Chapter 7, Problem 20RE

To determine

To calculate: The exact value of the expression $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ .

Answer to Problem 20RE

The exact value of the expression $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ is “not defined”.

Explanation of Solution

Given information:

The expression is $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ .

Formula used:

For the sine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Calculation:

Consider the expression $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ .

Observe that the value $1.6$ is not lies between $-1$ and $1$ .

Recall that for the sine function and its inverse is related as,

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Therefore, the exact value of the expression $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ cannot be calculated.

Thus, the exact value of the expression $\sin \left[{\sin }^{-1}\left(1.6\right)\right]$ is “not defined”.