# How do you find the value of cos ((3pi)/8) using the double or half angle formula?

Jul 2, 2016

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

#### Explanation:

Trig table, unit circle, and property of complementary arcs -->
$\cos \left(\frac{3 \pi}{8}\right) = \cos \left(- \frac{\pi}{8} + \left(4 \frac{\pi}{8}\right)\right) = \cos \left(- \frac{\pi}{8} + \frac{\pi}{2}\right) =$
$= \sin \left(\frac{\pi}{8}\right) .$
Find sin (pi/8) by using trig identity:
$\cos 2 a = 1 - 2 {\sin}^{2} a$
$\cos \left(\frac{2 \pi}{8}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} = 1 - 2 {\sin}^{2} \left(\frac{\pi}{8}\right)$
$2 {\sin}^{2} \left(\frac{\pi}{8}\right) = 1 - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2}$
${\sin}^{2} \left(\frac{\pi}{8}\right) = \frac{2 - \sqrt{2}}{4}$
$\sin \left(\frac{\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}$ (because $\sin \left(\frac{\pi}{8}\right)$ is positive.
Finally,
$\cos \left(\frac{3 \pi}{8}\right) = \sin \left(\frac{\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}$