# How do you find the exact values of sin(u/2), cos(u/2), tan(u/2) using the half angle formulas given cosu=3/5, 0<u<pi/2?

Jan 9, 2017

$\sin \left(\frac{u}{2}\right) = \frac{\sqrt{5}}{3}$
$\cos \left(\frac{u}{2}\right) = \frac{2 \sqrt{5}}{5}$
$\tan \left(\frac{u}{2}\right) = \frac{1}{2}$

#### Explanation:

Use trig identities:
$2 {\cos}^{2} \left(\frac{u}{2}\right) = 1 + \cos 2 u$
$2 {\sin}^{2} \left(\frac{u}{2}\right) = 1 - \cos 2 u$
In this case:
$2 {\cos}^{2} \left(\frac{u}{2}\right) = 1 + \frac{3}{5} = \frac{8}{5}$
${\cos}^{2} \left(\frac{u}{2}\right) = \frac{8}{10} = \frac{4}{5}$
$\cos \left(\frac{u}{2}\right) = \pm \frac{2}{\sqrt{5}} = \pm \frac{2 \sqrt{5}}{5}$
Since u is in Quadrant I, then $\cos \left(\frac{u}{2}\right)$ is positive.
$2 {\sin}^{2} \left(\frac{u}{2}\right) = 1 - \frac{3}{5} = \frac{2}{5}$
${\sin}^{2} \left(\frac{u}{2}\right) = \frac{2}{10} = \frac{1}{5}$
$\sin \left(\frac{u}{2}\right) = \pm \frac{1}{\sqrt{5}} = \pm \frac{\sqrt{5}}{5}$
Since u is in Quadrant I, then $\sin \left(\frac{u}{2}\right)$ is positive.
$\tan \left(\frac{u}{2}\right) = \frac{\sin}{\cos} = \left(\frac{\sqrt{5}}{5}\right) \left(\frac{5}{2 \sqrt{5}}\right) = \frac{1}{2}$