# A triangle has sides A, B, and C. The angle between sides A and B is (2pi)/3 and the angle between sides B and C is pi/12. If side B has a length of 2, what is the area of the triangle?

Jul 24, 2017

The area of the triangle is $= 0.63 {u}^{2}$

#### Explanation:

The angles are

$\hat{C} = \frac{2}{3} \pi$

$\hat{A} = \frac{1}{12} \pi$

Therefore,

$\hat{B} = \pi - \left(\frac{2}{3} \pi + \frac{1}{12} \pi\right) = \pi - \left(\frac{8}{12} \pi + \frac{1}{12} \pi\right) = \frac{3}{12} \pi = \frac{1}{4} \pi$

The side $b = 2$

We apply the sine rule to the triangle $\Delta A B C$

$\frac{a}{\sin} \hat{A} = \frac{b}{\sin} \hat{B}$

$\frac{a}{\sin} \left(\frac{1}{12} \pi\right) = \frac{2}{\sin} \left(\frac{1}{4} \pi\right)$

$a = 2 \sin \frac{\frac{1}{12} \pi}{\sin} \left(\frac{1}{4} \pi\right) = 0.73$

The area of the triangle is

$a r e a = \frac{1}{2} a b \sin \hat{C}$

$= \frac{1}{2} \cdot 2 \cdot 0.73 \cdot \sin \left(\frac{2}{3} \pi\right)$

$= 0.63 {u}^{2}$