Find the principal value of each of the following:

Question:

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) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)$

Solution:

(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) $\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\sin ^{-1}(1)=\sin ^{-1}\left[\sin \left(\frac{\pi}{2}\right)\right]=\frac{\pi}{2}$

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