(i) $\sin ^{-1}\left(\sin \frac{\pi}{6}\right)$
(ii) $\sin ^{-1}\left(\sin \frac{7 \pi}{6}\right)$
(iii) $\sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)$
(iv) $\sin ^{-1}\left(\sin \frac{13 \pi}{7}\right)$
(v) $\sin ^{-1}\left(\sin \frac{17 \pi}{8}\right)$
(vi) $\sin ^{-1}\left\{\left(\sin -\frac{17 \pi}{8}\right)\right\}$
(vii) $\sin ^{-1}(\sin 3)$
(viii) $\sin ^{-1}(\sin 4)$
(ix) $\sin ^{-1}(\sin 12)$
(x) $\sin ^{-1}(\sin 2)$
We know
$\sin \left(\sin ^{-1} \theta\right)=\theta$ if $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$
(i) We have
$\sin ^{-1}\left(\sin \frac{\pi}{6}\right)=\frac{\pi}{6}$
(ii) We have
$\sin ^{-1}\left(\sin \frac{7 \pi}{6}\right)=\sin ^{-1}\left\{\sin \left(\pi+\frac{\pi}{6}\right)\right\}$
$=\sin ^{-1}\left(\sin -\frac{\pi}{6}\right)$
$=-\frac{\pi}{6}$
(iii) We have
$\sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)=\sin ^{-1}\left\{\sin \left(\pi-\frac{\pi}{6}\right)\right\}$
$=\sin ^{-1}\left(\sin \frac{\pi}{6}\right)$
$=\frac{\pi}{6}$
(iv) We have
$\sin ^{-1}\left(\sin \frac{13 \pi}{7}\right)=\sin ^{-1}\left\{\sin \left(2 \pi-\frac{\pi}{7}\right)\right\}$
$=\sin ^{-1}\left(\sin -\frac{\pi}{7}\right)$
$=-\frac{\pi}{7}$
(v) We have
$\sin ^{-1}\left(\sin \frac{17 \pi}{8}\right)=\sin ^{-1}\left\{\sin \left(2 \pi+\frac{\pi}{8}\right)\right\}$
$=\sin ^{-1}\left(\sin \frac{\pi}{8}\right)$
$=\frac{\pi}{8}$
(vi) We have
$\sin ^{-1}\left(\sin -\frac{17 \pi}{8}\right)=\sin ^{-1}\left(-\sin \frac{17 \pi}{8}\right)$
$=\sin ^{-1}\left\{-\sin \left(2 \pi+\frac{\pi}{8}\right)\right\}$
$=\sin ^{-1}\left(-\sin \frac{\pi}{8}\right)$
$=\sin ^{-1}\left(\sin -\frac{\pi}{8}\right)$
$=-\frac{\pi}{8}$
(vii) We have
$\sin ^{-1}(\sin 3)=\sin ^{-1}\{\sin (\pi-3)\}$
$=\pi-3$
(viii)We have
$\sin ^{-1}(\sin 4)=\sin ^{-1}\{\sin (\pi-4)\}$
$=\pi-4$
(ix) We have
$\sin ^{-1}(\sin 12)=\sin ^{-1}\{\sin (-\pi+12)\}$
$=12-\pi$
(x) We have
$\sin ^{-1}(\sin 2)=\sin ^{-1}\{\sin (\pi-2)\}$
$=\pi-2$