# A triangle has sides A, B, and C. The angle between sides A and B is (7pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 6, what is the area of the triangle?

Mar 16, 2018

color(red)(A_t ~~ 5.2 sq units

#### Explanation:

Given : $\hat{C} = \frac{7 \pi}{12} , \hat{A} = \frac{\pi}{12} , \hat{B} = \pi - \frac{7 \pi}{12} - \frac{\pi}{12} = \frac{\pi}{3} , b = 6$

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{6}{\sin} \left(\frac{\pi}{3}\right) = \frac{c}{\sin} \left(\frac{7 \pi}{12}\right)$

$a = \frac{6 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{3}\right) \approx 1.79$

Area ${A}_{t} = \left(\frac{1}{2}\right) a b \sin C = \left(\frac{1}{2}\right) \cdot 1.79 \cdot 6 \sin \left(\frac{7 \pi}{12}\right)$

${A}_{t} \approx 5.2$ sq units