Question

Find the principal value of ${\cos }^{-1}(-\frac{1}{\sqrt{2}}).$

Answer 1

Let $y={\cos }^{-1}(-\frac{1}{\sqrt{2}})$
$\Rightarrow \cos y=-\frac{1}{\sqrt{2}}$
We know that the range of the principal value branch of ${\cos }^{-1}x$ is $[0,\pi ]$.
$\Rightarrow \cos y=\cos \left(\frac{3\pi }{4}\right)$
$\therefore y=\frac{3\pi }{4}$
Hence, the principal value of ${\cos }^{-1}(-\frac{1}{\sqrt{2}})$ is $\frac{3\pi }{4}.$