Question

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

Answer 1

We have;

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

$={\cos }^{-1}\left(\frac{1}{2}\right)+2{\sin }^{-1}\left(\frac{-1}{2}\right)$ $[\therefore {\sin }^{-1}(-x)=-{\sin }^{-1}x;xϵ[-1,1\left]\right]$

$=\frac{\pi }{2}+{\sin }^{-1}\left(\frac{1}{2}\right)$ $[\therefore {\sin }^{-1}+{\cos }^{-1}x=\frac{\pi }{2}xϵ[-1,1\left]\right]$

$=\frac{\pi }{2}+(2n\pi +\frac{\pi }{6})$

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

Hence, principal value of ${\cos }^{-1}\left(\frac{1}{2}\right)-2{\sin }^{-1}\left(\frac{-1}{2}\right)=2n\pi +\frac{2\pi }{3}$