# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 16, what is the area of the triangle?

Jun 30, 2017

The area of the triangle is $= 32 {u}^{2}$

#### Explanation:

The angle between side $A$ and $C$ is

$= \pi - \left(\frac{5}{12} \pi + \frac{1}{12} \pi\right) = \frac{6}{12} \pi = \frac{1}{2} \pi$

This is a right angle triangle.

$\sin \left(\frac{1}{2} \pi\right) = 1$

We apply the sine rule

$\frac{16}{\sin} \left(\frac{1}{2} \pi\right) = \frac{A}{\sin} \left(\frac{1}{12} \pi\right)$

$A = 16 \sin \left(\frac{1}{12} \pi\right)$

$\frac{16}{1} = \frac{C}{\sin} \left(\frac{5}{12} \pi\right)$

$C = 16 \sin \left(\frac{5}{12} \pi\right)$

The area of the triangle is

$= \frac{1}{2} A C \sin \left(\frac{1}{2} \pi\right) = \frac{1}{2} \cdot 16 \sin \left(\frac{1}{12} \pi\right) \cdot 16 \sin \left(\frac{5}{12} \pi\right)$

$= 128 \cdot 0.25$

$= 32$