# Write the difference between maximum and minimum values

Question:

Write the difference between maximum and minimum values of $\sin ^{-1} x$ for $x \in[-1,1]$.

Solution:

The maximum value of $\sin ^{-1} x$ in $x \in[-1,1]$ is at 1 .

So, the maximum value is

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

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

$=\frac{\pi}{2}$

Again, the minimum value is at -1.
Thus, the minimum value is

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

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

$=-\frac{\pi}{2}$

So, the difference between the maximum and the minimum value is

$\frac{\pi}{2}-\left(-\frac{\pi}{2}\right)=\pi$