# How do you do simplify sin(cos^(-1)(1/5))?

Nov 17, 2015

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

#### Explanation:

Consider a right angled triangle with hypotenuse $5$ and one leg $1$. Then the other leg will have length $\sqrt{{5}^{2} - {1}^{2}} = \sqrt{24} = 2 \sqrt{6}$.

So if $\cos \left(\theta\right) = \frac{1}{5}$ then $\sin \left(\theta\right) = \frac{2 \sqrt{6}}{5}$

So:

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

${\cos}^{2} \left(\theta\right) + {\sin}^{2} \left(\theta\right) = 1$
Let $\theta = {\cos}^{- 1} \left(\frac{1}{5}\right)$
Then $\cos \left(\theta\right) = \frac{1}{5}$ and we find:
sin(theta) = sqrt(1-cos^2(theta)) = sqrt(1-(1/5)^2)) = sqrt(24/25) = (2sqrt(6))/5