Q. 12 In Problems 9鈥�23, find the exa... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 12 (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.
$\cos^{- 1} (\cos \frac{15 蟺}{7})$

Short Answer

Expert verified

The exact value is $\frac{蟺}{7}$ .

Step by step solution

Step 1. Given Information

We are given a composite function $\cos^{- 1} (\cos \frac{15 蟺}{7})$ .

We need to find its exact value.

Step 2. Concept Used

The cosine function and its inverse, form these properties

$\cos^{- 1} (\cos x) = x$ where $0 鈮� x 鈮� 蟺$

and

$\cos (\cos^{- 1} x) = x$ where $- 1 鈮� x 鈮� 1$

Step 3. Find the exact value

The angle $\begin{array}{rcl} \frac{15 蟺}{7} \end{array}$ does not lie in the interval $[0, 蟺]$ , so we cannot use the property $\cos^{- 1} (\cos x) = x$ .

Now as cosine function is cyclic with a period of $2 蟺$ so we have $\begin{array}{rcl} \cos \end{array} x \begin{array}{rcl} = \end{array} \begin{array}{rcl} \cos (2 蟺 + x) \end{array}$ . Now $\begin{array}{rcl} \cos \frac{15 蟺}{7} \end{array}$ can be written as

$\begin{array}{rcl} \cos \frac{15 蟺}{7} & = & \cos (2 蟺 + \frac{蟺}{7}) \\ = & \cos \frac{蟺}{7} \end{array}$

Now, $\frac{蟺}{7}$ lies in the interval $[0, 蟺]$ . So using the property formed by cosine and its inverse function we have

$\begin{array}{rcl} \cos^{- 1} (\cos \frac{15 蟺}{7}) & = & \cos^{- 1} (\cos \frac{蟺}{7}) \\ = & \frac{蟺}{7} \end{array}$

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91影视

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.
$\cos^{- 1} (\cos \frac{15 蟺}{7})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Find the exact value

One App. One Place for Learning.

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91影视

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. cos-1cos15蟺7

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Find the exact value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Statistics

Discrete Mathematics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

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.
$\cos^{- 1} (\cos \frac{15 蟺}{7})$