Question

Write the principal value of ${\cos }^{-1}\left(\cos \frac{2\mathrm{\pi }}{3}\right)+{\sin }^{-1}\left(\sin \frac{2\mathrm{\pi }}{3}\right)$

Answer 1

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

∴ ${\cos }^{-1}\left(\cos \frac{2\mathrm{\pi }}{3}\right)+{\sin }^{-1}\left(\sin \frac{2\mathrm{\pi }}{3}\right)=\mathrm{\pi }$