Course: 3.7 相平和和差异公式

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Sine 平衡和差异公式
You've gotten quite good at knowing the values of trig functions. So much so that you and your friends play a game before class everyday to see who can get the most trig functions of different angles correct. However, your friend Jane keeps getting the trig functions of more angles right. You're amazed by her memory, until she smiles one day and tells you that she's been fooling you all this time.
::你非常了解三角函数的价值。如此之多, 你和你朋友每天都在上课前玩一个游戏, 看看谁可以得到不同角度的最三角函数。然而, 你朋友简总是把更多角度的三角函数弄对了。你对她的记忆感到惊讶, 直到有一天她笑了, 告诉你她一直在欺骗你。

"What you do you mean?" you say.
::"你是什么意思?" 你说

"I have a trick that lets me calculate more functions in my mind by breaking them down into sums of angles." she replies.
::"我有一个把戏,让我计算出我脑中更多的函数, 把它们分成许多角度。"她回答说。

You're really surprised by this. And all this time you thought she just had an amazing memory!
::一直以来你都觉得她记忆犹新!

"Here, let me show you," she says. She takes a piece of paper out and writes down:
::她说,她拿出一张纸,写下:

$\begin{align*}\sin \frac{7\pi}{12}\end{align*}$
::第712条

"This looks like an unusual value to remember for a trig function. So I have a special rule that helps me to evaluate it by breaking it into a sum of different numbers."
::“这看起来是一个不寻常的价值,值得记住一个三角函数。所以我有一条特殊规则, 帮助我通过把它破解成不同数字的总和来评估它。”

Sine Sum and Difference Formulas
::Sine 平衡和差异公式

Our goal here is to figure out a formula that lets you break down a the sine of a sum of two angles (or a difference of two angles) into a simpler formula that lets you use the sine of only one argument in each term.
::我们的目标是找出一个公式,让你将两个角度(或两个角度的差别)之和的正弦分解成一个简单的公式,使每个术语只使用一个参数的正弦。

To find
::要查找

$\begin{align*}\sin(a + b)\end{align*}$
:a+b)

:

$\begin{aligned} \sin (a + b) & = \cos [\frac{π}{2} - (a + b)] & Set θ = a + b \\ = \cos [(\frac{π}{2} - a) - b] & Distribute the negative \\ = \cos (\frac{π}{2} - a) \cos b + \sin (\frac{π}{2} - a) \sin b & Difference Formula for cosines \\ = \sin a \cos b + \cos a \sin b & Co-function Identities \end{aligned}$
$\begin{align*}\sin(a + b) & = \cos \left [\frac{\pi}{2} - (a+b) \right ] && \text{Set}\ \theta = a + b\\ & = \cos \left [\left (\frac{\pi}{2} - a \right ) - b \right ]&& \text{Distribute the negative} \\ & = \cos \left (\frac{\pi}{2} - a \right ) \cos b + \sin \left (\frac{\pi}{2} - a \right ) \sin b && \text{Difference Formula for cosines} \\ & = \sin a \cos b + \cos a \sin b && \text{Co-function Identities}\end{align*}$
::=============================================================================================================================(a))===b+==============================b==================================================b++++++++===================================================================================================================================================================================================================================================

In conclusion, $\begin{align*}\sin(a + b) = \sin a \cos b + \cos a \sin b\end{align*}$ , which is the sum formula for sine.
::总而言之,sin(a+b)=sinacosb+cosçasinb,这是正弦之和公式。

To obtain the identity for $\begin{align*}\sin(a - b)\end{align*}$ :
:a-b) 获取罪证的身份:

$\begin{aligned} \sin (a - b) & = \sin [a + (- b)] \\ = \sin a \cos (- b) + \cos a \sin (- b) & Use the sine sum formula \\ \sin (a - b) & = \sin a \cos b - \cos a \sin b & Use \cos (- b) = \cos b, and \sin (- b) = - \sin b \end{aligned}$
$\begin{align*}\sin(a - b) & = \sin[a + (-b)] \\ & = \sin a \cos(-b) + \cos a \sin (-b) && \text{Use the sine sum formula} \\ \sin(a - b) & = \sin a \cos b - \cos a \sin b && \text{Use}\ \cos(-b) = \cos b,\ \text{and}\ \sin(-b) = -\sin b\end{align*}$
:a-b) =sin[a+(-b)] =sinacos(b) +cosasin(b) 使用正数公式(a-b) =sinacosb- cosasin(b) uses cos(b) =cosb, 和sin(b) 和sin(b) sinb

In conclusion, $\begin{align*}\sin(a - b) = \sin a \cos b - \cos a \sin b\end{align*}$ , so, this is the difference formula for sine.
::总而言之,sin(a-b)=sinacosb-cosasinb,所以,这是正弦的差别公式。

Using the Sine Sum and Difference Formula
::使用 Sine 和差异公式

1. Find the exact value of $\begin{align*}\sin \frac{5 \pi}{12}\end{align*}$
::1. 查明罪的确切价值512

Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.
::回顾有多个角度可以增减到相等的角度。选择您觉得更满意的公式。

$\begin{aligned} \sin \frac{5 π}{12} & = \sin (\frac{3 π}{12} + \frac{2 π}{12}) \\ = \sin \frac{3 π}{12} \cos \frac{2 π}{12} + \cos \frac{3 π}{12} \sin \frac{2 π}{12} \\ \sin \frac{5 π}{12} & = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} \\ = \frac{\sqrt{6} + \sqrt{2}}{4} \end{aligned}$
$\begin{align*}\sin \frac{5 \pi}{12} & = \sin \left (\frac{3 \pi}{12} + \frac{2 \pi}{12} \right ) \\ & = \sin \frac{3 \pi}{12} \cos \frac{2 \pi}{12} + \cos \frac{3 \pi}{12} \sin \frac{2 \pi}{12} \\ \sin \frac{5 \pi}{12} & = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} \\ & = \frac{\sqrt{6} + \sqrt{2}}{4}\end{align*}$
::512=sin(312+212) =sin}312cos212+cos312sin212sin=512=22×32+22x12=6+24

2. Given $\begin{align*}\sin \alpha = \frac{12}{13}\end{align*}$ , where $\begin{align*}\alpha\end{align*}$ is in Quadrant II, and $\begin{align*}\sin \beta = \frac{3}{5}\end{align*}$ , where $\begin{align*}\beta\end{align*}$ is in Quadrant I, find the exact value of $\begin{align*}\sin(\alpha + \beta)\end{align*}$ .
::2. 鉴于A在Quadrant II中的罪 1213, 和e在Quadrant I中的罪 35, 找出罪的确切价值。

To find the exact value of $\begin{align*}\sin(\alpha + \beta)\end{align*}$ , here we use $\begin{align*}\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\end{align*}$ . The values of $\begin{align*}\sin \alpha\end{align*}$ and $\begin{align*}\sin \beta\end{align*}$ are known, however the values of $\begin{align*}\cos \alpha\end{align*}$ and $\begin{align*}\cos \beta\end{align*}$ need to be found.
::为了找到罪的确切价值, 我们在这里使用罪罪 =sin cos) cos sin 。罪( 罪) 和罪( 罪) 的价值观是众所周知的, 但是需要找到 cos 和 cos 的价值观。

Use $\begin{align*}\sin^2 \alpha + \cos^2 \alpha = 1\end{align*}$ , to find the values of each of the missing cosine values.
::使用 sin2cos21 查找每个缺失的余弦值的值。

For $\begin{align*}\cos a : \sin^2 \alpha + \cos^2 \alpha = 1\end{align*}$ , substituting $\begin{align*}\sin \alpha = \frac{12}{13}\end{align*}$ transforms to $\begin{align*}\left (\frac{12}{13} \right )^2 + \cos^2 \alpha = \frac{144}{169} + \cos^2 \alpha = 1\end{align*}$ or $\begin{align*}\cos^2 \alpha = \frac{25}{169} \cos \alpha = \pm \frac{5}{13}\end{align*}$ , however, since $\begin{align*}\alpha\end{align*}$ is in Quadrant II, the cosine is negative, $\begin{align*}\cos \alpha = - \frac{5}{13}\end{align*}$ .
::对于cosáa:sin2cos21, 取代sin21213变换为(12132+cos2144169+cos21或cos225169cos513),

For $\begin{align*}\cos \beta\end{align*}$ use $\begin{align*}\sin^2\beta + \cos^2\beta = 1\end{align*}$ and substitute $\begin{align*}\sin \beta = \frac{3}{5}, \left (\frac{3}{5} \right )^2 + \cos^2 \beta = \frac{9}{25} + \cos^2 \beta = 1\end{align*}$ or $\begin{align*}\cos^2 \beta = \frac{16}{25}\end{align*}$ and $\begin{align*}\cos \beta = \pm \frac{4}{5}\end{align*}$ and since $\begin{align*}\beta\end{align*}$ is in Quadrant I, $\begin{align*}\cos \beta = \frac{4}{5}\end{align*}$
::使用 sin2cos21 替代 sin35, (35) 2+cos2925+cos21 或cos21625 和cos%45 并自 β 位于Quadrant I, cos45

Now the sum formula for the sine of two angles can be found:
::现在可以找到两个角度的正弦值的和公式 :

$\begin{aligned} \sin (α + β) & = \frac{12}{13} \times \frac{4}{5} + (- \frac{5}{13}) \times \frac{3}{5} or \frac{48}{65} - \frac{15}{65} \\ \sin (α + β) & = \frac{33}{65} \end{aligned}$
$\begin{align*}\sin (\alpha + \beta)& = \frac{12}{13} \times \frac{4}{5} + \left (- \frac{5}{13} \right ) \times \frac{3}{5}\ \text{or}\ \frac{48}{65} - \frac{15}{65} \\ \sin (\alpha + \beta)& = \frac{33}{65}\end{align*}$
:) = 1213×45+(- 513) ×35 或 4865- 1565sin () = 3365

3. Find the exact value of $\begin{align*}\sin 15^\circ\end{align*}$
::3. 找出罪的确切价值 . . . . . . . .

Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.
::回顾有多个角度可以增减到相等的角度。选择您觉得更满意的公式。

$\begin{aligned} \sin 15^{\circ} & = \sin (45^{\circ} - 30^{\circ}) \\ = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \\ \sin 15^{\circ} & = (.707) \times (.866) + (.707) \times (.5) \\ = (.612262) \times (.3535) \\ = .2164 \end{aligned}$
$\begin{align*}\sin 15^\circ & = \sin \left (45^\circ - 30^\circ \right ) \\ & = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \\ \sin 15^\circ & = (.707) \times (.866) + (.707) \times (.5) \\ & = (.612262) \times (.3535) \\ &= .2164\end{align*}$
::== 612262x(3535)=2 164=2164。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about using the sine sum formula .
::早些时候,有人给了你一个使用正数公式的问题。

With the sine sum formula, you can break the sine into easier to calculate quantities:
::使用正弦数公式,您可以将正弦数折成更容易计算数量:

$\begin{aligned} \sin \frac{7 π}{12} & = \sin (\frac{4 π}{12} + \frac{3 π}{12}) \\ = \sin (\frac{π}{3} + \frac{π}{4}) \\ = \sin (\frac{π}{3}) \cos (\frac{π}{4}) + c o s (\frac{π}{3}) s i n (\frac{π}{4}) \\ = (\frac{\sqrt{3}}{2}) (\frac{\sqrt{2}}{2}) + (\frac{1}{2}) (\frac{\sqrt{2}}{2}) \\ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \\ = \frac{\sqrt{6} + \sqrt{2}}{4} \end{aligned}$
$\begin{align*} \sin \frac{7\pi}{12} &= \sin \left( \frac{4\pi}{12} + \frac{3\pi}{12}\right)\\ &=\sin \left( \frac{\pi}{3} + \frac{\pi}{4}\right)\\ &=\sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) + cos(\frac{\pi}{3})sin(\frac{\pi}{4})\\ &=\left(\frac{\sqrt{3}}{2} \right) \left(\frac{\sqrt{2}}{2} \right) + \left(\frac{1}{2} \right) \left(\frac{\sqrt{2}}{2} \right)\\ &=\frac{\sqrt{6}}{4} +\frac{\sqrt{2}}{4}\\ &=\frac{\sqrt{6}+ \sqrt{2}}{4}\\ \end{align*}$
::-====================================================================================================================================================================================6+24======================================================================================================================================================================================================================================================================================================================================

Example 2
::例2

Find the exact value for $\begin{align*}\sin 345^\circ\end{align*}$
::查找 sin345 的准确值@ action

$\begin{aligned} \sin 345^{\circ} & = \sin (300^{\circ} + 45^{\circ}) = \sin 300^{\circ} \cos 45^{\circ} + \cos 300^{\circ} \sin 45^{\circ} \\ = - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = - \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4} \end{aligned}$
$\begin{align*}\sin 345^\circ & = \sin (300^\circ + 45^\circ) = \sin 300^\circ \cos 45^ \circ + \cos 300^\circ \sin 45^\circ \\ & = - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = - \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4}\end{align*}$
::345444444264

Example 3
::例3

Find the exact value for $\begin{align*}\sin \frac{17 \pi}{12}\end{align*}$
::查找 sin 1712 的确切值

$\begin{aligned} \sin \frac{17 π}{12} & = \sin (\frac{9 π}{12} + \frac{8 π}{12}) = \sin (\frac{3 π}{4} + \frac{2 π}{3}) = \sin \frac{3 π}{4} \cos \frac{2 π}{3} + \cos \frac{3 π}{4} \sin \frac{2 π}{3} \\ = \frac{\sqrt{2}}{2} \cdot (- \frac{1}{2}) + - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = - \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{- \sqrt{2} - \sqrt{6}}{4} \end{aligned}$
$\begin{align*}\sin \frac{17 \pi}{12} & = \sin \left (\frac{9 \pi}{12} + \frac{8 \pi}{12} \right ) = \sin \left (\frac{3\pi}{4} + \frac{2\pi}{3} \right ) = \sin \frac{3\pi}{4} \cos \frac{2\pi}{3} + \cos \frac{3\pi}{4} \sin \frac{2\pi}{3} \\ & = \frac{\sqrt{2}}{2} \cdot (-\frac{1}{2}) + - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{-\sqrt{2} - \sqrt{6}}{4}\end{align*}$
::-=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

Example 4
::例4

If $\begin{align*}\sin y = - \frac{5}{13}\end{align*}$ , $\begin{align*}y\end{align*}$ is in quad III, and $\begin{align*}\sin z = \frac{4}{5}\end{align*}$ , $\begin{align*}z\end{align*}$ is in quad II find $\begin{align*}\sin(y + z)\end{align*}$
::如果siny 513,y在四角三,sinz=45,z在四角二中发现sin(y+z)

If $\begin{align*}\sin y = - \frac{5}{13}\end{align*}$ and in Quadrant III, then cosine is also negative. By the , the second leg is $\begin{align*}12 (5^2 + b^2 = 13^2)\end{align*}$ , so $\begin{align*}\cos y = - \frac{12}{13}\end{align*}$ . If the $\begin{align*}\sin z = \frac{4}{5}\end{align*}$ and in Quadrant II, then the cosine is also negative. By the Pythagorean Theorem, the second leg is $\begin{align*}3 (4^2 + b^2 = 5^2)\end{align*}$ , so $\begin{align*}\cos = - \frac{3}{5}\end{align*}$ . To find $\begin{align*}\sin(y + z)\end{align*}$ , plug this information into the sine sum formula.
::如果 siny 513 和在 Quadrant III 中, 则 Cosine 也为负值。到 , 第二腿为 12 (52+b2=132), 所以 cos y 1213。如果 sinz=45 和 Quadrant II 中, 则cosine 也为负值。到 Pythagoren 理论中, 第二腿为 3( 342+b2=52), 所以 cos =35。要找到 sin( y+z), 请将此信息插入正数公式中。

$\begin{aligned} \sin (y + z) & = \sin y \cos z + \cos y \sin z \\ = - \frac{5}{13} \cdot - \frac{3}{5} + - \frac{12}{13} \cdot \frac{4}{5} = \frac{15}{65} - \frac{48}{65} = - \frac{33}{65} \end{aligned}$
$\begin{align*}\sin (y + z) & = \sin y \cos z + \cos y \sin z \\ & = - \frac{5}{13} \cdot - \frac{3}{5} + - \frac{12}{13} \cdot \frac{4}{5} = \frac{15}{65} - \frac{48}{65} = - \frac{33}{65}\end{align*}$
:y+z)=sinycosz+cosysinz 51335121345=1565-48653365

Review
::回顾

Find the exact value for each sine expression.
::查找每个正弦表达式的准确值。

$\begin{align*}\sin75^\circ\end{align*}$
::75

$\begin{align*}\sin105^\circ\end{align*}$
::一九零五年... 一九零五年... 一九零五年...

$\begin{align*}\sin165^\circ\end{align*}$
::一九五五一五一一五一一五一一五一一一一五一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一

$\begin{align*}\sin255^\circ\end{align*}$
::确实如此... 255... _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}\sin-15^\circ\end{align*}$
::-15 -15 _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Write each expression as the sine of an angle.
::将每个表达式写为角度的正弦值。

$\begin{align*}\sin46^\circ\cos20^\circ+\cos46^\circ\sin20^\circ\end{align*}$
::四十六点六十二分二十点四十六分二十点八十分

$\begin{align*}\sin3x\cos2x-\cos3x\sin2x\end{align*}$
::3xcos%2x-cos%3xsin%2xx

$\begin{align*}\sin54^\circ\cos12^\circ+\cos54^\circ\sin12^\circ\end{align*}$
::来来来来来去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去

$\begin{align*}\sin29^\circ\cos10^\circ-\cos29^\circ\sin10^\circ\end{align*}$
::{\fn黑体\fs20\shad2\2aH82\3aH20\4aH33\fscx95\3cH592001\be1}那么... {\fn黑体\fs20\shad2\2aH82\3aH20\4aH33\fscx95\3cH592001\be1}... {\fn黑体\fs20\shad2\2aH82\3aH20\4aH33\fscx95\3cH592001\be1}... {\fn黑体\fs20\shad2\2aH82\3aH20\4aH33\fscx95\3cH592001\be1}

$\begin{align*}\sin4y\cos3y+\cos4y\sin3y\end{align*}$
::

Prove that $\begin{align*}\sin(x-\frac{\pi}{2})=-\cos(x)\end{align*}$
::证明有罪证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过

Suppose that x, y, and z are the three angles of a triangle. Prove that $\begin{align*}\sin(x+y)=\sin(z)\end{align*}$
::假设 x, y, 和 z 是三角形的三个角度。证明 sin * (x+y) =sin * (z)

Prove that $\begin{align*}\sin(\frac{\pi}{2}-x)=\cos(x)\end{align*}$
::证明 sin * (2-x) =cos * (x)

Prove that $\begin{align*}\sin(x+\pi)=-\sin(x)\end{align*}$
::证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过证明罪过

Prove that "> $\begin{align*}\sin(x-y)+\sin(x+y)=2\sin(x)\cos(y)\end{align*}$
::证明 sin(x-y)+sin(x+y)=2sin(x)cos

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

Sine 平衡和差异公式

Sine Sum and Difference Formulas ::Sine 平衡和差异公式

Using the Sine Sum and Difference Formula ::使用 Sine 和差异公式

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)

Sine Sum and Difference Formulas
::Sine 平衡和差异公式

Using the Sine Sum and Difference Formula
::使用 Sine 和差异公式

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)