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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. 27 (page 495)

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\sin (- \frac{蟺}{8})$

Short Answer

Expert verified

The exact value of the expression $\sin (- \frac{蟺}{8})$ by using Half angle formula is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$

Step by step solution

01

Step 1. Given

An expression $\sin (- \frac{蟺}{8})$

To find the exact value of the expression

02

Step 2. Half angle formula

The half angle formula is,

$\sin \frac{胃}{2} = 卤 \sqrt{\frac{1 - \cos 胃}{2}}$

03

Step 3. Consider an angle

Consider

$\sin 胃 = \sin \frac{蟺}{4} 胃 = \frac{蟺}{4} \frac{胃}{2} = \frac{\frac{蟺}{4}}{2} = \frac{蟺}{8}$

04

Step 4. Use the half-angle formula

By Half-angle formula,

$\sin \frac{胃}{2} = 卤 \sqrt{\frac{1 - \cos 胃}{2}} \sin (- \frac{蟺}{8}) = 卤 \sqrt{\frac{1 - \cos (- \frac{蟺}{4})}{2}} = 卤 \sqrt{\frac{1 - \cos \frac{蟺}{4}}{2}} = 卤 \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = 卤 \sqrt{\frac{2 - \sqrt{2}}{4}} = 卤 \frac{\sqrt{2 - \sqrt{2}}}{2}$

05

Step 5. Find the exact value

Since $\frac{蟺}{8}$ is in first quadrant,

$\sin \frac{蟺}{8} = \frac{\sqrt{2 - \sqrt{2}}}{2}$

So, $\sin (- \frac{蟺}{8}) = - \frac{\sqrt{2 - \sqrt{2}}}{2}$

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Most popular questions from this chapter

express each product as a sum containing only sines or only cosines

$\sin \frac{蠎}{2} \cos \frac{5 蠎}{2}$

If $\tan 伪 = x + 1$ and role="math" localid="1646425111199" $\tan 尾 = x - 1,$ show that

role="math" localid="1646426027733" $2 c o t (伪 - b) = x^{2}$

Find the value of the number C:

$\frac{1}{2} \sin^{2} x + C = - \frac{1}{4} \cos (2 x)$

Express the product $\cos 3 胃 \cos 5 胃$ as a sum containing only sines or only cosines.

Rewrite in terms of sine and cosine functions: $c o t (胃) s e c (胃)$

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