# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/2, the angle between sides B and C is (5pi)/12, and the length of B is 15, what is the area of the triangle?

Jun 3, 2018

Area of triangle A_t = color(violet)(419.85 sq units

#### Explanation:

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

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

Given : $b = 15 , \hat{C} = \frac{\pi}{2} , \hat{A} = \frac{5 \pi}{12} , \hat{B} = \pi - \frac{\pi}{2} - \frac{5 \pi}{12} = \frac{\pi}{12}$

It’s a right triangle and hence ${A}_{t} = \left(\frac{1}{2}\right) a b$ as $\sin C = \sin \left(\frac{\pi}{2}\right) = 1$

$a = \frac{b \sin A}{\sin} B = \frac{15 \cdot \sin \left(\frac{5 \pi}{12}\right)}{\sin} \left(\frac{\pi}{12}\right) = 55.98$

A_t = (1/2) * 55.98 * 15 = color (violet)(419.85