Q. 13 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. 13 (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.
$\sin^{- 1} [\sin (- \frac{8 蟺}{9})]$

Short Answer

Expert verified

The exact value is $- \frac{蟺}{9}$ .

Step by step solution

Step 1. Given Information

We are given a composite function $\sin^{- 1} [\sin (- \frac{8 蟺}{9})]$ .

We need to find its exact value

Step 2. Concept Used

The sine function and its inverse, form these properties

$\sin^{- 1} (\sin x) = x$ where $- \frac{蟺}{2} 鈮� x 鈮� \frac{蟺}{2}$

and

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

Step 3. Modify the angle

As the angle $- \frac{8 蟺}{9}$ does not lie in the interval $[- \frac{蟺}{2}, \frac{蟺}{2}]$ so we cannot use the property $\sin^{- 1} (\sin x) = x$ .

Now, we know $\sin (蟺 - x) = \sin x$ . So

$\begin{array}{rcl} \sin (蟺 - \frac{8 蟺}{9}) & = & \sin \frac{8 蟺}{9} \\ \sin \frac{蟺}{9} & = & \sin \frac{8 蟺}{9} \end{array}$

Now, using this we can write $- \sin x = \sin (- x)$

$\begin{array}{rcl} - \sin \frac{蟺}{9} & = & - \sin \frac{8 蟺}{9} \\ \sin (- \frac{蟺}{9}) & = & \sin (- \frac{8 蟺}{9}) \end{array}$

Step 4. Find the exact value

Now, $- \frac{蟺}{9}$ lies in the interval $[- \frac{蟺}{2}, \frac{蟺}{2}]$ , so we can use the property formed by sine and its inverse function

$\begin{array}{rcl} \sin^{- 1} [\sin (- \frac{8 蟺}{9})] & = & \sin^{- 1} [\sin (- \frac{蟺}{9})] \\ = & - \frac{蟺}{9} \end{array}$

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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.
$\sin^{- 1} [\sin (- \frac{8 蟺}{9})]$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Modify the angle

Step 4. Find the exact value

One App. One Place for Learning.

Most popular questions from this chapter

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Pure Maths

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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.sin-1sin-8蟺9

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Modify the angle

Step 4. Find the exact value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

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.
$\sin^{- 1} [\sin (- \frac{8 蟺}{9})]$