Question

Find the value of ${\cos }^{-1}\cos \frac{7\pi }{6}.$

Answer 1

The range of ${\cos }^{-1}\left(\cos x\right)\in \left[0.\pi \right]$
$\Rightarrow {\cos }^{-1}\cos \left(\frac{7\pi }{6}\right)\ne \frac{7\pi }{6}$
As $\frac{7\pi }{6}\notin \left[0.\pi \right]\Rightarrow {\cos }^{-1}\cos \left(\frac{7\pi }{6}\right)={\cos }^{-1}\cos (\pi +\frac{\pi }{6})={\cos }^{-1}(-\cos \frac{\pi }{6})=\pi -{\cos }^{-1}\left(\cos \frac{\pi }{6}\right)=\pi -\frac{\pi }{6}=\frac{5\pi }{6}$