Question

Find the principal values of the following questions:

$ta{n}^{-1}(-\sqrt{3})$

Answer 1

Let $ta{n}^{-1}(-\sqrt{3})=\theta \Rightarrow tan\theta =-\sqrt{3}$
We know that the range of principal value of $ta{n}^{-1}\theta$ is $(-\frac{\pi }{2},\frac{\pi }{2})$.
$\therefore tan\theta =-\sqrt{3}=-tan\frac{\pi }{3}=tan(-\frac{\pi }{3})(\because tan(-\theta )=-tan\theta )$
$\Rightarrow \theta =-\frac{\pi }{3}$, where $\theta ϵ(-\frac{\pi }{2},\frac{\pi }{2})\Rightarrow ta{n}^{-1}(-\sqrt{3})=-\frac{\pi }{3}$
Hence, principal value of $ta{n}^{-1}(-\sqrt{3})is-\frac{\pi }{3}$