Question

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

Answer 1

Solve using trigonometric rules:

The given inverse trigonometric function is ${\tan }^{-1}\left(-\sqrt{3}\right)$.

Let, ${\tan }^{-1}\left(-\sqrt{3}\right)=y$

$\Rightarrow$ $\left(-\sqrt{3}\right)=\tan y$ $\left[\because {\tan }^{-1}x=y\Rightarrow x=\mathrm{tany}\right]$

$\Rightarrow$ $\tan y=-\sqrt{3}$

As we know, the range of principal value of ${\tan }^{-1}\mathrm{x}$ lies in $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

$\Rightarrow$ $\tan y=-\tan \frac{\mathrm{\pi }}{3}$

$\Rightarrow$ $\tan y=\tan \left(-\frac{\mathrm{\pi }}{3}\right)$ $\left[\because tan\left(-\theta \right)=-tan\theta \right]$

$\Rightarrow$ $y=-\frac{\mathrm{\pi }}{3}$

So, the principal value of ${\tan }^{-1}\left(-\sqrt{3}\right)$ is $-\frac{\mathrm{\pi }}{3}$.

Hence, the required principal value is $-\frac{\mathrm{\pi }}{3}$.