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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. 52 (page 504)

In Problems $45 - 52$ , find the exact value of each expression.
$\sin \frac{5 蟺}{8}$ .

Short Answer

Expert verified

The exact expression for $\sin \frac{5 蟺}{8}$ is, $\sqrt{\frac{1 + \sqrt{2}}{2 \sqrt{2}}}$ .

Step by step solution

01

Step 1 Given expression is,

$\sin \frac{5 蟺}{8}$ .

Let role="math" localid="1646649909794" $\sin \frac{5 蟺}{8} = \sin 112 . 5^{鈭�}$ .

role="math" localid="1646649902351" $\sin 112 . 5^{鈭�} = \sin (90^{鈭�} + 22 . 5^{鈭�})$

We know that, $\sin (A + B) = \sin A \cos B + \cos A \sin B$

02

Step 2 Apply the formula in the expression.

$\sin (90^{鈭�} + 22 . 5^{鈭�}) = \sin 90^{鈭�} \cos 22 . 5^{鈭�} + \cos 90^{鈭�} \sin 22 . 5^{鈭�} = (1) \cos 22 . 5^{鈭�} + (0) \sin 22 . 5^{鈭�} = \cos 22 . 5^{鈭�}$

Now find the value of $\cos 22 . 5^{鈭�}$ by using the Half angle formula.

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

03

Step 3 Since 22.5∘ lies in the first quadrant, it value should be in positive.

$\cos 22 . 5^{鈭�} = \cos \frac{45^{鈭�}}{2} = \sqrt{\frac{1 + \cos 45^{鈭�}}{2}} = \sqrt{\frac{1}{2} (1 + \frac{1}{\sqrt{2}})} = \sqrt{\frac{1 + \sqrt{2}}{2 \sqrt{2}}}$

Hence, $\sin \frac{5 蟺}{8} = \sqrt{\frac{1 + \sqrt{2}}{2 \sqrt{2}}}$ .

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Most popular questions from this chapter

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

$\cos 3 蠎 \cos 4 蠎$

Establish each identity.

$\frac{\sin^{3} 胃 + \cos^{3} 胃}{1 - 2 \cos^{2} 胃} = \frac{s e c 胃 - \sin 胃}{\tan 胃 - 1}$

(a) $\sin \frac{蟺}{4} 路 \cos \frac{蟺}{3} =______$

(b) $\tan \frac{蟺}{4} - \sin \frac{蟺}{6} =________$

Find the value of the number C:

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

Rewrite over a common denominator:

$\frac{\sin (胃) + \cos (胃)}{\cos (胃)} + \frac{\sin (胃) + \cos (胃)}{\sin (胃)}$

See all solutions

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