Question

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

Answer 1

The principal value of $co{s}^{-1}x$ lies in $[0,\pi ]$ and that of $si{n}^{-1}x$ lies in $[-\frac{\pi }{2},\frac{\pi }{2}]$

$\Rightarrow co{s}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$ and $si{n}^{-1}(-\frac{1}{2})=\frac{-\pi }{6}$

Substituting these values in the final equation, we get

$\frac{\pi }{3}-2\times \frac{-\pi }{6}$

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

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