# 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 1, what is the area of the triangle?

Feb 15, 2018

Area of right Delta ABC = color(green)(0.125

#### Explanation:

Given $\hat{A} = \frac{\pi}{12} , b = 1 , \hat{C} = \frac{5 \pi}{13}$

Third angle $\hat{B} = \pi - \frac{5 p u}{12} - \frac{\pi}{12} = \frac{\pi}{2}$

Hence it’s a right triangle.

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

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{1}{\sin} \left(\frac{\pi}{2}\right) = \frac{c}{\sin} \left(\frac{5 \pi}{12}\right) = 1$, since $\sin \left(\frac{\pi}{2}\right) = 1$

$a = \sin \left(\frac{\pi}{12}\right) = 0.2588$

$c = \sin \left(\frac{5 \pi}{12}\right) = 0.9659$

Area of right Delta ABC = (1/2) a c = (1/2) * 0.2588 * 0.9659 = color(green)(0.125