Question

The principal value of ${\text{cosec}}^{-1}(-2)$ is

[1 mark]

• A. $-\frac{\pi }{6}$
• B. $\frac{5\pi }{6}$
• C. $-\frac{\pi }{3}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is A $-\frac{\pi }{6}$

Let ${\text{cosec}}^{-1}(-2)=x$
$\Rightarrow \text{cosec}x=-2$
$\Rightarrow \text{cosec}x=\text{cosec}(-\frac{\pi }{6})$
We know that the range of principal value branch of ${\text{cosec}}^{-1}$ is $[-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}$ and $-\frac{\pi }{6}\in [-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}$
$\therefore$ Principal value of ${\text{cosec}}^{-1}(-2)$ is $-\frac{\pi }{6}.$