A value of ^-1[(^-123)] is

Question

A value of $	an^{-1}\left[\sin\left(\cos^{-1}\sqrt{\dfrac23}ight)ight]$ is

Accepted Answer

Let $\theta=\cos^{-1}\sqrt{\dfrac23}$, so $\cos\theta=\sqrt{\dfrac23}$, $\sin\theta=\sqrt{1-\dfrac23}=\dfrac{1}{\sqrt3}$.

$$\tan^{-1}\left[\sin\theta\right]=\tan^{-1}\left(\frac{1}{\sqrt3}\right)=\frac{\pi}{6}.$$

Solutions · 1