# Use a double or half angle formula to determine value of cos(13π/12)?

May 5, 2018

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

#### Explanation:

$\cos \left(\frac{13 \pi}{12}\right) = \cos \left(\frac{\pi}{12} + \pi\right) = - \cos \left(\frac{\pi}{12}\right)$
Find $\cos \left(\frac{\pi}{12}\right)$ by using trig identity:
$2 {\cos}^{2} a = 1 + \cos 2 a$.
In this case:
$2 {\cos}^{2} \left(\frac{\pi}{12}\right) = 1 + \cos \left(\frac{\pi}{6}\right) = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$
${\cos}^{2} \left(\frac{\pi}{12}\right) = \frac{2 + \sqrt{3}}{4}$
$\cos \left(\frac{\pi}{12}\right) = \frac{2 + \sqrt{3}}{2}$ --> ($\cos \left(\frac{\pi}{12}\right)$ is positive).
Finally,
$\cos \left(\frac{13 \pi}{12}\right) = - \cos \left(\frac{\pi}{12}\right) = - \frac{2 + \sqrt{3}}{2}$