8.8小结：解析三角

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• 选择节小结：解析三角

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  小结：解析三角

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  Chapter Summary
  ::章次摘要

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  In this chapter, we learned about:
  ::在本章中,我们了解到:

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  • Trigonometric Identities
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    • Reciprocal Identities  
      $$\begin{array}{r}\sin \theta =\frac{1}{\csc \theta }\cos \theta =\frac{1}{\sec \theta }\tan \theta =\frac{1}{\cot \theta }\\ \csc \theta =\frac{1}{\sin \theta }\sec \theta =\frac{1}{\cos \theta }\cot \theta =\frac{1}{\tan \theta }\end{array}$$
       
      ::= 1 csc = 1 sec = 1 sec = 1 sec = 1 sec = 1 csc = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = 1 sec = oc = 1 sec = = 1 sec = = 1 sec = = 1 = 1 sec = 1 = 1 sec = = 1 = 1 sec = = 1 = 1 sic = 1 = 1 sic = = 1 sec = = 1 = 1 si = 1 sic = * = = 1 sic = = = 1 si si si = * = = 1 = 1 = 1 seccc = = * = = = = 1 = 1 = 1 = 1 = 1 = 1 succc = = = = 1 = 1 = 1 = * = 1 = * = * = * = * = * = * = * = 1 = 1 = * = = = = = 1 = 1 = 1 = 1 = 1 = * = = = = * = = = = = 1 = * cccccccccccccccccc = * = * = = = = = = = = = = = = = = = = = * = = = = = = = = = = = = = = * = * = *
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    • Quotient Identities 
      $$\begin{array}{r}\tan \theta =\frac{\sin \theta }{\cos \theta }\cot \theta =\frac{\cos \theta }{\sin \theta }\end{array}$$
      
      ::引言名人 日 日 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪 罪
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    • Cofunction Identities 
      $$\begin{array}{r}\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta \sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta \cot \left(\frac{\pi }{2}-\theta \right)=\tan \theta \\ \sec \left(\frac{\pi }{2}-\theta \right)=\csc \theta \csc \left(\frac{\pi }{2}-\theta \right)=\sec \theta \tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta \end{array}$$
      
      ::身份 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪 =罪
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    • Even-Odd Trigonometric Identities 
      $$\begin{array}{rl}\cos (-\theta )& =\cos \theta \sec (-\theta )=\sec \theta \\ \sin (-\theta )& =-\sin \theta \csc (-\theta )=-\csc \theta \\ \tan (-\theta )& =-\tan \theta \cot (-\theta )=-\cot \theta \end{array}$$
      
      ::even-Od Trigologities cos = csc
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    • Pythagorean Trigonometric Identities  
      $$\begin{array}{r}{\sin }^{2}x+{\cos }^{2}x=1\\ 1+{\cot }^{2}x={\csc }^{2}x\\ {\tan }^{2}x+1={\sec }^{2}x\end{array}$$
       
      ::Pythagoren Trigonorian 三一特征 原罪 2 x + cos 2 x = 1 1 + Cot 2 x = csc 2 x tan 2 x + 1 = 秒 2 × x x x = 秒 2 x
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    • Difference Identities 
      $$\begin{array}{r}\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta \\ \sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\ \tan (\alpha -\beta )=\frac{\sin (\alpha -\beta )}{\cos (\alpha -\beta )}=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\end{array}$$
      
      ::= 罪 1
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    • Sum Identities 
      $$\begin{array}{r}\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\ \tan (\alpha +\beta )=\frac{\sin (\alpha +\beta )}{\cos (\alpha +\beta )}=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\end{array}$$
       
      ::=罪 =罪 =罪 =罪 =罪 =罪 =罪 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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    • Double Angle Identities  
      $$\begin{array}{r}\sin 2x=2\sin x\cos x\cos 2x={\cos }^{2}x-{\sin }^{2}x\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}\end{array}$$
       
      ::2 x = 2 sin * * x = 2 sin * * x cos * = 2 x cos * 2 x = cos 2 x = cos 2 * x x - 罪 2 x tan * 2 x = 2 tan * x 1 - tan 2 * * * * * = 2 = 2 tan * x 1 - tan * * * * * * = 2 x tan * * * = 2 x tan * * * * * * = 2 x tan * * * * * * * * = = 2 x tan * * * * * * * = = 2 x tan * * * * * * x = = 2 x than = 2 x tan * * * * * * x x x x x x x x x x x * * * * * * = = 2 x y = 2 x tan * * * x = = tan * * * * * * * * * * = * * * * * * * = * * * = = * = = = = = = = = = * = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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    • Half Angle Identities  
      $$\begin{array}{r}\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}\tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}\end{array}$$
       
      ::半角度辨别罪 x 2 = 1 = 1 = 2 2 x 2 = 1 + 2 + x 2 tan x 2 = 1 - x 1 + x x x x x x x x x x x x x x x x x
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    • Power Reducing Identities 
      $$\begin{array}{r}{\sin }^{2}x=\frac{1-\cos 2x}{2}{\cos }^{2}x=\frac{1+\cos 2x}{2}{\tan }^{2}x=\frac{1-\cos 2x}{1+\cos 2x}\end{array}$$
      
      ::减量功率 减少功率 2 = 1 = 1 = 2 x 2 = 2 2 = 2 x 2 = 2 x 2 x = 1 + CO = 2 x 2 x 2 tan 2 × x = 1 - 2 x 1 = = 2 x 1 = 1 = = 2 x 1 + CO = 2 x 2 x x = 2 x = = 1 = 2 = = = 2 x 1 = = = 2 x 1 = = = = 2 x 1 = = = 2 x 1 = = 2 x 1 = = = 2 x 1 = = 2 x 1 + + = 2 x 2 x x x x = 2 x = = 1 = 1 = = = = 2 x 2 x = 2 x = 2 x = 2 = = 2 = = 2 = 2 = 2 = = 2 = 2 = = = = = 1 = = = = = = 1 = 1 = = = = = = 1 = = = = = = = = = = 1 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
    
    ::* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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  • How to Solve   a Trigonometric Equation
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    • Solve for the variable as you would a linear equation. 
      ::以线性方程的方式解决变量 。
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    • Use inverse trigonometric functions to isolate the variable.
      ::使用逆三角函数分隔变量。
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    • Use Pythagorean Trigonometric, sum and difference, and other trigonometric identities to get a single trigonometric function.
      ::使用毕达哥里安三角测量、总和和差异 以及其他三角特征 来获得单一三角函数
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    • Use sum and difference, double angle, and half angle identities to get a single angle.
      ::使用和差异, 双角度, 和半角度身份来获得一个角度 。
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    • Use algebraic techniques, such as factoring or squaring, to further simplify the expressions.
      ::使用代数技术,如乘数或方位法,进一步简化表达式。
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    • Use the unit circle to find the initial solution or to determine values within the specified interval. There may be more than one value.
      ::使用单位圆查找初始解决方案或确定指定间隔内的值。可能有一个以上的值。
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    • Provide an appropriate set of solutions. 
      ::提供一套适当的解决办法。
    
    ::如何解析变量的三角方程式, 和线性方程式一样。 使用反三角方程式函数来孤立变量。 使用 Pytagoren Trigonorian Trigonology、 sum 和 difference 和其他三角方程式特性来获得单三角方程式函数。 使用和差异、 双角度 和半角度特性来获得单一角度 。 使用代数法技术, 如乘数或方程, 来进一步简化表达式 。 使用单位圆来找到初始解决方案或在指定间隔内确定值 。 可能有不止一个值 。 提供一套合适的解决方案 。

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  Review
  ::回顾

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  Try the following cumulative review problems to practice the concepts in this chapter:
  ::尝试下列累积审查问题来实践本章中的概念: