Section: 双重、一半和减少功率 | 6.4 双重、一半和减少功率

Section outline

These identities are significantly more involved and less intuitive than previous identities. By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase. Each identity in this concept is named aptly. Double angles work on finding $\begin{aligned} \sin 80^{\circ} \end{aligned}$ if you already know $\begin{aligned} \sin 40^{\circ} \end{aligned}$ . Half angles allow you to find $\begin{aligned} \sin 15^{\circ} \end{aligned}$ if you already know $\begin{aligned} \sin 30^{\circ} \end{aligned}$ . Power reducing identities allow you to find $\begin{aligned} \sin^{2} 15^{\circ} \end{aligned}$ if you know the of $\begin{aligned} 30^{\circ} \end{aligned}$ .
::这些身份比先前的身份更具有参与性,更不直观。通过练习和操作这些先进的身份, 您的工具箱和流利的替换和验证将会增加。这一概念中的每一个身份都会被适当命名。如果您已经知道 $40 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

What is $\begin{aligned} \sin^{2} 15^{\circ} \end{aligned}$ ?
::什么是罪215?

Double Angle, Half Angle, and Power Reducing Identities
::双角、半角、减少功率

Double Angle Identities
::双角度特征

The double angle identities are proved by applying the sum and difference identities. They are left as review problems. These are the double angle identities.
::双角身份通过使用总和和差异身份来证明,它们被留作审查问题,即双角身份。

$\begin{aligned} \sin 2 x = 2 \sin x \cos x \end{aligned}$
::sin *% 2x=2sin *xcos *xxx

$\begin{aligned} \cos 2 x = \cos^{2} x - \sin^{2} x \end{aligned}$
::cos% 2x=cos2_x-sin2_sin2_xx

$\begin{aligned} \tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x} \end{aligned}$
::tan2x=2tanx1-tan2xx

Half Angle Identities
::半角特征

The half angle identities are a rewritten version of the power reducing identities. The proofs are left as review problems.
::半角身份是重写能力降低身份的版本。证据被留作审查问题。

$\begin{aligned} \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} \end{aligned}$
::______222222222222222222222

$\begin{aligned} \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \end{aligned}$
::cosx21+cosx2

$\begin{aligned} \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \end{aligned}$
::tanx2\\\\\\\ \ \ \ \ \ \ \ \ \ x1 \ + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Power Reducing Identities
::减少功率

The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers. The proofs are left as examples and review problems.
::降低身份的功率使您可以写出以较小功率平方表示的三角函数。这些证据作为实例和审查问题。

$\begin{aligned} \sin^{2} x = \frac{1 - \cos 2 x}{2} \end{aligned}$
::sin2 =2 =1 - cos=2x2 =1 - cos=2x2 =2x2

$\begin{aligned} \cos^{2} x = \frac{1 + \cos 2 x}{2} \end{aligned}$
::cos2% x=1+cos=2x2 =1+cos=2x2

$\begin{aligned} \tan^{2} x = \frac{1 - \cos 2 x}{1 + \cos 2 x} \end{aligned}$
::tan2% x=1 - cos=2x1+cos=2x

Power reducing identities are most useful when you are asked to rewrite expressions such as $\begin{aligned} \sin^{4} x \end{aligned}$ as an expression without powers greater than one. While $\begin{aligned} \sin x \cdot \sin x \cdot \sin x \cdot \sin x \end{aligned}$ does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.
::当您被要求重写 sin4 {x} 等表达式时, 功率降低身份是非常有用的。 sin}}sin{xsin}xxsin}xsin}xsin}xsin}xsin}x}}xsin}x 这样的表达式在技术上可以在必要时简化该表达式, 但您应该尝试将术语加起来, 而不是一起乘。

\begin{aligned} \sin^{4} x & = (\sin^{2} x)^{2} \\ = {(\frac{1 - \cos 2 x}{2})}^{2} \\ = \frac{1 - 2 \cos 2 x + \cos^{2} 2 x}{4} \\ = \frac{1}{4} (1 - 2 \cos 2 x + \frac{1 + \cos 4 x}{2}) \end{aligned}

::sin4x = (sin2x) 2= (1 - cos2x2) 2= 1 - 2cos2x+cos2 2x4=14(1 - 2cos2x+1+cos4x2)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find $\begin{aligned} \sin^{2} 15^{\circ} \end{aligned}$ . In order to fully identify $\begin{aligned} \sin^{2} 15^{\circ} \end{aligned}$ you need to use the power reducing formula.
::早些时候, 您被要求找到 sin2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

\begin{aligned} \sin^{2} x & = \frac{1 - \cos 2 x}{2} \\ \sin^{2} 15^{\circ} & = \frac{1 - \cos 30^{\circ}}{2} = \frac{1}{2} - \frac{\sqrt{3}}{4} \\ = \frac{2 - \sqrt{3}}{4} \end{aligned}

::=2=1 -cos=2x2sin2}=15=1 -cos=30=2=12 -34=2 -34

Example 2
::例2

Write the following expression with only $\begin{aligned} \sin x \end{aligned}$ and $\begin{aligned} \cos x \end{aligned}$ : $\begin{aligned} \sin 2 x + \cos 3 x \end{aligned}$ .
::仅用 sinx 和 cosx 写下以下表达式: sin2x+cos3x 。

\begin{aligned} \sin 2 x + \cos 3 x & = 2 \sin x \cos x + \cos (2 x + x) \\ = 2 \sin x \cos x + \cos 2 x \cos x - \sin 2 x \sin x \\ = 2 \sin x \cos x + (\cos^{2} x - \sin^{2} x) \cos x - (2 \sin x \cos x) \sin x \\ = 2 \sin x \cos x + \cos^{3} x - \sin^{2} x \cos x - 2 \sin^{2} x \cos x \\ = 2 \sin x \cos x + \cos^{3} x - 3 \sin^{2} x \cos x \end{aligned}

::2x 2xxxxxxxxxxxxxxxxxxxxxxxxxxxx(cos2xx2xxxxxxxxxxxx(cos2xxxxxxxxxxxxxxxxxx(cos2xxxxxxxxxxxx)x(2sin*xxxxxxxxxxxxxxxxxxxxxx%2xxxxxxinxxxxxxxxxxx(xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx2xxxxx2xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Example 3
::例3

Use half angles to find an exact value of $\begin{aligned} \tan {22.5}^{\circ} \end{aligned}$ without using a calculator.
::使用半角度在不使用计算器的情况下查找 tan\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\没有计算。

$\begin{aligned} \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \end{aligned}$
::tanx2\\\\\\\ \ \ \ \ \ \ \ \ \ x1 \ + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

$\begin{aligned} \tan {22.5}^{\circ} & = \tan \frac{45^{\circ}}{2} = \pm \sqrt{\frac{1 - \cos 45^{\circ}}{1 + \cos 45^{\circ}}} = \pm \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} = \pm \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{\frac{2}{2} + \frac{\sqrt{2}}{2}}} = \pm \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} \\ = \pm \sqrt{\frac{(2 - \sqrt{2})^{2}}{2}} \end{aligned}$

::221212111114122222222222222222222222222222222222222222222222222222222222222222

Example 4
::例4

Prove the power reducing identity for sine.
::证明无时不刻刻能降低身份的能力。

$\begin{aligned} \sin^{2} x = \frac{1 - \cos 2 x}{2} \end{aligned}$
::sin2 =2 =1 - cos=2x2 =1 - cos=2x2 =2x2

Using the double angle identity for cosine:
::使用余弦的双角身份 :

\begin{aligned} \cos 2 x & = \cos^{2} x - \sin^{2} x \\ \cos 2 x & = (1 - \sin^{2} x) - \sin^{2} x \\ \cos 2 x & = 1 - 2 \sin^{2} x \end{aligned}

::COs% 2x=cos2_x-sin2_xxcos%2x=(1-sin2_x)-sin2_xxxxx%2x=1-2sin2xxx

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.
::此表达式相当于双角身份的表达式,通常被视为另一种形式。

Example 5
::例5

Simplify the following identity: $\begin{aligned} \sin^{4} x - \cos^{4} x \end{aligned}$ .
::简化以下身份:sin4x-cos4x。

Here are the steps:
::以下是步骤:

\begin{aligned} \sin^{4} x - \cos^{4} x & = (\sin^{2} x - \cos^{2} x) (\sin^{2} x + \cos^{2} x) \\ = - (\cos^{2} x - \sin^{2} x) \\ = - \cos 2 x \end{aligned}

cos2)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Summary

Double Angle Identities:

$\begin{aligned} \sin 2 x = 2 \sin x \cos x \\ \cos 2 x = \cos^{2} x - \sin^{2} x \\ \tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x} \end{aligned}$

::双角度 : sin @ 2x=2sin @xcos @xcos=2x=cos2}%x-sin2}%xtan}%2x=2tan}x1_tan2}xx=2tan}x1_tan2}x

Half Angle Identities:

$\begin{aligned} \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} \\ \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \\ \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \end{aligned}$

::半角度特性 : sinx21- cosx2cosx2x21+cos=x2tan}x21 -cos=x1+cos*x

Power Reducing Identities:

$\begin{aligned} \sin^{2} x = \frac{1 - \cos 2 x}{2} \\ \cos^{2} x = \frac{1 + \cos 2 x}{2} \\ \tan^{2} x = \frac{1 - \cos 2 x}{1 + \cos 2 x} \end{aligned}$

::降低功率的功率 : sin2\\\ x=1 - cos=2x2cos2\\ xx=1+cos=2x2tan2\\\ xx=1 -cos=2x1+cos%2x1+cos2x2xxx

Review
::回顾

Prove the following identities.
::证明以下身份。

1. $\begin{aligned} \sin 2 x = 2 \sin x \cos x \end{aligned}$
::一罪2x=2sinxcosxxxxx

2. $\begin{aligned} \cos 2 x = \cos^{2} x - \sin^{2} x \end{aligned}$
::2. COs2x=cos22x-sin2xx

3. $\begin{aligned} \tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x} \end{aligned}$
::3. tan_2x=2tan_2x1-tan2xx

4. $\begin{aligned} \cos^{2} x = \frac{1 + \cos 2 x}{2} \end{aligned}$
::4. cos2x=1+cos=2x2

5. $\begin{aligned} \tan^{2} x = \frac{1 - \cos 2 x}{1 + \cos 2 x} \end{aligned}$
::5. tan2x=1-cos=2x1+cos=2x

6. $\begin{aligned} \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} \end{aligned}$
::6. sin *x2 *1 -cos *x2

7. $\begin{aligned} \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \end{aligned}$
::7. cosx21+cosx2

8. $\begin{aligned} \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \end{aligned}$
::8. tanx21-cosx1+cosx

9. $\begin{aligned} \csc 2 x = \frac{1}{2} \csc x \sec x \end{aligned}$
::9. csc=2x=12csc=xsec=xsec=xx

10. $\begin{aligned} \cot 2 x = \frac{\cot^{2} x - 1}{2 \cot x} \end{aligned}$
::10. comt2x=cot2}2x-12cotx

Find the value of each expression using half angle identities.
::使用半角度身份查找每个表达式的值。

11. $\begin{aligned} \tan 15^{\circ} \end{aligned}$
::11 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

12. $\begin{aligned} \tan {22.5}^{\circ} \end{aligned}$
::12 tan22.5____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

13. $\begin{aligned} \sec {22.5}^{\circ} \end{aligned}$
::13秒22.5

14. Show that $\begin{aligned} \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \end{aligned}$ .
::14. 显示 tan_x2=1 -cos_xsin_xxxxx。

15. Using your knowledge from the answer to question 14, show that $\begin{aligned} \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \end{aligned}$ .
::15. 利用您对问题14的答复中的知识, 显示 tan_x2=sin_x1+cos_x。

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

Double Angle, Half Angle, and Power Reducing Identities ::双角、 半角、 减少功率

Double Angle Identities ::双角度特征

Half Angle Identities ::半角特征

Power Reducing Identities ::减少功率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)