6.3 总数和差异

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  合计和差异

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  With your knowledge of special angles like the of $\begin{array}{r}{30}^{\circ }\end{array}$  and $\begin{array}{r}{45}^{\circ }\end{array}$ , you can find the sine and cosine of $\begin{array}{r}{15}^{\circ }\end{array}$ , the difference of $\begin{array}{r}{45}^{\circ }\end{array}$  and $\begin{array}{r}{30}^{\circ }\end{array}$ , and $\begin{array}{r}{75}^{\circ }\end{array}$ , the sum of $\begin{array}{r}{45}^{\circ }\end{array}$  and $\begin{array}{r}{30}^{\circ }\end{array}$ .  Using what you know about the unit circle and the sum and difference identities, how do you determine $\begin{array}{r}\sin {15}^{\circ }\end{array}$  and $\begin{array}{r}\sin {75}^{\circ }\end{array}$ ?
  ::以你对30和45等特殊角度的知识, 你能找到15的正弦和正弦, 45和30之间的差数, 和75的差数, 和45和30之间的差数。使用你对单位圆的了解, 和总和和和差异的身份, 您如何确定 15 和sin 75 ?

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  Sum and Difference Identities
  ::合计和差异

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  First look at the derivation of the cosine difference identity :
  ::首先看余弦差异特性的衍生:

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  $\begin{array}{r}\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{array}$
  :)=coscossinsinsin

  

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  Start by drawing two arbitrary angles $\begin{array}{r}\alpha \end{array}$  and $\begin{array}{r}\beta \end{array}$ .  In the image above $\begin{array}{r}\alpha \end{array}$  is the angle in red and $\begin{array}{r}\beta \end{array}$  is the angle in blue.  The difference $\begin{array}{r}\alpha -\beta \end{array}$  is noted in black as $\begin{array}{r}\theta \end{array}$ .  The reason why there are two pictures is because the image on the right has the same angle $\begin{array}{r}\theta \end{array}$  in a rotated position.  This will be useful to work with because the length of $\begin{array}{r}\overline{AB}\end{array}$  will be the same as the length of $\begin{array}{r}\overline{CD}\end{array}$ .
  ::开始绘制两个任意角度 α 和 。 在 α 上方的图像中, 以红色为角度, β 以蓝色为角度。 区别 \\\\ 以黑色表示为 \ 。 原因是右侧的图像在旋转位置上有着相同的角度 \ 。 这将有用, 因为 AB 的长度将与 CD 的长度相同 。

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  $$\begin{array}{rl}\overline{AB}& =\overline{CD}\\ \sqrt{(\cos \alpha -\cos \beta {)}^2+(\sin \alpha -\sin \beta {)}^2}& =\sqrt{(\cos \theta -1{)}^2+(\sin \theta -0{)}^2}\\ (\cos \alpha -\cos \beta {)}^2+(\sin \alpha -\sin \beta {)}^2& =(\cos \theta -1{)}^2+(\sin \theta -0{)}^2\end{array}$$

  
  ::AB2+(cos)2+(sinsin)2=(cos1)2+(sin0)(2(coscos)2+(sinsin)2=(cos1)2+(sin0)2

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  $$\begin{array}{rl}(\cos \alpha {)}^2-2\cos \alpha \cos \beta +(\cos \beta {)}^2+(\sin \alpha {)}^2-2\sin \alpha \sin \beta +(\sin \beta {)}^2& =(\cos \theta -1{)}^2+(\sin \theta {)}^2\end{array}$$

  
  :cos)2 -2cos(cos)2+(sin)2-2-2sin(sin)2=(cos1)2+(sin)2

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  $$\begin{array}{rl}2-2\cos \alpha \cos \beta -2\sin \alpha \sin \beta & =(\cos \theta {)}^2-2\cos \theta +1+(\sin \theta {)}^2\\ 2-2\cos \alpha \cos \beta -2\sin \alpha \sin \beta & =1-2\cos \theta +1\\ -2\cos \alpha \cos \beta -2\sin \alpha \sin \beta & =-2\cos \theta \\ \cos \alpha \cos \beta +\sin \alpha \sin \beta & =\cos \theta \\ & =\cos (\alpha -\beta )\end{array}$$

  
  ::2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -1 -1 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2 -2

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  You can use this identity to prove the cosine of a sum identity . First, start with the cosine of a difference and make a substitution.  Then use the odd-even identity.
  ::您可以使用此身份来证明一个相加身份的余弦。 首先, 从一个差异的余弦开始, 然后进行替换。 然后使用奇数身份 。

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  $\begin{array}{r}\cos \alpha \cos \beta +\sin \alpha \sin \beta =\cos (\alpha -\beta )\end{array}$
  : )

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  Let $\begin{array}{r}\gamma =-\beta \end{array}$
  ::让我们... ... Let let ___________________________________________________________________________________________________________________________________________

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  $$\begin{array}{rl}\cos \alpha \cos (-\gamma )+\sin \alpha \sin (-\gamma )& =\cos (\alpha +\gamma )\\ \cos \alpha \cos \gamma -\sin \alpha \sin \gamma & =\cos (\alpha +\gamma )\end{array}$$

  
  ::

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  The proofs for sine and tangent are left to the videos and examples.  They are listed here for your reference.  Cotangent , secant and cosecant are excluded because you can use to get those once you have sine, cosine and tangent. 
  ::用于正切和正切的证明留待视频和示例中。 它们在此列出供您参考。 共切、 分离和余切被排除, 因为您可以使用来获取那些有正弦、 顺弦和正切的证明 。

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  Summary
  ::摘要

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  • $\begin{array}{r}\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{array}$
    :)=coscossinsinsin
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  • $\begin{array}{r}\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \end{array}$
    :)=sincoscoscossinsin
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  • $\begin{array}{r}\tan (\alpha \pm \beta )=\frac{\sin (\alpha \pm \beta )}{\cos (\alpha \pm \beta )}=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\end{array}$
    ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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  The order of the plus or minus signs is important because for cosine of a sum, the negative sign is used on the other side of the identity.  This is the opposite of sine of a sum, where a positive sign is used on the other side of the identity. 
  ::增加或减号的顺序很重要, 因为对于一个相余的数, 负号在身份的另一方使用。 这是一个正数的正数的反面, 在身份的另一方使用正数的正数 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to evaluate $\begin{array}{r}\sin {15}^{\circ }\end{array}$  and $\begin{array}{r}\sin {75}^{\circ }\end{array}$  exactly without a calculator. To do this you need to use the sine of a difference and sine of a sum.
  ::早些时候,有人要求你完全在没有计算器的情况下评估 sin 15 和 sin 75 。 要做到这一点,你需要使用差异的正弦和正弦。

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  $$\begin{array}{rl}\sin ({45}^{\circ }-{30}^{\circ })& =\sin {45}^{\circ }\cos {30}^{\circ }-\cos {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot \frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\\ \sin ({45}^{\circ }+{30}^{\circ })& =\sin {45}^{\circ }\cos {30}^{\circ }+\cos {45}^{\circ }\sin {30}^{\circ }=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\end{array}$$

  
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  Example 2
  ::例2

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  Find the exact value of $\begin{array}{r}\tan {15}^{\circ }\end{array}$  without using a calculator.
  ::在不使用计算器的情况下查找 tan% 15 的准确值 。

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    $\begin{array}{r}\tan {15}^{\circ }=\tan ({45}^{\circ }-{30}^{\circ })=\frac{\tan {45}^{\circ }-\tan {30}^{\circ }}{1+\tan {45}^{\circ }\tan {30}^{\circ }}=\frac{1-\frac{\sqrt{3}}{3}}{1+1\cdot \frac{\sqrt{3}}{3}}=\frac{3-\sqrt{3}}{3+\sqrt{3}}\end{array}$
  
  ::~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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  A final solution will not have a radical in the denominator.  In this case multiplying through by the conjugate of the denominator will eliminate the radical.
  ::最终解决方案在分母上不会有激进的分母。 在这种情况下,通过分母的组合而倍增的分母将消除激进。

  $$\begin{array}{rl}& =\frac{(3-\sqrt{3})\cdot (3-\sqrt{3})}{(3+\sqrt{3})\cdot (3-\sqrt{3})}\\ & =\frac{(3-\sqrt{3}{)}^2}{9-3}\\ & =\frac{(3-\sqrt{3}{)}^2}{6}\end{array}$$

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  Example 3
  ::例3

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  Evaluate the expression exactly without using a calculator.
  ::准确评价表达式而不使用计算器 。

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  $\begin{array}{r}\cos {50}^{\circ }\cos 5^{\circ }+\sin {50}^{\circ }\sin 5^{\circ }\end{array}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么?

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  Once you know the general form of the sum and difference identities then you will recognize this as cosine of a difference.
  ::一旦知道总和和差异身份的一般形式 你就会承认这是差异的余弦

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  $\begin{array}{r}\cos {50}^{\circ }\cos 5^{\circ }+\sin {50}^{\circ }\sin 5^{\circ }=\cos ({50}^{\circ }-5^{\circ })=\cos {45}^{\circ }=\frac{\sqrt{2}}{2}\end{array}$
  ::====================================================================================================================================== ================================================================================================================================

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  Example 4  
  ::例4

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  Use a sum or difference identity to find an exact value of $\begin{array}{r}\cot \left(\frac{5\pi }{12}\right)\end{array}$ .
  ::使用一个总和或差异身份来查找 cot( 512) 的准确值 。

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  Start with the definition of cotangent as the inverse of tangent. 
  ::以正切值定义为反正切值开始 。

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  $$\begin{array}{rl}\cot \left(\frac{5\pi }{12}\right)& =\frac{1}{\tan \left(\frac{5\pi }{12}\right)}\\ & =\frac{1}{\tan \left(\frac{9\pi }{12}-\frac{4\pi }{12}\right)}\\ & =\frac{1}{\tan ({135}^{\circ }-{60}^{\circ })}\\ & =\frac{1+\tan {135}^{\circ }\tan {60}^{\circ }}{\tan {135}^{\circ }-\tan {60}^{\circ }}\\ & =\frac{1+(-1)\cdot \sqrt{3}}{(-1)-\sqrt{3}}\\ & =\frac{(1-\sqrt{3})}{(-1-\sqrt{3})}\\ & =\frac{(1-\sqrt{3}{)}^2}{(-1+\sqrt{3})\cdot (1-\sqrt{3})}\\ & =\frac{(1-\sqrt{3}{)}^2}{-(1-3)}\\ & =\frac{(1-\sqrt{3}{)}^2}{2}\end{array}$$

  
  :512) = 1tan(512) = 1tan(912-412) = 1tan(13560) = 1+tan(13560) (1) (6) (61) +(-1) 3(1) - 3=(1-3) (1-3) (2)(1-3) (1-3) □(3) □(3) □(1-3) =(1-3) 2-(1-3) =(1-3) =(1-3) 3) 22

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  Example 5
  ::例5

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  Prove the following identity:
  ::证明以下身份:

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  $\begin{array}{r}\frac{\sin (x-y)}{\sin (x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}\end{array}$
  :x-y)sin(x+y)=tanx-tanytanx+tany

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  Here are the steps:
  ::以下是步骤:

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  $$\begin{array}{rl}\frac{\sin (x-y)}{\sin (x+y)}& =\frac{\tan x-\tan y}{\tan x+\tan y}\\ \frac{\sin x\cos y-\cos x\sin y}{\sin x\cos y+\cos x\sin y}& =\\ \frac{\sin x\cos y-\cos x\sin y}{\sin x\cos y+\cos x\sin y}\cdot \frac{\left(\frac{1}{\cos x\cdot \cos y}\right)}{\left(\frac{1}{\cos x\cdot \cos y}\right)}& =\\ \frac{\left(\frac{\sin x\cancel{\cos y}}{\cos x\cdot \cancel{\cos y}}\right)-\left(\frac{\cancel{\cos x}\sin y}{\cancel{\cos x}\cdot \cos y}\right)}{\left(\frac{\sin x\cancel{\cos y}}{\cos x\cdot \cancel{\cos y}}\right)+\left(\frac{\cancel{\cos x}\sin y}{\cancel{\cos x}\cdot \cos y}\right)}& =\\ \frac{\tan x-\tan y}{\tan x+\tan y}& =\end{array}$$

  
  :x+Y) = tanxxx x x x x x x x x x y= x x x x x xx xxx }+ xxxx x xx (1cosxxxx x x x x s y) = (cosxx sinx x x x x }}) + (cosxx xxx x x x }= (cusxxxxxxxxx }=

  Summary

  播放段落 For arbitrary angles $\begin{array}{r}\alpha \end{array}$  and $\begin{array}{r}\beta ,\end{array}$  the sum and difference identities are: 播放段落 $\begin{array}{r}\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{array}$   :)=coscossinsinsin 播放段落 $\begin{array}{r}\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \end{array}$   :)=sincoscoscossinsin 播放段落 $\begin{array}{r}\tan (\alpha \pm \beta )=\frac{\sin (\alpha \pm \beta )}{\cos (\alpha \pm \beta )}=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\end{array}$   ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ::对于任意角度的α和β,其总和和区别身份是:cos()=coscoscossinsinsinsinsin()=sincoscoscossin tan()=tan()1}()sinsinsin}tan()=sin()sin}()=sin(){cos){()()=tan{tan}1}}}()=sincosççosinsin}sinsin tan(){}}=sin()=sin{()sin}()=sin(){(){()(()()(()()()()()()()())(()(()(}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}*}}}}}}}}}**************************************

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  Review
  ::回顾

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  Find the exact value for each expression by using a sum or difference identity.
  ::使用总和或差异身份来查找每个表达式的准确值。

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  1. $\begin{array}{r}\cos {75}^{\circ }\end{array}$
  ::1. 承担第75-75-______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  2.  $\begin{array}{r}\cos {105}^{\circ }\end{array}$
  ::2. COS105_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  3.  $\begin{array}{r}\cos {165}^{\circ }\end{array}$
  ::3,cos165

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  4.  $\begin{array}{r}\sin {105}^{\circ }\end{array}$
  ::4 sin 105______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  5.  $\begin{array}{r}\sec {105}^{\circ }\end{array}$
  ::5 秒 105

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  6.  $\begin{array}{r}\tan {75}^{\circ }\end{array}$
  ::6. 锡 75_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  7. Prove the sine of a sum identity.
  ::7. 证明真实身份的必要条件。

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  8. Prove the tangent of a sum identity.
  ::8. 证明一个总特征的相切性。

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  9. Prove the tangent of a difference identity.
  ::9. 证明差异认同的相切性。

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  10. Evaluate without a calculator: $\begin{array}{r}\cos {50}^{\circ }\cos {10}^{\circ }-\sin {50}^{\circ }\sin {10}^{\circ }\end{array}$ .
  ::10. 无计算器的评价:cos50cos10sin50sin10。

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  11. Evaluate without a calculator: $\begin{array}{r}\sin {35}^{\circ }\cos 5^{\circ }-\cos {35}^{\circ }\sin 5^{\circ }\end{array}$ .
  ::11. 评估没有计算器:sin_35cos_5cos_35sin_55。

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  12. Evaluate without a calculator: $\begin{array}{r}\sin {55}^{\circ }\cos 5^{\circ }+\cos {55}^{\circ }\sin 5^{\circ }\end{array}$ .
  ::12. 评估没有计算法:sin55cos55cos55sin555。

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  13. If $\begin{array}{r}\cos \alpha \cos \beta =\sin \alpha \sin \beta \end{array}$ , then what does $\begin{array}{r}\cos (\alpha +\beta )\end{array}$  equal?
  ::13. 如果cosççosísinsin,那么cos()等于什么?

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  14. Prove that $\begin{array}{r}\tan \left(x+\frac{\pi }{4}\right)=\frac{1+\tan x}{1-\tan x}\end{array}$ .
  ::14. 证明Tan(x4)=1+tanx1-tanx。

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  15. Prove that $\begin{array}{r}\sin (x+\pi )=-\sin x\end{array}$ .
  ::15. 证明这一罪行。

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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 。