Question

Find the principal value of each of the following:

(i) ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
(ii) ${\sin }^{-1}\left(\cos \frac{2\mathrm{\pi }}{3}\right)$
(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) ${\sin }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)$
(vi) ${\sin }^{-1}\left(\tan \frac{5\mathrm{\pi }}{4}\right)$

Answer 1

(i) ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)={\sin }^{-1}\left[\sin \left(-\frac{\mathrm{\pi }}{3}\right)\right]=-\frac{\mathrm{\pi }}{3}$
(ii) ${\sin }^{-1}\left(\cos \frac{2\mathrm{\pi }}{3}\right)={\sin }^{-1}\left(-\frac{1}{2}\right)={\sin }^{-1}\left[\sin \left(-\frac{\mathrm{\pi }}{6}\right)\right]=-\frac{\mathrm{\pi }}{6}$
(iii) ${\sin }^{-1}\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)={\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{12}\right)=\frac{\mathrm{\pi }}{12}$
(iv) ${\sin }^{-1}\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)={\sin }^{-1}\left(\sin \frac{5\mathrm{\pi }}{12}\right)=\frac{5\mathrm{\pi }}{12}$
(v) ${\sin }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)={\sin }^{-1}\left(-\frac{\sqrt{2}}{2}\right)={\sin }^{-1}\left[\sin \left(-\frac{\mathrm{\pi }}{4}\right)\right]=-\frac{\mathrm{\pi }}{4}$
(vi) ${\sin }^{-1}\left(\tan \frac{5\mathrm{\pi }}{4}\right)=={\sin }^{-1}\left(1\right)={\sin }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{2}\right)\right]=\frac{\mathrm{\pi }}{2}$