# How do you evaluate cos^-1(sin((5pi)/6)) without a calculator?

Sep 15, 2016

${\cos}^{- 1} \left(\sin \left(\frac{5 \pi}{6}\right)\right) = \frac{\pi}{3}$

#### Explanation:

We know that $\sin \left(\frac{\pi}{2} + \theta\right) = \cos \theta$

Hence $\sin \left(\frac{5 \pi}{6}\right) = \sin \left(\frac{3 \pi + 2 \pi}{6}\right) = \sin \left(\frac{3 \pi}{6} + \frac{2 \pi}{6}\right)$

= $\sin \left(\frac{\pi}{2} + \frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$

Hence ${\cos}^{- 1} \left(\sin \left(\frac{5 \pi}{6}\right)\right) = {\cos}^{- 1} \left(\frac{1}{2}\right) = \frac{\pi}{3}$