Chapter 6.8, Problem 11CFU

To determine

To state: true or false, and give a counterexample in case of false statement.

Answer to Problem 11CFU

True

Explanation of Solution

Given information:

  $\sin \left({\text{Sin}}^{-1}x\right)=x\text{ for }-1\le x\le 1$

Calculation:

Put the value for $-1\le x\le 1$ in the given function and solve as follows:

  $\begin{array}{c}\sin \left({\text{Sin}}^{-1}\left(-1\right)\right)=-1\left[\text{For }x=-1\right]\\ \sin \left(-\frac{\pi }{2}\right)=-1\\ -\sin \frac{\pi }{2}=-1\\ -1=-1\\ \\ \sin \left({\text{Sin}}^{-1}1\right)=1\left[\text{For }x=1\right]\\ \sin \left(\frac{\pi }{2}\right)=1\\ 1=1\\ \\ \sin \left({\text{Sin}}^{-1}0\right)=0\left[\text{For }x=0\right]\\ \sin \left(0\right)=0\\ 0=0\end{array}$

Since the input of $x$ for the given condition gives the expected output. Thus the statement is true.