# What is cos (arcsin (5/13))?

Jul 21, 2015

$\frac{12}{13}$

#### Explanation:

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

$\epsilon$ simply represents an angle.

This means that we are looking for color(red)cos(epsilon)!

If $\epsilon = \arcsin \left(\frac{5}{13}\right)$ then,

$\implies \sin \left(\epsilon\right) = \frac{5}{13}$

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

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

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

Dec 6, 2015

$\frac{12}{13}$

#### Explanation:

First, see $\arcsin \left(\frac{5}{13}\right)$. This represents the ANGLE where $\sin = \frac{5}{13}$.

That is represented by this triangle:

Now that we have the triangle that $\arcsin \left(\frac{5}{13}\right)$ is describing, we want to figure out $\cos \theta$. The cosine will be equal to the adjacent side divided by the hypotenuse, $15$.

Use the Pythagorean Theorem to determine that the adjacent side's length is $12$, so $\cos \left(\arcsin \left(\frac{5}{13}\right)\right) = \frac{12}{13}$.