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Q 16.

Expert-verified

Precalculus Enhanced with Graphing Utilities

Found in: Page 457

Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

In Problems 9–36, find the exact value of each expression.
$\sec (\tan^{- 1} \sqrt{3})$

The value of the expression $\sec (\tan^{- 1} \sqrt{3}) = 2$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given expression is:

$\sec (\tan^{- 1} \sqrt{3})$

Let's take $θ = \tan^{- 1} \sqrt{3}$ ,

$\tan θ = \sqrt{3}$

$- \frac{π}{2} \leq θ \leq \frac{π}{2}$ is the domain of the function.

Step 2. Find the value of θ.

Initially, we have function $\tan^{- 1}$ and the bounds of tan function are going to be in the interval $[- \frac{π}{2}, \frac{π}{2}]$ . It represents the range of $\tan^{- 1}$ .

By using the unit circle, we can say that $θ$ must be equal to $\frac{π}{3}$ , so that we can get $\sqrt{3}$ .

$\tan θ = \sqrt{3} \tan^{- 1} \sqrt{3} = \frac{π}{3}$

Step 3. Find the exact value of the expression.

$\sec (\tan^{- 1} \sqrt{3}) = \sec (\frac{π}{3}) \sec (θ) = \frac{1}{\cos θ} \sec (\frac{π}{3}) = \frac{1}{\cos (\frac{π}{3})}$

By using the unit circle, we know that $\cos (\frac{π}{3}) = \frac{1}{2}$ , then

$\sec (\frac{π}{3}) = \frac{1}{\frac{1}{2}} = 2$

Therefore, $\sec (\tan^{- 1} \sqrt{3}) = 2$ .

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