Question

Find the principal value of tan-1$\bigg(-\sqrt3\bigg)$

Answer 1

Let tan-1$\big(-\sqrt3\big)=y$
Then ,$tan \,y = -\sqrt3=-tan\Big(\frac {\pi}{3}\Big)=tan \Big(\frac{-\pi}{3}\Big)$
We know that the range of the principal value branch of tan-1 is
$\bigg(\frac {-\pi}{2},\frac {\pi}{2}\bigg) and\, tan \bigg(\frac{-\pi}{3}\bigg)is \,-\sqrt3.$
Therefore, the principal value of tan-1$(-\sqrt3)$ is $-\frac{\pi}{3}$