# 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 15, what is the area of the triangle?

Aug 10, 2018

color(maroon)(A_t = 1/2 a b sin C ~~ 35.66 sq. units

#### Explanation:

$\hat{A} = \frac{\pi}{4} , \hat{C} = \frac{\pi}{12} , \hat{B} = \pi - \frac{\pi}{4} - \frac{\pi}{12} = \frac{2 \pi}{3} , b = 15$

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

$a = \frac{b \sin A}{\sin} B = \frac{15 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{4}\right) \approx 5.49$

Area of $\Delta = {A}_{t} = \frac{1}{2} a b \sin C$

color(maroon)(A_t = 1/2 * 5.49 * 15 sin ((2pi)/3) ~~ 35.66 sq. units