# 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 6, what is the area of the triangle?

Feb 17, 2018

Area of right triangle A_t = color (red)(4.5

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{5 \pi}{12} , b = 6$

$\hat{B} = \pi - \frac{\pi}{12} - \frac{5 \pi}{12} = \frac{\pi}{2}$

It’s a right triangle.

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

$a = \frac{6 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{2}\right) = 1.5529$

$c = 6 \cdot \sin \left(\frac{5 \pi}{12}\right) = 5.7956$

Area of right triangle A_t = (1/2) a c = (1/2) * 1.5529 * 5.7956 = 4.5#