# A triangle has sides A, B, and C. If the angle between sides A and B is (3pi)/8, the angle between sides B and C is (pi)/3, and the length of B is 6, what is the area of the triangle?

Jun 28, 2018

color(crimson)(A_t = (1/2) a b sin C ~~ 18.55 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{3} , \hat{C} = \frac{3 \pi}{8} , \hat{C} = \pi - \frac{\pi}{3} - \frac{3 \pi}{8} = \frac{7 \pi}{24} , b = 6$

Applying the Law of Sines,

$a = \frac{b \sin A}{\sin} B = \frac{6 \cdot \sin \left(\frac{\pi}{3}\right)}{\sin} \left(\frac{7 \pi}{24}\right) \approx 6.55$

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

color(crimson)(A_t = (1/2) 6.55 * 6 * sin ((3pi) / 8) ~~ 18.55 " sq units"