How do you use the half angle identity to find cos 105?

Nov 6, 2014

Half Angle Formula

$\cos \left(\setminus \frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

First, since ${105}^{\circ}$ is the 2nd quadrant, cosine is negative, so by the half angle formula above,

$\cos \left({105}^{\circ}\right) = \cos \left(\frac{{210}^{\circ}}{2}\right) = - \sqrt{\frac{1 + \cos \left({210}^{\circ}\right)}{2}}$

by $\cos \left({210}^{\circ}\right) = - \frac{\sqrt{3}}{2}$,

$= - \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = - \sqrt{\frac{2 - \sqrt{3}}{4}} = - \frac{\sqrt{2 - \sqrt{3}}}{2}$

I hope that this was helpful.