# What does arcsin(cos ((5pi)/6))  equal?

Mar 13, 2016

$= - \frac{\pi}{3}$

#### Explanation:

"principal value" of the arcsin function means it's between
$- \frac{\pi}{2} \le \theta \le + \frac{\pi}{2}$

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

for least positve value
$\arcsin \left(\cos \left(5 \frac{\pi}{6}\right)\right) = \arcsin \left(\cos \left(\frac{\pi}{2} + \frac{\pi}{3}\right)\right) = \arcsin \left(- \sin \left(\frac{\pi}{3}\right)\right) = \arcsin \sin \left(\pi + \frac{\pi}{3}\right) = 4 \frac{\pi}{3}$

Mar 18, 2016

$\frac{4 \pi}{3} , \frac{5 \pi}{3}$

#### Explanation:

Trig table gives -->
$\cos \left(\frac{5 \pi}{6}\right) = - \frac{\sqrt{3}}{2}$
Find $\arcsin \left(- \frac{\sqrt{3}}{2}\right)$
Trig unit circle, and trig table give -->
$\sin x = - \frac{\sqrt{3}}{2}$ -->2 solutions -->
arc $x = - \frac{\pi}{3}$ and arc $x = \frac{4 \pi}{3}$
Answers: $\left(\frac{4 \pi}{3}\right)$ and $\left(\frac{5 \pi}{3}\right)$--> (co-terminal with $\left(- \frac{\pi}{3}\right)$