5.0 Trigonometric Functions of Half Angles

$$\begin{equation} \begin{aligned} (i)\sin \frac{A}{2} = \sqrt {\frac{{(s - b)(s - c)}}{{bc}}} ,\sin \frac{B}{2} = \sqrt {\frac{{(s - c)(s - a)}}{{ca}}} ,\sin \frac{C}{2} = \sqrt {\frac{{(s - a)(s - b)}}{{ab}}} \\ (ii)\cos \frac{A}{2} = \sqrt {\frac{{s(s - a)}}{{bc}}} ,\cos \frac{B}{2} = \sqrt {\frac{{s(s - b)}}{{ca}}} ,\cos \frac{C}{2} = \sqrt {\frac{{s(s - c)}}{{ab}}} \\ (iii)\tan \frac{A}{2} = \sqrt {\frac{{(s - b)(s - c)}}{{s(s - a)}}} = \frac{\Delta }{{s(s - a)}} = \frac{{(s - b)(s - c)}}{\Delta } \\\end{aligned} \end{equation} $$ where $s = \frac{{a + b + c}}{2}$ is the perimeter of triangle and $\Delta $ is the area of triangle.

$$(iv)\sin A = \frac{2}{{bc}}\sqrt {s(s - a)(s - b)(s - c)} = \frac{{2\Delta }}{{bc}}$$