# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/3, the angle between sides B and C is (7pi)/12, and the length of B is 7, what is the area of the triangle?

Aug 7, 2016

$= 79.16$

#### Explanation:

This is a triangle where side $B = 7$ is opposite of the
Angle [pi-(pi/3+7pi/12)]=pi-11pi/12=pi/12
Therefore
$\frac{B}{\sin} \left(\frac{\pi}{12}\right) = \frac{C}{\sin} \left(\frac{\pi}{3}\right)$
or
$\frac{7}{\sin} \left(\frac{\pi}{12}\right) = \frac{C}{\sin} \left(\frac{\pi}{3}\right)$
or
$C = 7 \sin \frac{\frac{\pi}{3}}{\sin \frac{\pi}{12}}$
or
$C = 7 \left(3.34\right)$
or
$C = 23.42$
$H e i g h t$ of the triangle is $= 7 \sin \left(\pi - 7 \frac{\pi}{12}\right) = 7 \sin \left(5 \frac{\pi}{12}\right) = 6.76$
Therefore
Area of the triangle$= \frac{1}{2} \left(C\right) \left(H e i g h t\right)$
$= \frac{1}{2} \left(23.42\right) \left(6.76\right)$
$= 79.16$