Question

Using principal values, evaluate the following
${\cos }^{-1}\left(\cos \frac{2\pi }{3}\right)$+ ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$

Answer 1

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

Figure 1 represents graph of ${\sin }^{-1}\sin x$

Figure 2 represents graph of ${\cos }^{-1}\cos x$

Hence, ${\cos }^{-1}\cos \frac{2\pi }{3}+{\sin }^{-1}\sin \frac{2\pi }{3}$

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

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