Question

Cos^-1 (cos 2pi/3)+sin^-1 (sin 2pi/3)

Answer 1

Dear Student,
$$\begin{gathered} {\cos }^{-1}\left(\cos \frac{2\mathrm{\pi }}{3}\right)+{\sin }^{-1}\left(\sin \frac{2\mathrm{\pi }}{3}\right) \\ ={\cos }^{-1}\left(-\frac{1}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right) \\ =\mathrm{\pi }-{\cos }^{-1}\left(\frac{1}{2}\right)+{\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)\left({\cos }^{-1}\left(-x\right)=\mathrm{\pi }-{\cos }^{-1}\left(\mathrm{x}\right)\right) \\ =\mathrm{\pi }-{\cos }^{-1}\left(\cos \frac{\mathrm{\pi }}{3}\right)+{\sin }^{-1}\left(\sin \frac{2\mathrm{\pi }}{3}\right) \\ =\mathrm{\pi }-\frac{\mathrm{\pi }}{3}+\frac{2\mathrm{\pi }}{3}=\frac{4\mathrm{\pi }}{3} \end{gathered}$$
Regards