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Q. Find the principal value $cosec^{-1} \left(\frac{-2}{\sqrt{3}}\right) $

Inverse Trigonometric Functions

Solution:

Let $cosec^{-1} \left(\frac{-2}{\sqrt{3}}\right) = \theta$
$\Rightarrow cosec\,\theta = \frac{-2}{\sqrt{3}}$
$= -cosec \frac{\pi}{3}$
$= cosec \left(\frac{-\pi }{3}\right)$
$\Rightarrow \theta = \frac{-\pi }{3} \in \left[\frac{-\pi }{2}, \frac{\pi }{2}\right] - \left\{0\right\}$
$\therefore $ Principal value of $cosec^{-1} \left(\frac{-2}{\sqrt{3}}\right) is \left(\frac{-\pi }{3}\right)$