Question

The principal value of ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is

Answer 1

The given inverse trigonometric function is ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

We want to find out its principal value$.$

If $\theta$ be the principal value of ${\sin }^{-1}x$ then $-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

Therefore$,$ If the principal value of ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ be $\theta$ then ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$$=\theta$

$Sin\theta =-\frac{\sqrt{3}}{2}$

$Sin\theta =\sin \left(-\frac{\mathrm{\pi }}{3}\right),\left[Sin\left(-\theta \right)=-\sin \theta \right]$

$\theta =-\frac{\mathrm{\pi }}{3}$ $,$ Since $-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

Hence, the principal value of ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is $-\frac{\mathrm{\pi }}{3}$.