Question

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.

$\sin ({\sin }^{-1}0.9)$

Short answer

The exact value is $0.9$.

Step-by-step solution

Step 1. Given Information 

We are given a composite function $\sin \left({\sin }^{-1}0.9\right)$.

We need to find its exact value.

Step 2. Concept used 

The sine function and its inverse, form these properties

${\sin }^{-1}\left(\sin x\right)=x$where$-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}$

and

$\sin \left({\sin }^{-1}x\right)=x$where$-1\le x\le 1$

Step 3. Find the exact value 

As $0.9$lies in the interval $\left[-1,1\right]$. So using the property formed by sine and its inverse function we have

$\sin \left({\sin }^{-1}0.9\right)=0.9$