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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. 89 (page 496)

In Problems 79-90, find the exact value of each expression.
$\sec (2 \tan^{- 1} \frac{3}{4})$

Short Answer

Expert verified

The exact value is $\frac{25}{7}$

Step by step solution

01

Given information

Given expression $\sec (2 \tan^{- 1} \frac{3}{4})$

02

Rewriting using inverse rule and calculating secant

Rewriting, we get

$胃 = \tan^{- 1} \frac{3}{4} \tan 胃 = \frac{3}{4} \sec (2 \tan^{- 1} \frac{3}{4}) = \sec 2 胃 \sec (2 \tan^{- 1} \frac{3}{4}) = = \frac{1}{\cos 2 胃}$

03

Using double angle formula and calculating

Calculating, we get

$\sec (2 \tan^{- 1} \frac{3}{4}) = \frac{1}{2 \cos^{2} 胃 - 1} \cos 胃 = \frac{4}{5} \sec (2 \tan^{- 1} \frac{3}{4}) = \frac{1}{2 {(\frac{4}{5})}^{2} - 1} \sec (2 \tan^{- 1} \frac{3}{4}) = \frac{1}{\frac{32}{25} - 1} \sec (2 \tan^{- 1} \frac{3}{4}) = \frac{25}{32 - 25} \sec (2 \tan^{- 1} \frac{3}{4}) = \frac{25}{7}$

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Most popular questions from this chapter

Multiply and simplify: $\frac{(1 + \tan (胃)) (1 + \tan (胃)) - s e c (胃)}{\tan (胃)}$

Establish each identity.

$\frac{1 - 2 \cos^{2} 胃}{\sin 胃 \cos 胃} = \tan 胃 - c o t 胃$

If $z = \tan \frac{伪}{2}$ , show that localid="1646416257885" $\cos 伪 = \frac{1 - z^{2}}{1 + z^{2}}$ .

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

$\sin 胃 - \sqrt{3} \cos 胃 = 1$

In Problems 9鈥�36, find the exact value of each expression.

$\sin (\cos^{- 1} \frac{1}{2})$

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