# How do you find the exact value of cos(arcsin(1/3))?

Aug 28, 2016

$\frac{2 \sqrt{2}}{3}$

#### Explanation:

Let $a = a r c \sin \left(\frac{1}{3}\right) \in Q 1$,

for the principal value, wherein cosine is positive.

The given expression is

$\cos a = \sqrt{1 - {\sin}^{2} a} = \sqrt{1 - \frac{1}{9}} = \frac{2 \sqrt{2}}{3}$.

If Q2 value of $a = {180}^{o} - Q 1 a$ is permitted, the answer will be

$\pm \left(2 \sqrt{2}\right) / 3$,

using

$\sin \left({180}^{o} - a\right) = \sin a$ but $\cos \left({180}^{o} - a\right) = - \cos a$.