# A triangle has sides A, B, and C. The angle between sides A and B is pi/3 and the angle between sides B and C is pi/12. If side B has a length of 5, what is the area of the triangle?

May 19, 2016

Area of triangle is $2.90$

#### Explanation:

The third angle opposite sides $A$ and $C$ is

$\pi - \frac{\pi}{3} - \frac{\pi}{12} = \left(12 - 4 - 1\right) \frac{\pi}{12} = 5 \frac{\pi}{12}$ and it has side $B = 5$ opposite it.

As side $A$ has angle opposite it $\frac{\pi}{12}$, using sine formula, we get

$\frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{B}{\sin} \left(\frac{5 \pi}{12}\right)$ or

$A = B \times \sin \frac{\frac{\pi}{12}}{\sin} \left(\frac{5 \pi}{12}\right) = 5 \times \frac{0.2588}{0.9659} = 1.34$

Hence area of triangle is $\frac{1}{2} \times 5 \times 1.34 \times \sin \left(\frac{\pi}{3}\right)$

= $\frac{1}{2} \times 5 \times 1.34 \times 0.866 = 2.90$