Question

principal value of sin inverse(sin 10) = ?

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{principal}\mathrm{value}\mathrm{branch}\mathrm{of}{\sin }^{-1}\mathrm{is}\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]. \\ \mathrm{so},{\sin }^{-1}\left(\sin 10\right)={\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{18}\right)=\frac{\mathrm{\pi }}{18}\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]. \\ \mathrm{so},\mathrm{the}\mathrm{principal}\mathrm{value}\mathrm{of}{\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{18}\right)=\frac{\mathrm{\pi }}{18} \end{gathered}$$