节: 使用总和和差异公式查找精确的三角值 | 14.12 使用总和和差异公式查找精确的三角值

章节大纲

You measure an angle with your protractor to be $\begin{align*}165^\circ\end{align*}$ . How could you find the exact sine of this angle without using a calculator?
::您用您的减速器测量角度为 165 {} 。您如何在不使用计算器的情况下找到这个角度的正弦值 ?

Sum and Difference Formulas
::总和和差额和差额公式

You know that $\begin{align*}\sin 30^\circ=\frac{1}{2}, \cos 135^\circ=-\frac{\sqrt{2}}{2}, \tan 300 ^\circ = -\sqrt{3},\end{align*}$ etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of $\begin{align*}30^\circ, 45^\circ,\end{align*}$ and $\begin{align*}60^\circ\end{align*}$ . Using the , we can find these exact trig values.
::您知道 sin3012,cos13522,tan3003,等等... 来自特殊的右三角形。在这个概念中,我们将学习如何找到除3045和603的倍数以外的角度的三角函数的精确值。使用这些参数,我们就能找到这些精确的三角值。

Sum and Difference Formulas
::总和和差额和差额公式

$\begin{aligned} \sin (a \pm b) & = \sin a \cos b \pm \cos a \sin b \\ \cos (a \pm b) & = \cos a \cos b \mp \sin a \sin b \\ \tan (a \pm b) & = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b} \end{aligned}$
$\begin{align*}\sin(a\pm b) &=\sin a \cos b \pm \cos a \sin b \\ \cos(a\pm b) &=\cos a \cos b \mp \sin a \sin b \\ \tan(a \pm b) &=\frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}\end{align*}$
:ab)=sinacosbcosbbsin(ab) =cosçacosbsinbsinbtan(ab) =tanaatanbb1natanatanatanbbbb}(ab)

Find the exact values using the Sum and Difference Formulas: $\begin{align*}\sin 75^\circ\end{align*}$
::使用总和和差异公式查找准确值:sin75

This is an example of where we can use the sine sum formula from above, $\begin{align*}\sin(a+ b)=\sin a \cos b+\cos a \sin b\end{align*}$ , where $\begin{align*}a = 45^\circ\end{align*}$ and $\begin{align*}b = 30^\circ\end{align*}$ .
::这个例子说明我们可以从上面使用正数公式,sin(a+b)=sinacosb+cosçasinb,a=45和b=30。

$\begin{aligned} \sin 75^{\circ} & = \sin (45^{\circ} + 30^{\circ}) \\ = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \\ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \\ = \frac{\sqrt{6} + \sqrt{2}}{4} \end{aligned}$
$\begin{align*}\sin 75^\circ &=\sin(45^\circ + 30 ^\circ) \\ &= \sin 45^\circ \cos 30^\circ +\cos 45^\circ \sin 30 ^\circ \\ &= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\ &= \frac{\sqrt{6}+\sqrt{2}}{4}\end{align*}$
:4530) (45303030304545453030302232+2212=6+24)

In general, $\begin{align*}\sin (a+b)\ne \sin a+\sin b\end{align*}$ and similar statements can be made for the other sum and difference formulas.
::一般而言,对于其他总和和差价公式,可以作sin(a+b)sina+sinb和类似的说明。

Find the exact values using the Sum and Difference Formulas: $\begin{align*}\cos \frac{11 \pi}{12}\end{align*}$
::使用总和和差异公式查找准确值: cos1112

For this problem, we could use either the sum or difference cosine formula, $\begin{align*}\frac{11\pi}{12}=\frac{2\pi}{3}+\frac{\pi}{4}\end{align*}$ or $\begin{align*}\frac{11\pi}{12}=\frac{7\pi}{6}-\frac{\pi}{4}\end{align*}$ . Let’s use the sum formula .
::对于这个问题,我们可以使用总和或差额的余弦配方,1112=234或1112=764。让我们使用总和配方。

$\begin{aligned} \cos \frac{11 π}{12} & = \cos (\frac{2 π}{3} + \frac{π}{4}) \\ = \cos \frac{2 π}{3} \cos \frac{π}{4} - \sin \frac{2 π}{3} \sin \frac{π}{4} \\ = - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \\ = - \frac{\sqrt{2} + \sqrt{6}}{4} \end{aligned}$
$\begin{align*}\cos \frac{11\pi}{12} &=\cos \left(\frac{2\pi}{3}+\frac{\pi}{4}\right) \\ &=\cos \frac{2\pi}{3}\cos \frac{\pi}{4}-\sin\frac{2\pi}{3}\sin \frac{\pi}{4} \\ &= -\frac{1}{2}\cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \\ &= -\frac{\sqrt{2}+\sqrt{6}}{4}\end{align*}$
::=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

Find the exact values using the Sum and Difference Formulas: $\begin{align*}\tan \left(-\frac{\pi}{12}\right)\end{align*}$
::使用总和和差异公式查找准确值: tan 12

This angle is the difference between $\begin{align*}\frac{\pi}{4}\end{align*}$ and $\begin{align*}\frac{\pi}{3}\end{align*}$ .
::此角度是 4 和 3 之间的差。

$\begin{aligned} \tan (\frac{π}{4} - \frac{π}{3}) & = \frac{\tan \frac{π}{4} - \tan \frac{π}{3}}{1 + \tan \frac{π}{4} \tan \frac{π}{3}} \\ = \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \end{aligned}$
$\begin{align*}\tan \left(\frac{\pi}{4}-\frac{\pi}{3}\right) &=\frac{\tan \frac{\pi}{4}-\tan \frac{\pi}{3}}{1+\tan \frac{\pi}{4} \tan \frac{\pi}{3}} \\ &=\frac{1-\sqrt{3}}{1+\sqrt{3}}\end{align*}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This angle is also the same as $\begin{align*}\frac{23 \pi}{12}\end{align*}$ . You could have also used this value and done $\begin{align*}\tan\left(\frac{\pi}{4}+\frac{5 \pi}{3}\right)\end{align*}$ and arrived at the same answer.
::此角度也与 2312 相同。您也可以使用此值, 完成 tan( 4+53) 并达成相同的答案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the exact value of $\begin{align*}\sin 165^\circ\end{align*}$ without using the calculator.
::早些时候,你被要求在不使用计算器的情况下找到sin"165"的准确价值。

We can use the sine sum formula, $\begin{align*}\sin(a+b)=\sin a \cos b+\cos a \sin b\end{align*}$ , where $\begin{align*}a = 120^\circ\end{align*}$ and $\begin{align*}b = 45^\circ\end{align*}$ .
::我们可以使用正弦数公式,sin(a+b)=sinacosb+cosasinb, 其中a=120和b=45。

$\begin{aligned} \sin 165^{\circ} & = \sin (120^{\circ} + 45^{\circ}) \\ = \sin 120^{\circ} \cos 45^{\circ} + \cos 120^{\circ} \sin 45^{\circ} \\ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{- 1}{2} \cdot \frac{\sqrt{2}}{2} \\ = \frac{\sqrt{6} - \sqrt{2}}{4} \end{aligned}$
$\begin{align*}\sin 165^\circ &=\sin(120^\circ + 45 ^\circ) \\ &= \sin 120^\circ \cos 45^\circ +\cos 120^\circ \sin 45 ^\circ \\ &= \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}+\frac{-1}{2} \cdot \frac{\sqrt{2}}{2} \\ &= \frac{\sqrt{6}-\sqrt{2}}{4}\\\end{align*}$
:12045) (12044444421202221222=624)

Example 2
::例2

Find the exact value of $\begin{align*}\cos 15^\circ\end{align*}$ .
::查找准确的 cos15 值。

$\begin{aligned} \cos 15^{\circ} & = \cos (45^{\circ} - 30^{\circ}) \\ = \cos 45^{\circ} \cos 30^{\circ} + \sin 45^{\circ} \sin 30^{\circ} \\ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \\ = \frac{\sqrt{6} + \sqrt{2}}{4} \end{aligned}$
$\begin{align*}\cos 15^\circ &=\cos(45^\circ - 30^\circ) \\ &= \cos 45^\circ \cos 30^\circ + \sin 45 ^\circ \sin 30 ^\circ \\ &= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\ &= \frac{\sqrt{6}+\sqrt{2}}{4}\end{align*}$
::==6+24 =6+24 ==6+24 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Example 3
::例3

Find the exact value of $\begin{align*}\tan 255^\circ\end{align*}$ .
::查找 tan 255 的确切值。

$\begin{aligned} \tan (210^{\circ} + 45^{\circ}) & = \frac{\tan 210^{\circ} + \tan 45^{\circ}}{1 - \tan 210^{\circ} \tan 45^{\circ}} \\ = \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3}} = \frac{\frac{\sqrt{3} + 3}{3}}{\frac{3 - \sqrt{3}}{3}} = \frac{\sqrt{3} + 3}{3 - \sqrt{3}} \end{aligned}$
$\begin{align*}\tan (210^\circ + 45^\circ) &=\frac{\tan 210^\circ+\tan 45^\circ}{1-\tan 210^\circ \tan 45^\circ} \\ &= \frac{\frac{\sqrt{3}}{3}+1}{1-\frac{\sqrt{3}}{3}}=\frac{\frac{\sqrt{3}+3}{3}}{\frac{3-\sqrt{3}}{3}}=\frac{\sqrt{3}+3}{3-\sqrt{3}}\end{align*}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Review
::回顾

Find the exact value of the following trig functions.
::查找以下三角函数的准确值。

$\begin{align*}\sin 15^\circ\end{align*}$
::15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 18岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁 15岁

$\begin{align*}\cos \frac{5\pi}{12}\end{align*}$
::CO=512

$\begin{align*}\tan 345^\circ\end{align*}$
::丹・345 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

$\begin{align*}\cos (-255^\circ)\end{align*}$
::COs(- 255)

$\begin{align*}\sin \frac{13 \pi}{12}\end{align*}$
::问题13和12

$\begin{align*}\sin \frac{17\pi}{12}\end{align*}$
::问题17=12

$\begin{align*}\cos 15^\circ\end{align*}$
::来来来来来 15 15

$\begin{align*}\tan (-15^\circ)\end{align*}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\begin{align*}\sin 345^\circ\end{align*}$
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Now, use $\begin{align*}\sin 15^\circ\end{align*}$ from #1, and find $\begin{align*}\sin 345^\circ\end{align*}$ . Do you arrive at the same answer? Why or why not?
::现在,请使用1号的sin15, 并找到 sin345。您是否达到相同的答案? 为什么或为什么不呢 ?

Using $\begin{align*}\cos 15^\circ\end{align*}$ from #7, find $\begin{align*}\cos 165^\circ\end{align*}$ . What is another way you could find $\begin{align*}\cos 165^\circ\end{align*}$ ?
::使用 # 7 的 cos 15 , 找到 cos 165 。另一种方法是什么? 能找到 cos 165 ?

Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
::描述您看到的这些“新”角度的正弦、连弦和正切之间的任何模式。

Using your calculator, find the $\begin{align*}\sin 142^\circ\end{align*}$ . Now, use the sum formula and your calculator to find the $\begin{align*}\sin 142^\circ\end{align*}$ using $\begin{align*}83^\circ\end{align*}$ and $\begin{align*}59^\circ\end{align*}$ .
::使用您的计算器, 找到 sin\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Use the sine difference formula to find $\begin{align*}\sin 142^\circ\end{align*}$ with any two angles you choose. Do you arrive at the same answer? Why or why not?
::使用正弦差公式查找 sin\\\\ 142\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Challenge Using $\begin{align*}\sin (a+b)=\sin a \cos b +\cos a \sin b\end{align*}$ and $\begin{align*}\cos (a+b)=\cos a \cos b - \sin a \sin b\end{align*}$ , show that $\begin{align*}\tan (a+b)=\frac{\tan a + \tan b}{1-\tan a \tan b}\end{align*}$ .
::使用sin(a+b) =sinacosb+cosasinb 和cos(a+b) =cosacosb-sinasinb 显示,tan(a+b) =tana+tanb1-tanatanatanatanb。

Review (Answers)
::回顾(答复)

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。

章节大纲

Sum and Difference Formulas ::总和和差额和差额公式

Sum and Difference Formulas ::总和和差额和差额公式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Sum and Difference Formulas
::总和和差额和差额公式

Sum and Difference Formulas
::总和和差额和差额公式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)