Chapter 7: Q. 13 (page 503)

In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.
$\sin^{- 1} [\sin (- \frac{8 π}{9})]$

Short Answer

Expert verified

The exact value is $- \frac{π}{9}$ .

Step by step solution

Step 1. Given Information

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

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 $- \frac{π}{2} \leq x \leq \frac{π}{2}$

and

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

Step 3. Modify the angle

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

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

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

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

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

Step 4. Find the exact value

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

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

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In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.
$\sin^{- 1} [\sin (- \frac{8 π}{9})]$

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.

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In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.sin-1sin-8π9

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

Geometry

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.
$\sin^{- 1} [\sin (- \frac{8 π}{9})]$