# A triangle has sides A,B, and C. If the angle between sides A and B is (3pi)/4, the angle between sides B and C is pi/12, and the length of B is 9, what is the area of the triangle?

May 24, 2018

Area of triangle color(maroon)(A_t = 14.83 sq units

#### Explanation:

Given : $\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{3 \pi}{4} , b = 9$

$\hat{B} = \pi - \frac{\pi}{12} - \frac{3 \pi}{4} = \frac{\pi}{6}$

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

$a = \frac{b \sin A}{\sin} B = \frac{9 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{6}\right)$

$a = 4.66$

Area of triangle ${A}_{t} = \left(\frac{1}{2}\right) a b \sin C$

${A}_{t} = \left(\frac{1}{2}\right) \cdot 4.66 \cdot 9 \cdot \sin \left(\frac{3 \pi}{4}\right)$

${A}_{t} = 14.83$ sq units