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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. 21 (page 495)

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{7 蟺}{8}$

Short Answer

Expert verified

The exact value of the expression $\tan \frac{7 蟺}{8}$ by using Half angle formula is $1 - \sqrt{2}$

Step by step solution

01

Step 1. Given

An expression $\tan \frac{7 蟺}{8}$

To find the exact value of the expression

02

Step 2. Half angle formula

The half angle formula is,

$\tan \frac{伪}{2} = 卤 \sqrt{\frac{1 - \cos 伪}{1 + \cos 伪}}$

03

Step 3. Consider an angle

Consider,

$\tan 伪 = \tan \frac{7 蟺}{8} 伪 = \frac{7 蟺}{8} \frac{伪}{2} = \frac{\frac{7 蟺}{4}}{2}$

Let $伪 = \frac{7 蟺}{4}$

We know that, $\tan \frac{伪}{2} = \frac{1 - \cos 伪}{\sin 伪}$

04

Step 4. Use the half-angle formula

By Half-angle formula,

$\tan \frac{伪}{2} = \frac{1 - \cos 伪}{\sin 伪}$

Compute for $\cos 伪, \sin 伪$

$\cos \frac{7 蟺}{4} = \cos (2 蟺 - \frac{蟺}{4}) = \cos (- \frac{蟺}{4}) = \cos \frac{蟺}{4} = \frac{\sqrt{2}}{2}$

$\sin \frac{7 蟺}{4} = \sin (2 蟺 - \frac{蟺}{4}) = \sin (- \frac{蟺}{4}) = - \sin \frac{蟺}{4} = - \frac{\sqrt{2}}{2}$

05

Step 5. Find the exact value

Since $\tan \frac{伪}{2} = \frac{1 - \cos 伪}{\sin 伪}$

$\tan \frac{\frac{7 蟺}{4}}{2} = \frac{1 - \frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = \frac{\frac{2 - \sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = - (\frac{2 - \sqrt{2}}{\sqrt{2}}) = 1 - \sqrt{2}$

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