# How do you find the exact value of cos( 2 arctan (5/12) )?

Jun 8, 2016

$\frac{119}{169}$

#### Explanation:

Use the cosine double angle formula:

$\cos \left(2 x\right) = 2 {\cos}^{2} \left(x\right) - 1$

Here, inside the cosine function, the angle $\arctan \left(\frac{5}{12}\right)$ is being doubled, so we see that

$\cos \left(2 \arctan \left(\frac{5}{12}\right)\right) = 2 {\cos}^{2} \left(\arctan \left(\frac{5}{12}\right)\right) - 1$

To find $\cos \left(\arctan \left(\frac{5}{12}\right)\right)$, draw a picture where $\tan \left(\beta\right) = \frac{5}{12}$, or where $5$ is the opposite side's length and $12$ is the adjacent side's length.

In this triangle, $13$ is the hypotenuse (through the Pythagorean Theorem), and $\cos \left(\beta\right) = \frac{12}{13}$. This also means that $\cos \left(\arctan \left(\frac{5}{12}\right)\right) = \frac{12}{13}$.

So we see that

$2 {\cos}^{2} \left(\arctan \left(\frac{5}{12}\right)\right) - 1 = 2 {\left(\frac{12}{13}\right)}^{2} - 1 = \frac{288}{169} - 1 = \frac{119}{169}$