Question

Find the principal value of $cose{c}^{-1}(-2)$.

Answer 1

Let $y={cosec}^{-1}(-2)$

$\Rightarrow cosecy=-2$ is negative

$\therefore cosecy$ lies in third quadrant or fourth quadrant.

$\therefore y=\pi +\frac{\pi }{6},2\pi -\frac{\pi }{6}$ [$\because cosec\frac{3\pi }{6}=-2$]

$\therefore$Principal values of $\pi +\frac{\pi }{6},2\pi -\frac{\pi }{6}$

or $\frac{6\pi +\pi }{6},\frac{12\pi -\pi }{6}$

or $\frac{7\pi }{6},\frac{11\pi }{6}$