# 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/12. If side B has a length of 10, what is the area of the triangle?

Mar 27, 2018

color(brown)(A_t = (1/2) * a * b * sin C = 10.6 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{\pi}{4} , b = 10$

$\hat{B} = \pi - \frac{\pi}{12} - \frac{\pi}{4} = \frac{2 \pi}{3}$

Applying the law of sines,

$a = \frac{b \cdot \sin A}{\sin} B = \frac{10 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{2 \pi}{3}\right) \approx 3 \text{ units}$

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

color(brown)(A_t = (1/2) * 3 * 10 * sin (pi/4) = 10.6 " sq units"