Find the principal value of each of the following:
(i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
(ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)$
(iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)$
(iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)$
(v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)$
(vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)$
(i) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{3}\right)\right]=-\frac{\pi}{3}$
(ii) $\sin ^{-1}\left(\cos \frac{2 \pi}{3}\right)=\sin ^{-1}\left(-\frac{1}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{6}\right)\right]=-\frac{\pi}{6}$
(iii) $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{\pi}{12}\right)=\frac{\pi}{12}$
(iv) $\sin ^{-1}\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\sin \frac{5 \pi}{12}\right)=\frac{5 \pi}{12}$
(v) $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)=\sin ^{-1}\left[\sin \left(-\frac{\pi}{4}\right)\right]=-\frac{\pi}{4}$
(vi) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\sin ^{-1}(1)=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}\right)\right]=\frac{\pi}{2}$