共名和反省

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While toying with a triangular puzzle piece, you start practicing your math skills to see what you can find out about it. You realize one of the interior angles of the puzzle piece is $\begin{align*}30^\circ\end{align*}$ , and decide to compute the trig functions associated with this angle. You immediately want to compute the cosine of the angle, but can only remember the values of your sine functions.
::在玩一个三角拼图片时, 您开始练习自己的数学技能, 以了解您能找到什么。 您意识到拼图片的一个内部角度是 30 , 并决定计算与此角度相关的三角函数 。 您立即要计算角的余弦, 但只能记住您正弦函数的值 。

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Is there a way to use this knowledge of sine functions to help you in your computation of the cosine function for $\begin{align*}30^\circ\end{align*}$ ?
::是否有方法可以使用这种对正弦函数的知识 来帮助您计算 30\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以帮助您计算 comsine函数的连弦函数的计算 30\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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Cofunction Identities and Reflection
::共名和反省

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In a right triangle, you can apply what are called " cofunction identities". These are called cofunction identities because the functions have common values. These identities are summarized below.
::在右三角形中, 您可以应用所谓的“ 函数身份 ” 。 这些特性被称为“ 函数身份 ” , 因为函数具有共同值。 这些特性概述如下 。

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$$\begin{align*}\sin \theta = \cos(90^\circ-\theta) && \cos \theta = \sin (90^\circ-\theta)\\ \tan \theta = \cot(90^\circ-\theta) && \cot \theta = \tan (90^\circ-\theta)\end{align*}$$
:90) (90) (90) (tan) (90) (tan) (cot) (90) (cot) (90) (cot) (90) (90)

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Let's take a look at some problems involving cofunction identities and reflection. 
::让我们来研究一下 与共用身份和思考有关的一些问题

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1. Find the value of $\begin{align*}\cos 120^\circ\end{align*}$ .
::1. 找出Cos120的值。

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Because this angle has a reference angle of $\begin{align*}60^\circ\end{align*}$ , the answer is $\begin{align*}\cos 120^\circ = -\frac{1}{2}\end{align*}$ .
::因为这个角度的参考角度为 60 , 答案是 cos 120 12 。

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2. Find the value of $\begin{align*}\cos (-120^\circ)\end{align*}$ .
::2. 查找 cos(- 120) 的值 。

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Because this angle has a reference angle of $\begin{align*}60^\circ\end{align*}$ , the answer is $\begin{align*}\cos (-120^\circ) = \cos 240^\circ = -\frac{1}{2}.\end{align*}$
::因为这个角度的参考角度为 60 , 答案是 cos (- 120 ) = cos 240 12 。

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3. Find the value of $\begin{align*}\sin 135^\circ\end{align*}$ .
::3. 找出罪的值 135______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Because this angle has a reference angle of $\begin{align*}45^\circ\end{align*}$ , the answer is $\begin{align*}\sin 135^\circ = \frac{\sqrt{2}}{2}\end{align*}$
::因为这个角度的参考角度是 45, 答案是 sin13522

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked if there is a way to use your knowledge of sine functions to help you in your computation of the cosine function.
::早些时候,有人问您是否有办法使用您对正弦函数的知识来帮助您计算余弦函数。

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Since you now know the cofunction relationships, you can use your knowledge of sine functions to help you with the cosine computation:
::既然你现在知道共管关系, 你可以使用你对正弦函数的知识 来帮助你计算余弦值:

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$\begin{align*}\cos 30^\circ = \sin (90^\circ - 30^\circ) = \sin (60^\circ) = \frac{\sqrt{3}}{2}\end{align*}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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Example 2
::例2

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 Find the value of $\begin{align*}\sin 45^\circ\end{align*}$ using a cofunction identity .
::使用共用身份查找 sin45 的值 。

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The sine of $\begin{align*}45^\circ\end{align*}$ is equal to $\begin{align*}\cos (90^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2}\end{align*}$ .
::45乘以45等于cos(9045)=cos4522。

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Example 3
::例3

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Find the value of $\begin{align*}\cos 45^\circ\end{align*}$ using a cofunction identity.
::使用共同功能身份查找 cos45 的值 。

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The cosine of $\begin{align*}45^\circ\end{align*}$ is equal to $\begin{align*}\sin (90^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}\end{align*}$ .
::45的余弦等于sin(9045)=sin4522。

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Example 4
::例4

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Find the value of $\begin{align*}\cos 60^\circ\end{align*}$ using a cofunction identity.
::使用共同功能身份查找 COs60 的值 。

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The cosine of $\begin{align*}60^\circ\end{align*}$ is equal to $\begin{align*}\sin (90^\circ - 60^\circ) = \sin 30^\circ = .5\end{align*}$ .
::60的余弦等于sin(9060)=sin305.

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Review
::回顾

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1. Find a value for $\begin{align*}\theta\end{align*}$ for which $\begin{align*}\sin \theta=\cos 15^\circ\end{align*}$ is true.
   ::找到一个值, 其中sincos15是真实的 。
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2. Find a value for $\begin{align*}\theta\end{align*}$ for which $\begin{align*}\cos \theta=\sin 55^\circ\end{align*}$ is true.
   ::查找“ ” 的值, Cossin_ 55是真实的 。
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3. Find a value for $\begin{align*}\theta\end{align*}$ for which $\begin{align*}\tan \theta=\cot 80^\circ\end{align*}$ is true.
   ::寻找“ ” 的值, 对于该值来说, tancot80是真实的 。
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4. Find a value for $\begin{align*}\theta\end{align*}$ for which $\begin{align*}\cot \theta=\tan 30^\circ\end{align*}$ is true.
   ::寻找“ ” 的值, 因为对于“ ” 来说是真实的 。
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5. Use cofunction identities to help you write the expression $\begin{align*}\tan 255^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
   ::使用 complay 身份来帮助您写入表达式 tan 255 , 以作为度量小于 45 的急性角度的函数 。
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6. Use cofunction identities to help you write the expression $\begin{align*}\sin 120^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
   ::使用 complay 身份来帮助您写入 sin 120 的表达式, 以作为度量小于 45 的急性角度的函数 。
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7. Use cofunction identities to help you write the expression $\begin{align*}\cos 310^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
   ::使用 complay 身份来帮助您写入表达式 cos310 , 以作为度量小于 45 的急性角度的函数 。
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8. Use cofunction identities to help you write the expression $\begin{align*}\cot 260^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
   ::使用 complay 身份来帮助您写入表达式 comt 260 , 以作为度量小于 45 的急性角度的函数 。
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9. Use cofunction identities to help you write the expression $\begin{align*}\cos 280^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
   ::使用 complay 身份来帮助您写入表达式 cos @ @ @ @ 280 @ , 以作为度量小于 45\\ 的急性角度的函数 。
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10. Use cofunction identities to help you write the expression $\begin{align*}\tan 60^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
    ::使用 complay 身份来帮助您写入表达式 tan60 , 以作为测量量小于 45 的急性角度的函数 。
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11. Use cofunction identities to help you write the expression $\begin{align*}\sin 100^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
    ::使用 complay 身份来帮助您写入 sin100 表达式, 以作为测量量小于 45 的急性角度的函数 。
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12. Use cofunction identities to help you write the expression $\begin{align*}\cos 70^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
    ::使用 complay 身份来帮助您写入表达式 cos 70 {} , 以作为度量小于 45 {} 的急性角度的函数 。
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13. Use cofunction identities to help you write the expression $\begin{align*}\cot 240^\circ\end{align*}$ as the function of an acute angle of measure less than $\begin{align*}45^\circ\end{align*}$ .
    ::使用 complay 身份来帮助您写入表达式 cot 240 , 以作为度量小于 45 的急性角度的函数 。
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14. Use a right triangle to prove that $\begin{align*}\sin \theta=\cos (90^\circ-\theta)\end{align*}$ .
    ::使用右三角形来证明 sincos(90) 。
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15. Use the sine and cosine cofunction identities to prove that $\begin{align*}\tan (90^\circ-\theta)=\cot \theta\end{align*}$ .
    ::使用正弦和余弦的共振身份来证明 tan(90)=cot。

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Review (Answers)
::回顾(答复)

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Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。