What is sin(arccos(5/13))?

Jul 21, 2015

$\frac{12}{13}$

Explanation:

First consider that : $\theta = \arccos \left(\frac{5}{13}\right)$

$\theta$ just represents an angle.

This means that we are looking for color(red)sin(theta)!

If $\theta = \arccos \left(\frac{5}{13}\right)$ then,

$\implies \cos \left(\theta\right) = \frac{5}{13}$

To find $\sin \left(\theta\right)$ We use the identity : ${\sin}^{2} \left(\theta\right) = 1 - {\cos}^{2} \left(\theta\right)$

=>sin(theta)=sqrt(1-cos^2(theta)

$\implies \sin \left(\theta\right) = \sqrt{1 - {\left(\frac{5}{13}\right)}^{2}} = \sqrt{\frac{169 - 25}{169}} = \sqrt{\frac{144}{169}} = \textcolor{b l u e}{\frac{12}{13}}$