# How do you find the exact values of sin(u/2), cos(u/2), tan(u/2) using the half angle formulas given sinu=5/13, pi/2<u<pi?

Jul 7, 2017

$\sin u = \frac{5}{13}$. First, find cos u.
${\cos}^{2} u = 1 - {\sin}^{2} u = 1 - \frac{25}{169} = \frac{144}{169}$ --> $\cos u = \pm \frac{12}{13}$.
Since u is in Quadrant 2, then, cos u < 0.
$\cos u = - \frac{12}{13}$.
To find $\cos \left(\frac{u}{2}\right)$, apply trig identity:
$2 {\cos}^{2} \left(\frac{u}{2}\right) = 1 + \cos u = 1 - \frac{12}{13} = \frac{1}{13}$
${\cos}^{2} \left(\frac{u}{2}\right) = \frac{1}{26}$ --> $\cos \left(\frac{u}{2}\right) = \pm \frac{1}{\sqrt{26}}$.
Sin u is in Quadrant 2, then $\frac{u}{2}$ is in Quadrant 1, and $\cos \left(\frac{u}{2}\right)$ is positive:
$\cos \left(\frac{u}{2}\right) = \frac{1}{\sqrt{26}} = \frac{\sqrt{26}}{26}$
To find $\sin \left(\frac{u}{2}\right)$, apply the trig identity:
$\sin u = 2 \sin \left(\frac{u}{2}\right) . \cos \left(\frac{u}{2}\right)$
$\sin \left(\frac{u}{2}\right) = \frac{\sin u}{2 \cos \left(\frac{u}{2}\right)} = \left(\frac{5}{13}\right) \left(\frac{\sqrt{26}}{2}\right) = \frac{5 \sqrt{26}}{26}$
$\tan \left(\frac{u}{2}\right) = \frac{\sin \left(\frac{u}{2}\right)}{\cos \left(\frac{u}{2}\right)} = \left(\frac{5 \sqrt{26}}{26}\right) \left(\sqrt{26}\right) = 5$