# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/4 and the angle between sides B and C is pi/6. If side B has a length of 8, what is the area of the triangle?

Mar 27, 2018

color(green)(Delta " " A_t = (1/2) a b sin C = 11.71 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{6} , \hat{C} = \frac{\pi}{4} , b = 8$

$\hat{B} = \pi - \frac{\pi}{6} - \frac{\pi}{4} = \frac{7 \pi}{12}$

Applying the Law of Sines,

$a = \frac{b \sin A}{\sin} B = \frac{8 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{7 \pi}{12}\right) = 4.14 \text{ units}$

color(green)(Delta " " A_t = (1/2) a b sin C = (1/2) * 4.14 * 8 * sin (pi/4) = 11.71 " sq units"