Question

Find the value of cos^-1 (cos 2pi / 3 ) + sin^-1 ( sin 2pi/3 )

Answer 1

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

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

$=\mathrm{\pi }-\frac{\mathrm{\pi }}{3}+\frac{2\mathrm{\pi }}{3}=\frac{4\mathrm{\pi }}{3}$