Find the principal value of each of the following :
Question:

Find the principal value of each of the following :

$\operatorname{cosec}^{-1}(-\sqrt{2})$

Solution:

$\operatorname{cosec}^{-1}(-\sqrt{2})=-\operatorname{cosec}^{-1}(\sqrt{2})\left[\right.$ Formula: $\left.\operatorname{cosec}^{-1}(-x)=-\operatorname{cosec}^{-1}(x)\right]$

$=-\frac{\pi}{4}$

This can also be solved as

$\operatorname{cosec}^{-1}(-\sqrt{2})$

Since cosec is negative in the third quadrant, the angle we are looking for will be in the third quadrant.

$=\pi+\frac{\pi}{4}$

$=\frac{5 \pi}{4}$