Question

Find the value of $\sin ^{-1}(\sin 7)$ and $\sin ^{-1}(\sin (-5))$.

Answer 1

SOLUTION : Let $y=\sin ^{-1}(\sin 7)$

$\sin ^{-1}(\sin 7) \neq 7 \text { as } 7 \notin\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \quad \because \quad 2 \pi<7<\frac{5 \pi}{2}$

graph of $y=\sin ^{-1}(\sin x)$ is as :

From the graph we can see that if $2 \pi \leq x \leq \frac{5 \pi}{2}$, then $y=\sin ^{-1}(\sin x)$ can be written as :

$y=x-2 \pi$

$\therefore \quad \sin ^{-1}(\sin 7)=7-2 \pi$