Course: 14.15 使用双角和半角公式查找精确的三角值

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使用双角和半角公式查找精确的三角值
You want to find the exact value of $\begin{align*}\tan \frac{3\pi}{8}\end{align*}$ . How could you find this value without using a calculator?
::您想要找到 tan38 的确切值。您如何在不使用计算器的情况下找到此值 ?

Double Angle and Half Angle Formulas
::双角和半角公式

In this concept, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles. Here we will introduce the Double-Angle $\begin{align*}(2a)\end{align*}$ and Half-Angle $\begin{align*}\left(\frac{a}{2}\right)\end{align*}$ Formulas.
::在此概念中, 我们将学习如何找到其他角度的一半或两倍的角的三角函数的精确值。在此我们将引入双角( 2a) 和半角( a2) 公式。

Double-Angle and Half-Angle Formulas
::双角和半角公式

$\begin{aligned} \cos 2 a & = \cos^{2} a - \sin^{2} a & \sin 2 a = 2 \sin a \cos a \\ = 2 \cos^{2} a - 1 & \tan 2 a = \frac{2 \tan a}{1 - \tan^{2} a} \\ = 1 - \sin^{2} a \\ \sin \frac{a}{2} & = \pm \sqrt{\frac{1 - \cos a}{2}} & \tan \frac{a}{2} = \frac{1 - \cos a}{\sin a} \\ \cos \frac{a}{2} & = \pm \sqrt{\frac{1 + \cos a}{2}} & = \frac{\sin a}{1 + \cos a} \end{aligned}$
$\begin{align*}\cos 2a&=\cos^2 a- \sin^2a && \sin 2a=2 \sin a \cos a \\ &=2 \cos^2 a-1 && \tan 2a=\frac{2 \tan a}{1- \tan^2 a} \\ &=1- \sin^2a \\ \sin \frac{a}{2}&= \pm \sqrt{\frac{1- \cos a}{2}} && \tan \frac{a}{2}=\frac{1- \cos a}{\sin a} \\ \cos \frac{a}{2}&= \pm \sqrt{\frac{1+ \cos a}{2}} && \qquad \ \ =\frac{\sin a}{1+ \cos a}\end{align*}$
::2a=cos2a=cos2a-sin2asin2a=2sinçacosa=2cos2a=2cos2aa=2cos2aa-1tan2a1-tan2a=1-sin2a2a2a1-cosía2a2a2a2a2=1cosçasinçacosa2a1+cos_a2=sina1+cosa2

The signs of $\begin{align*}\sin \frac{a}{2}\end{align*}$ and $\begin{align*}\cos \frac{a}{2}\end{align*}$ depend on which quadrant $\begin{align*}\frac{a}{2}\end{align*}$ lies in. For $\begin{align*}\cos 2a\end{align*}$ and $\begin{align*}\tan \frac{a}{2}\end{align*}$ any formula can be used to solve for the exact value.
::sina2 和 cosa2 的符号取决于 A2 的方位值。对于 os2a 和 tana2 , 任何公式都可以用来解析准确值。

Let's find the exact value of $\begin{align*}\cos \frac{\pi}{8}\end{align*}$ .
::让我们来找找Cos8的准确价值。

$\begin{align*}\frac{\pi}{8}\end{align*}$ is half of $\begin{align*}\frac{\pi}{4}\end{align*}$ and in the first quadrant.
::8是4的一半在第一个象限。

$\begin{aligned} \cos (\frac{1}{2} \cdot \frac{π}{4}) & = \sqrt{\frac{1 + \cos \frac{π}{4}}{2}} \\ = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \\ = \sqrt{\frac{1}{2} \cdot \frac{2 + \sqrt{2}}{2}} \\ = \frac{\sqrt{2 + \sqrt{2}}}{2} \end{aligned}$
$\begin{align*}\cos \left(\frac{1}{2} \cdot \frac{\pi}{4}\right)&=\sqrt{\frac{1+ \cos \frac{\pi}{4}}{2}} \\ &=\sqrt{\frac{1+ \frac{\sqrt{2}}{2}}{2}} \\ &=\sqrt{\frac{1}{2} \cdot \frac{2+\sqrt{2}}{2}} \\ &=\frac{\sqrt{2+ \sqrt{2}}}{2}\end{align*}$
:124) = 1+cos42= 1+222=122+22=2+22

Now, let's find the exact value of $\begin{align*}\sin 2a\end{align*}$ if $\begin{align*}\cos a=- \frac{4}{5}\end{align*}$ and $\begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}$ .
::现在,让我们找到sin2a的准确价值如果cosa45 和 32a<2。

To use the sine double-angle formula, we also need to find $\begin{align*}\sin a\end{align*}$ , which would be $\begin{align*}\frac{3}{5}\end{align*}$ because $\begin{align*}a\end{align*}$ is in the $\begin{align*}4^{th}\end{align*}$ quadrant.
::要使用正弦双角公式, 我们还需要找到sina, 这将是35, 因为一个是在第四象限。

$\begin{aligned} \sin 2 a & = 2 \sin a \cos a \\ = 2 \cdot \frac{3}{5} \cdot - \frac{4}{5} \\ = - \frac{24}{25} \end{aligned}$
$\begin{align*}\sin 2a&=2 \sin a \cos a \\ &=2 \cdot \frac{3}{5} \cdot - \frac{4}{5} \\ &=- \frac{24}{25}\end{align*}$
::-=================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

Finally, let's find the exact value of $\begin{align*}\tan 2a\end{align*}$ for $\begin{align*}a\end{align*}$ from the previous problem.
::最后,让我们从上一个问题中找到 tan2a 的确切价值。

Use $\begin{align*}\tan a=\frac{\sin a}{\cos a}=\frac{\frac{3}{5}}{- \frac{4}{5}}=- \frac{3}{4}\end{align*}$ to solve for $\begin{align*}\tan 2a\end{align*}$ .
::使用 tana = sinçacosa = 35 - 4534 来解答 tan2a 。

$\begin{aligned} \tan 2 a = \frac{2 \cdot - \frac{3}{4}}{1 - {(- \frac{3}{4})}^{2}} = \frac{- \frac{3}{2}}{\frac{7}{16}} = - \frac{3}{2} \cdot \frac{16}{7} = - \frac{24}{7} \end{aligned}$
$\begin{align*}\tan 2a=\frac{2 \cdot - \frac{3}{4}}{1- \left(- \frac{3}{4}\right)^2}=\frac{- \frac{3}{2}}{\frac{7}{16}}= - \frac{3}{2} \cdot \frac{16}{7}=- \frac{24}{7}\end{align*}$
::2a=2341 -(-34)23271632167247

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the value of $\begin{align*}\tan \frac{3\pi}{8}\end{align*}$ without a calculator.
::早些时候,有人要求你找到 tan38的值而不使用计算器。

$\begin{align*}\frac{3\pi}{8} = \frac{1}{2} \cdot \frac{3\pi}{4}\end{align*}$ so we can use the formula $\begin{align*}tan \frac{a}{2} =\frac{\sin a}{1+ \cos a}\end{align*}$ for $\begin{align*}a=\frac{3\pi}{4}\end{align*}$
::38=1234 这样我们就可以使用公式 tana2=sina1+cosa 来表示 a=34

$\begin{aligned} \tan \frac{3 π}{8} = \frac{\sin \frac{3 π}{4}}{1 + \cos \frac{3 π}{4}} \\ = \frac{\frac{\sqrt{2}}{2}}{1 + \frac{- \sqrt{2}}{2}} \end{aligned}$
$\begin{align*}\tan \frac{3\pi}{8} =\frac {\sin \frac{3\pi}{4}}{1+ \cos \frac{3\pi}{4}}\\ =\frac {\frac{\sqrt{2}}{2}}{{1 +\frac{-\sqrt{2}}{2}}}\end{align*}$
::======================================================================================================================================================================================================================================================================================================

If we simplify this expression, we get $\begin{align*}\sqrt{2} + 1\end{align*}$ .
::如果我们简化这个表达式, 我们就会得到 2+1 。

Example 2
::例2

Find the exact value of $\begin{align*}\cos \left(-\frac{5 \pi}{8}\right)\end{align*}$ .
::查找 cos(- 58) 的确切值。

$\begin{align*}- \frac{5 \pi}{8}\end{align*}$ is in the $\begin{align*}3^{rd}\end{align*}$ quadrant.
::-5 -8位于第3象限

$\begin{aligned} - \frac{5 π}{8} = \frac{1}{2} (- \frac{5 π}{4}) \to \cos \frac{1}{2} (- \frac{5 π}{4}) = - \sqrt{\frac{1 + \cos (- \frac{5 π}{4})}{2}} \\ = - \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{1}{2} \cdot \frac{2 - \sqrt{2}}{2}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \end{aligned}$
$\begin{align*}- \frac{5 \pi}{8}=\frac{1}{2} \left(- \frac{5 \pi}{4}\right) \rightarrow \cos \frac{1}{2} \left(- \frac{5 \pi}{4}\right)=- \sqrt{\frac{1+ \cos \left(- \frac{5 \pi}{4}\right)}{2}} \\ =-\sqrt{\frac{1- \frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{1}{2} \cdot \frac{2-\sqrt{2}}{2}}=\frac{\sqrt{2- \sqrt{2}}}{2}\end{align*}$
::-58=12(- 54) 12(- 54) 1+cos(- 54) 21-2-222=122-22=2-22

Example 3
::例3

Given the function $\begin{align*}\cos a=\frac{4}{7}\end{align*}$ and $\begin{align*}0 \le a < \frac{\pi}{2}\end{align*}$ , find $\begin{align*}\sin 2a\end{align*}$ .
::根据函数 cosa=47 和 0a2, 找到 sin2a。

First, find $\begin{align*}\sin a\end{align*}$ . $\begin{align*}4^2+y^2=7^2\rightarrow y=\sqrt{33}\end{align*}$ , so $\begin{align*}\sin a=\frac{\sqrt{33}}{7}\end{align*}$
::首先,找到sina. 42+y2=72y=33,所以sina=337

$\begin{align*}\sin 2a=2 \cdot \frac{\sqrt{33}}{7} \cdot \frac{4}{7}=\frac{8 \sqrt{33}}{49}\end{align*}$
::=233747=83349

Example 4
::例4

Given the function $\begin{align*}\cos a=\frac{4}{7}\end{align*}$ and $\begin{align*}0 \le a < \frac{\pi}{2}\end{align*}$ , f ind $\begin{align*}\tan \frac{a}{2}\end{align*}$ .
::根据函数 cosa=47 和 0a2, 找到 tana2 。

You can use either $\begin{align*}\tan \frac{a}{2}\end{align*}$ formula.
::您可以使用 tana2 公式。

$\begin{aligned} \tan \frac{a}{2} = \frac{1 - \frac{4}{7}}{\frac{\sqrt{33}}{7}} = \frac{3}{7} \cdot \frac{7}{\sqrt{33}} = \frac{3}{\sqrt{33}} = \frac{\sqrt{33}}{11} \end{aligned}$
$\begin{align*}\tan \frac{a}{2}=\frac{1- \frac{4}{7}}{\frac{\sqrt{33}}{7}}=\frac{3}{7} \cdot \frac{7}{\sqrt{33}}=\frac{3}{\sqrt{33}}=\frac{\sqrt{33}}{11}\end{align*}$
::tana2=1-47337=37733=333=3311

Review
::回顾

Find the exact value of the following angles.
::查找以下角度的确切值。

$\begin{align*}\sin 105^\circ\end{align*}$
::一九零五年... 一九零五年... 一九零五年...

$\begin{align*}\tan \frac{\pi}{8}\end{align*}$
::tan8

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

$\begin{align*}\cos 165^\circ\end{align*}$
::1651151111111111111111111111111221222222222222222222222222111122211112212211121111211211122

$\begin{align*}\sin \frac{3 \pi}{8}\end{align*}$
::三、三、八、九、九、九、九、九、九、八、十、十、十、十、十、十、十、八、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、

$\begin{align*}\tan \left(- \frac{\pi}{12}\right)\end{align*}$
::tan 12

$\begin{align*}\sin \frac{11 \pi}{8}\end{align*}$
::性别 11_11_8_11_11_8_11_11_11_11_8_11_11_11_11_11_11_11_11_11_11_11_11_11_11_11_8_11_11_11_11_11_11_11_11_8_11_11_11_11_11_8_11_11_11_8_11_11_8_11_11_8_11_11_11_8_11_8_11_11_8_11_11_8_8_11_11_8_11_8_8_11_11_8_8_11_11_8_8_11_11_8_11_8_8_11_8_11_8_8_11_8_8_11_8_8_8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

$\begin{align*}\cos \frac{19 \pi}{12}\end{align*}$
::COS1912

The $\begin{align*}\cos a= \frac{5}{13}\end{align*}$ and $\begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}$ . Find:
::Cosa=513 和 32a<2。查找 :

$\begin{align*}\sin 2a\end{align*}$
::二、第2款a

$\begin{align*}\cos \frac{a}{2}\end{align*}$
::COsa2

$\begin{align*}\tan \frac{a}{2}\end{align*}$
::塔纳亚2color

$\begin{align*}\cos 2a\end{align*}$
::COs%2a COs=%2a

The $\begin{align*}\sin a=\frac{8}{11}\end{align*}$ and $\begin{align*}\frac{\pi}{2} \le a < \pi\end{align*}$ . Find:
::罪状=811和2QA 查找:

$\begin{align*}\tan 2a\end{align*}$
::丹丹 2a

$\begin{align*}\sin \frac{a}{2}\end{align*}$
::疾病2

$\begin{align*}\cos \frac{a}{2}\end{align*}$
::COsa2

$\begin{align*}\sin 2a\end{align*}$
::二、第2款a

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。

Section outline

General

使用双角和半角公式查找精确的三角值

Double Angle and Half Angle Formulas ::双角和半角公式

Double-Angle and Half-Angle Formulas ::双角和半角公式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Double Angle and Half Angle Formulas
::双角和半角公式

Double-Angle and Half-Angle Formulas
::双角和半角公式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)