## Half/Double/Multiple angle formula

### Half angle formula:

$$1.\ \sin \frac A2 = \pm \sqrt \frac{(1- \cos A)}{2}$$
$$2.\ \cos \frac A2 = \pm \sqrt \frac{(1+ \cos A)}{2}$$
$$3.\ \tan \frac A2 = \pm \sqrt \frac{(1- \cos A)}{(1+ \cos A)}= \frac{(1- \cos A)}{\sin A} = \frac {\sin A}{(1+ \cos A)} = \text{cosec} \ A – \cot A$$
$$4.\ \cot \frac A2 = \pm \sqrt \frac{(1+ \cos A)}{(1- \cos A)}= \frac{(1+ \cos A)}{\sin A} = \frac {\sin A}{(1- \cos A)} = \text{cosec} \ A + \cot A$$

### Solution:

$\mathrm{sin}3A=3\left(\frac{3}{5}\right)–4{\left(\frac{3}{5}\right)}^{3}$
$=\frac{9}{5}–4\left(\frac{27}{125}\right)$
$=\frac{9}{5}–\frac{108}{125}$
$=\frac{9}{5}×\frac{25}{25}–\frac{108}{125}$
$=\frac{225}{125}–\frac{108}{125}$
$=\frac{\left(225–108\right)}{125}$
$=\frac{117}{125}$

### Solution:

$$\sin \frac A2 = \pm \sqrt \frac{(1- \cos A)}{2}$$
$$\sin \frac {60}{2} = \pm \sqrt \frac{(1- \cos 60^\circ)}{2}$$
$$\sin 30^\circ = \pm \sqrt \frac{1- \frac 12}{2}$$
$$\sin 30^\circ = \pm \sqrt \frac 14$$
$\mathrm{sin}30°=0.5$