WEBVTT

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如果在街道上我走过来

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问你-

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我不想写得那么复杂

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sinπ/4等于多少

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很明显 我们讨论的是弧度

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要么记住结果

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要么画出单位圆

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画得不太标准

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你们明白意思就行

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π/4弧度

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也就是45°

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半径为1

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那么sin可以定义为

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单位圆上的纵坐标

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只需要知道这个值

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就可以了

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这个角是45°

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将三角形画得大一些

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三角形像这样

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这是45°

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这是45°

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这是90°

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可以求解45°45°90°三角形

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斜边长度为1

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这两个边设为x

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长度相等

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是等腰三角形

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底角相等

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那么

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x方+x方等于1的平方

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也就是等于1

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2倍的x方等于1

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x方等于1/2

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x等于1/2的平方根

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等于1/根2

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可以化为有理数

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乘以√2/√2

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得到x等于√2/2

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这里高是√2/2

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如果想知道这个长度

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也是√2/2

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我们只关心这个高度

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因为正弦值 sinπ/4

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就是这个高度

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纵坐标值

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等于√2/2

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这些都是复习

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在单位圆那一集视频中学过

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但是如果是其他人

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另一天我走过来

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向你请教

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arcsin√2/2等于多少

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arcsin是什么？

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你们会感到困惑

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你们可能会想 我知道一个角度的正弦值

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但这个是Sal定义的一个新的

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三角函数

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你们应该意识到

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当单词前缀是arc时

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有时指的是反正弦

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可以简写为

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sin-1√2/2

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等于多少？

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其实问题是

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正弦值是根2/2

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对应角度是多少

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也就是什么角度的

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正弦值是√2/2

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这些表述可以重写为

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我们来写一下

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这些表述可以重写为

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sin?=√2/2

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我认为这个问题

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更简单一些

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sin?=√2/2

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我说过sinπ/4等于

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√2/2

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在这种情况下

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我知道sinπ/4

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等于√2/2

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所以这个问号等于π/4

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也可以重写为

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arcsin√2/2等于

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π/4

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你们或许会说- 复习一下

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给定一个值

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我是指对于给定角度

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求其正弦值

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值是确定的

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但是你们倾向于-

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看一下

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从这里复习

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π/4是正确的（注意这里老师误说成π/2了）

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45°是正确的

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如果不断累加360°

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或者不断增加2π

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这些值都是正确的

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因为这些值对应

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单位圆上同一个点

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确实如此

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所有这些值

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都是正确的结果

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对吧？

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因为如果对任意这些角度求正弦值

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对不断累加360°后的角度

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求其正弦值

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都会得到√2/2

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这是个问题

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不应该存在这样的函数

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如果写成函数

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不应该存在函数f(x)

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同一x对应多个函数值

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x可以取π/4

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可以取π/4+2π

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也可以取π/4+4π

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为得到正确的函数

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为使反正弦函数有意义

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需要限定取值范围

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将取值范围

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限定为最普遍的情况

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现在限定取值范围

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换句话说

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定义域是什么？

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如果求arcsin

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对x求arcsin

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设arcsin等于θ

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定义域是什么？

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x的取值范围是什么？

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x可以取多少？

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对任意角度求正弦值

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得到的结果在1到-1之间

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所以x取值

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-1≤x

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x≤1

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这就是定义域

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为使函数存在

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需要限定取值范围

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可能的取值

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需要限定取值范围

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对于arcsin

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一般限定在第一和

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第四象限

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将取值角度限定在

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单位圆上这一区域内

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所以θ取值

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θ≤π/2

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-π/2≤θ

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由此可知arcsin是多少了

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再解一道题

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换个地方来写

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再求一个arcsin

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问题是

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arcsin(-√3/2)

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如果你记得

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那么马上可以得到sinx

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或者sinθ=-√3/2

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问题解决了

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但是我没记住结果

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我就需要画出单位圆

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帮助求解arcsin

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只需要画出单位圆的

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第一和第四象限

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这是y轴

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这是x轴

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x和y

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题目对应的点在哪里？

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某角度的正弦值

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等于-√3/2

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意味着单位圆上点的纵坐标

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等于-√3/2

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就是这个点

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纵坐标是-√3/2

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这是对应的点

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对应的角度是多少？

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想一下

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纵坐标是-√3/2

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就是这个角

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这个角是负的

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因为沿顺时针方向在x轴下方

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为求解

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画一个小三角形

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换种颜色

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这个三角形

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用蓝色画

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将三角形放大

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像这样

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这是θ

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这是θ

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这个边长是多少？

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也就是纵坐标

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可以称为

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是√3/2

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其实是负值 因为是在x轴下面

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但只是为表征这个角

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我们知道这个角是负的

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当看到√3/2时

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希望你们会想到这是30°60°90°三角形

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这个边是√3/2

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这个边是1/2

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当然 这个边是1

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因为单位圆

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半径为1

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30°60°90°三角形

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对边是√3/2

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θ=60°

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这个角是30°

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我们得到θ=60°

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这是其大小

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因为在x轴下面

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所以是-60°

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所以θ=-60°

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如果化为弧度

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还不够好

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可以乘以100-

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抱歉 乘以π弧度比上180°

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消去角度

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得到θ=-π/3

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弧度

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那么现在

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可以得出结论

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arcsin(-√3/2)

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等于-π/3弧度

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或者写成sin-1(-√3/2)等于

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-π/3弧度

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为验证这个结果

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我们用计算器算一下

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设为弧度模式

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检查一下

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点击Per second mode键

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弧度模式下

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希望得到正确的结果

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计算反正弦

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sin-1

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（-√3/2）

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等于-1.04

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可知

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等于-1.04rad

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那么π/3应该等于1.04

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验算一下

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-π/3

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等于多少？

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结果相同

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计算器给出相同的结果

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但是计算器有缺点

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因为计算器不能显示出

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这就是-π/3