Question:
Write the value of $\sin ^{-1}\left(\sin \left(-600^{\circ}\right)\right)$.
Solution:
We know that $\sin ^{-1}(\sin x)=x$.
Now,
$\sin ^{-1}\left\{\sin \left(-600^{\circ}\right)\right\}=\sin ^{-1}\left\{\sin \left(720^{\circ}-600^{\circ}\right)\right\}$
$=\sin ^{-1}\left\{\sin \left(120^{\circ}\right)\right\}$
$=\sin ^{-1}\left\{\sin \left(180^{\circ}-120^{\circ}\right)\right\} \quad[\because \sin x=\sin (\pi-x)]$
$=\sin ^{-1}\left(\sin 60^{\circ}\right)$
$=60^{\circ}$
$\therefore \sin ^{-1}\left\{\sin \left(-600^{\circ}\right)\right\}=60^{\circ}$
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