# 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 72, what is the area of the triangle?

Jun 3, 2018

Area of the triangle ${A}_{t} = \textcolor{c r i m s o n}{748.32}$

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \left(7 \pi\right) : 12 , \hat{B} = \frac{\pi}{3} , b = 72$

Law of Sines $\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C$

$A r e a = {A}_{t} = \left(\frac{1}{2}\right) a b \sin C$

$a = \frac{b \sin A}{\sin} B = \frac{72 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{3}\right) = 21.52$

${A}_{t} = \left(\frac{1}{2}\right) \cdot 21.52 \cdot 72 \cdot \sin \left(\frac{7 \pi}{12}\right) = \textcolor{c r i m s o n}{748.32}$