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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. 18 (page 505)

use sum, difference, product, or half-angle formulas to find the exact value of each expression
$\tan 75 掳$

Short Answer

Expert verified

$\tan 75 掳 = 2 + \sqrt{3}$

Step by step solution

01

Step 1. Given expression

$\tan 75 掳$

02

Step 2. Using half angle formula

We know :

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

Using it:
$\tan (\frac{150}{2}) = 卤 \sqrt{\frac{1 - \cos 150 掳}{1 + \cos 150 掳}} \tan (75) = \sqrt{\frac{1 - \frac{- \sqrt{3}}{2}}{1 + \frac{- \sqrt{3}}{2}}} \tan (75) = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$

03

Step 3.Rationalizing

$\tan (75 掳) = (\sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}) . (\sqrt{\frac{2 + \sqrt{3}}{2 + \sqrt{3}}}) \tan (75 掳) = \frac{2 + \sqrt{3}}{\sqrt{2^{2} - {(\sqrt{3})}^{2}}} \tan (75 掳) = 2 + \sqrt{3}$

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Most popular questions from this chapter

True or False. $\sin (2 胃)$ has two equivalent forms:

$2 \sin u \cos u$ and $\sin^{2} 胃 - \cos^{2} 胃$

Establish each identity.

$s e c 胃 - \cos 胃 = \sin 胃 \tan 胃$

solve each equation on the interval $0 鈮� 胃 < 2 蟺$

$\sin 胃 - \sqrt{3} \cos 胃 = 1$

Find the value of the number C:

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

Use the fact that

$\cos \frac{蟺}{12} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$

to find $\sin \frac{蟺}{24}, \cos \frac{蟺}{24}$ .

See all solutions

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