Question

Find the principal value of ${\tan }^{-1}(-\sqrt{3}).$

Answer 1

Let ${\tan }^{-1}(-\sqrt{3})=y$.

Then, $\tan y=\frac{\pi }{3}$.
We know that the range of the principal value of ${\tan }^{-1}x$ is $(-\frac{\pi }{2},\frac{\pi }{2})$ and $\tan (-\frac{\pi }{3})=-\sqrt{3}$.

Therefore, the principal value of ${\tan }^{-1}(-\sqrt{3})$ is $-\frac{\pi }{3}$