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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 29 (page 485)

In problem 23-32,find the exact value of the each expression
$\sin \frac{蟺}{12} \cos \frac{7 蟺}{12} - \cos \frac{蟺}{12} \sin \frac{7 蟺}{12}$

Short Answer

Expert verified

The exact value of $\sin \frac{蟺}{12} \cos \frac{7 蟺}{12} - \cos \frac{蟺}{12} \sin \frac{7 蟺}{12} = - 1$

Step by step solution

01

Step 1.Given information

The given expression $\sin \frac{蟺}{12} \cos \frac{7 蟺}{12} - \cos \frac{蟺}{12} \sin \frac{7 蟺}{12}$

02

Step 2.Finding exact value of the given expression

The indicated expression is in the form of the right side of the formula

$\sin (伪 - 尾) = \sin 伪 \cos 尾 - \cos 伪 \sin 尾 w i t h 伪 = \frac{蟺}{12} a n d 尾 = \frac{7 蟺}{12}$

Now,evaluate the expression using the difference formula of the sine as follows:

$\sin \frac{蟺}{12} \cos \frac{7 蟺}{12} - \cos \frac{蟺}{12} \sin \frac{7 蟺}{12} = \sin (\frac{蟺}{12} - \frac{7 蟺}{12}) = \sin (\frac{蟺 - 7 蟺}{12}) = \sin (- \frac{6 蟺}{12}) = \sin (- \frac{蟺}{2})$

03

Step 3.Simplify the expression using Even-odd identities

$\sin (- \frac{蟺}{2}) = - \sin (\frac{蟺}{2})$

= -1

Since,sine is an odd function, $\sin (- 胃) = - \sin 胃$

since, $\sin (\frac{蟺}{2}) = 1$

Therefore the exact value of the expression $\sin 20 掳 \cos 10 掳 + \cos 20 掳 \sin 10 掳$ is

\sin \frac{蟺}{12} \cos \frac{7 蟺}{12} - \cos \frac{蟺}{12} \sin \frac{7 蟺}{12} = - 1

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