how to evaluate trig functions with double and half angle identities

the double and half angle trig identities are useful in evaluating more stuff

double angle identities

these are derived from the angle sum identities

$\sin (2a)=2\cdot \sin (a)\cdot \cos (a)$

$\cos (2a)=2\cdot {\cos }^2(a)-1$

$\tan (2a)=\frac{2\tan (a)}{1-{\tan }^2(a)}$

half angle identities

these are derived from the double angle identities

$\sin (\frac{a}{2})=\pm \sqrt{\frac{1-\cos (a)}{2}}$

$\cos (\frac{a}{2})=\pm \sqrt{\frac{1+\cos (a)}{2}}$

$\tan (\frac{a}{2})=\frac{\sin (a)}{1+\cos (a)}$

worked example: given $\cos (a)=\frac{1}{6}$, find $\sin (2a)$
$\sin (2a)=2\cdot \sin (a)\cdot \cos (a)$
use a triangle to use $\cos (a)=\frac{1}{6}$ to find $\sin (a)$