# How do you simplify cos(arcsin(1/3))?

Feb 4, 2017

$\cos \left(\arcsin \left(\frac{1}{3}\right)\right) = \frac{2 \sqrt{2}}{3}$

#### Explanation:

$\arcsin x$ is the angle $\theta$ for which $\sin \theta = \frac{1}{3}$

or $\theta = \arcsin \left(\frac{1}{3}\right)$

As $\sin \theta = \frac{1}{3}$, $\cos \theta = \sqrt{1 - {\left(\frac{1}{3}\right)}^{2}} = \sqrt{\frac{8}{9}} = \frac{2 \sqrt{2}}{3}$

Now as $\cos \theta = \frac{2 \sqrt{2}}{3}$ and $\theta = \arcsin \left(\frac{1}{3}\right)$

$\cos \left(\arcsin \left(\frac{1}{3}\right)\right) = \frac{2 \sqrt{2}}{3}$