# 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 21, what is the area of the triangle?

Mar 24, 2018

color(brown)(" Area of Triangle " A_t = (1/2) a c = 55.16 "sq units"

#### Explanation:

$\hat{C} = \frac{5 \pi}{12} , \hat{A} = \frac{\pi}{12} , \hat{B} = \pi - \left(\frac{5 \pi}{12} + \frac{\pi}{12}\right) = \frac{\pi}{2} , b = 21$

According to Law of Sines,

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{21}{\sin} \left(\frac{\pi}{2}\right) = \frac{c}{\sin} \left(\frac{5 \pi}{12}\right)$

$a = 21 \sin \left(\frac{\pi}{12}\right) = 5.44$

$c = 21 \sin \left(\frac{5 \pi}{12}\right) = 20.28$

Area of Triangle " A_t = (1/2) a c = (1/2) * 5.44 * 20.28 = 55.16 "sq units"#