mo>
                  <mi>k</mi>
                  <mo>∈</mo>
                  <mrow>
                    <mi mathvariant="bold">Z</mi>
                  </mrow>
                  <mo data-mjx-texclass="CLOSE">)</mo>
                </mrow>
              </math>， <i>x</i>∈-π，π，则<i>x</i>=____.
            </p>
            <p>
              3.已知<i>m</i>＜0，角<i>α</i>的终边经过点<i>P</i>（-3<i>m</i>，4<i>m</i>），求sin
              <i>α</i>+2cos <i>α</i>的值.
            </p>
            <p>4.已知tan（π-<i>α</i>）=3，求下列各式的值.</p>
            <p>（1） 2sin <i>α</i>·cos <i>α</i>；</p>
            <p>
              （2）
              <math display="0">
                <mfrac>
                  <mrow>
                    <msup>
                      <mi>sin</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>2</mn>
                    <mi>sin</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>⋅</mo>
                    <mi>cos</mi>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                  </mrow>
                  <mrow>
                    <mn>4</mn>
                    <msup>
                      <mi>cos</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>−</mo>
                    <mn>3</mn>
                    <msup>
                      <mi>sin</mi>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msup>
                    <mo data-mjx-texclass="NONE">⁡</mo>
                    <mi>α</mi>
                    <mo>+</mo>
                    <mn>1</mn>
                  </mrow>
                </mfrac>
              </math>.
            </p>
            <p>5.用“五点法”画出下列函数的图像，并写出它们的周期.</p>
            <p>（1） <i>y</i>=-5+2sin <i>x</i>；</p>
            <p>（2） <i>y</i>=5-cos <i>x</i>.</p>
          </div>
        </div>
      </div>
    </div>