Question

What is the principal value of

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

Answer 1

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

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

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

$a=\frac{2\pi }{3}+\frac{\pi }{3}=\pi$

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