# How do you use the half angle identity to find exact value of Sin (3pi/8)?

Oct 5, 2015

Find $\sin \left(\frac{3 \pi}{8}\right)$

Ans: $\frac{\sqrt{2 + \sqrt{2}}}{2}$

#### Explanation:

This is the popular way to solve this kind of math question.
Call $\sin a = \sin \left(\frac{3 \pi}{8}\right)$
$\cos 2 a = \cos \left(\frac{6 \pi}{8}\right) = \cos \left(\frac{3 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$
Apply the trig identity: $\cos 2 a = 1 - 2 {\sin}^{2} a$
$- \frac{\sqrt{2}}{2} = 1 - 2 {\sin}^{2} a$
$2 {\sin}^{2} a = 1 + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2}$
${\sin}^{2} a = \frac{2 + \sqrt{2}}{4}$
$\sin a = \sin \left(\frac{3 \pi}{8}\right) = \pm \frac{\sqrt{2 + \sqrt{2}}}{2.}$
Since the arc $\left(\frac{3 \pi}{8}\right)$ is located in Quadrant I, its sin is positive, then
$\sin \left(\frac{3 \pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}$
Check by calculator.
$\sin \left(\frac{3 \pi}{8}\right) = \sin 67.5 \mathrm{de} g = 0.92$
$\frac{\sqrt{2 + \sqrt{2}}}{2} = \frac{1.84}{2} = 0.92$. OK