Question

Write the principal value of ${\sin }^{-1}\left\{\cos \left({\sin }^{-1}\frac{1}{2}\right)\right\}$

Answer 1

$$\begin{gathered} {\sin }^{-1}\left\{\cos \left({\sin }^{-1}\frac{1}{2}\right)\right\}={\sin }^{-1}\left\{\cos \left[{\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{3}\right)\right]\right\} \\ ={\sin }^{-1}\left[\cos \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\sin }^{-1}\left[\frac{1}{2}\right] \\ ={\sin }^{-1}\left[\sin \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$