# A triangle has sides A, B, and C. The angle between sides A and B is (3pi)/4 and the angle between sides B and C is pi/12. If side B has a length of 4, what is the area of the triangle?

Sep 3, 2016

$S = 4 \sqrt{4 - \sqrt{6}}$

#### Explanation:

The angle between A and C is $\pi - 3 \frac{\pi}{4} - \frac{\pi}{12} = \frac{\pi}{6}$
Acording theorem of sin length C: $\frac{C}{\sin} \left(3 \frac{\pi}{4}\right) = \frac{4}{\sin} \left(\frac{\pi}{6}\right)$
$C = 4 \sqrt{2}$
Area of the triangle: $S = B C \sin \frac{\frac{\pi}{12}}{2}$
$\sin \left(\frac{\pi}{12}\right) = \sqrt{\frac{1 - \cos \left(\frac{\pi}{6}\right)}{2}}$
$\sin \left(\frac{\pi}{12}\right) = \frac{\sqrt{2 - \sqrt{3}}}{2}$