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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 503)

In Problems 9鈥�23, find the exact value, if any, of each composite function. If there is no value, say it is 鈥渘ot defined.鈥� Do not use a calculator.
$s e c (\tan^{- 1} \frac{\sqrt{3}}{3})$

Short Answer

Expert verified

The exact value of the composite function is $\frac{2 \sqrt{3}}{3} .$

Step by step solution

01

Step 1. Given Information

The given composite function is $s e c (\tan^{- 1} \frac{\sqrt{3}}{3}) .$

We have to find the exact value.

02

Step 2. Finding the value

To find the exact value, let $x = \tan^{- 1} \frac{\sqrt{3}}{3} .$

Now, the domain of the tan function is $- \frac{蟺}{2} < x < \frac{蟺}{2}$

The value of $\tan^{- 1} \frac{\sqrt{3}}{3} = \frac{蟺}{6} .$

So, $x = \frac{蟺}{6}$ because the value lies in the domain of the tan function.

03

Step 3. Solve the composite function

Now, substitute the xvalue in the given function.

Thus,

$s e c (\tan^{- 1} \frac{\sqrt{3}}{3}) = s e c (\frac{蟺}{6}) = \frac{2}{\sqrt{3}} R a t i o n a l i z e t h e d e n o m i n a t o r = \frac{2}{\sqrt{3}} 脳 \frac{\sqrt{3}}{\sqrt{3}} = \frac{2 \sqrt{3}}{3}$

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