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

Apr 10, 2018

:.color(brown)(A_t = (1/2) * 4.39 * 12 * sin ((2pi)/3) = 22.81 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{2 \pi}{3} , \hat{B} = \pi - \frac{\pi}{12} - \frac{2 \pi}{3} = \frac{\pi}{4} , b = 12$

As per the Law of Sines,

$\textcolor{p u r p \le}{\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C}$

$\therefore a = \frac{b \cdot \sin A}{\sin} B = \frac{12 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{4}\right) = 4.39$

color(indigo)("Area of " Delta " " A_t = (1/2) * a * b * sin C

:.color(brown)(A_t = (1/2) * 4.39 * 12 * sin ((2pi)/3) = 22.81 " sq units"