# How do you find the exact value of sin^-1(cos(pi/3))?

Use ${\sin}^{- 1} \sin x = x$
Here, ${\sin}^{- 1} \cos \left(\frac{\pi}{3}\right) = {\sin}^{- 1} \sin \left(\frac{\pi}{2} - \frac{\pi}{3}\right) = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6}$.
The general value for the angle whose sine = $\cos \left(\frac{\pi}{3}\right) = \sin \left(\frac{\pi}{6}\right)$ is
$n \pi + {\left(- 1\right)}^{n} \frac{\pi}{6} , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$