# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 36, what is the area of the triangle?

Apr 10, 2018

:.color(brown)(A_t = (1/2) * a * c = (1/2) * 9.32 * 34.77 = 162.06 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{5 \pi}{12} , \hat{B} = \pi - \frac{\pi}{12} - \frac{5 \pi}{12} = \frac{\pi}{2} , b = 36$

It's a right triangle with $\hat{B} = {90}^{\circ}$

As per the Law of Sines,

$\textcolor{p u r p \le}{\frac{a}{\sin} A = \frac{b}{\sin} B = \frac{c}{\sin} C}$

$\therefore a = \frac{b \cdot \sin A}{\sin} B = \frac{36 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{2}\right) = 9.32$

$\therefore c = \frac{b \cdot \sin C}{\sin} B = \frac{36 \cdot \sin \left(\frac{5 \pi}{12}\right)}{\sin} \left(\frac{\pi}{2}\right) = 34.77$

:.color(brown)(A_t = (1/2) * a * c = (1/2) * 9.32 * 34.77 = 162.06 " sq units"