# 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 3, what is the area of the triangle?

Jul 10, 2017

The area is $= 1.04 {u}^{2}$

#### Explanation:

The third angle of the triangle is

$= \pi - \left(\frac{\pi}{3} + \frac{\pi}{12}\right) = \pi - \frac{5}{12} \pi = \frac{7}{12} \pi$

We apply the sine rule to the triangle

$\frac{A}{\sin} \left(\frac{1}{12} \pi\right) = \frac{B}{\sin} \left(\frac{7}{12} \pi\right)$

$\frac{A}{\sin} \left(\frac{1}{12} \pi\right) = \frac{3}{\sin} \left(\frac{7}{12} \pi\right) = 3.11$

$A = 3.11 \cdot \sin \left(\frac{1}{12} \pi\right) = 0.8$

The area of the triangle is

$= \frac{1}{2} A B \sin \left(\frac{1}{3} \pi\right) = 0.5 \cdot 0.8 \cdot 3 \cdot \sin \left(\frac{1}{3} \pi\right) = 1.04$