Question

Find the principal value of $co{s}^{-1}\left(cos\frac{7\pi }{6}\right)$.

Answer 1

${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)\ne \frac{7\pi }{6}$ because $\frac{7\pi }{6}$ does not lie between $0$ and x.

${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)={\cos }^{-1}\left\{\cos (2\pi -\frac{5\pi }{6})\right\}$

$={\cos }^{-1}\left(\cos \frac{5\pi }{6}\right)$

$=\frac{5\pi }{6}$.