AAS Theorem

$\begin{figure}\begin{center}\BoxedEPSF{AASTheorem.epsf}\end{center}\end{figure}$

Specifying two angles and and a side uniquely determines a Triangle with Area

$\begin{displaymath} K = {a^2\sin B\sin C\over 2\sin A} = {a^2\sin B\sin(\pi-A-B)\over 2\sin A}. \end{displaymath}$

(1)

The third angle is given by

$\begin{displaymath} C=\pi-A-B, \end{displaymath}$

(2)

since the sum of angles of a Triangle is 180° ( $\pi$ Radians). Solving the Law of Sines

$\begin{displaymath} {a\over\sin A}={b\over\sin B} \end{displaymath}$

(3)

for

gives

$\begin{displaymath} b=a{\sin B\over\sin A}. \end{displaymath}$

(4)

Finally,

$\displaystyle c$	$\textstyle =$	$\displaystyle b\cos A+a\cos B=a(\sin B\cot A+\cos B)$	(5)
	$\textstyle =$	$\displaystyle a\sin B(\cot A+\cot B).$	(6)