# 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 12, what is the area of the triangle?

Feb 2, 2017

Area = 18 units

#### Explanation:

The given triangle is sketched as in the figure above. Angle B would be $\pi - \frac{\pi}{12} - \frac{5 \pi}{12} = \frac{\pi}{2}$

Side b is 12. It is the hypotenuse because it is opposite the right angle B.

For area, base and altitude is required. In this case it is side 'a' and side 'c'.

Side 'c' = $12 \cos \left(\frac{\pi}{12}\right)$

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

Area = $\frac{1}{2} \left(12 \sin \left(\frac{\pi}{12}\right) \cdot 12 \cos \left(\frac{\pi}{12}\right)\right)$
=$72 \sin \left(\frac{\pi}{12}\right) \cos \left(\frac{\pi}{12}\right)$ =$36 \left(2 \sin \left(\frac{\pi}{12}\right) \cos \left(\frac{\pi}{12}\right)\right)$= 36 sin pi/6= 36$\frac{1}{2}$= 18

Area = 18 units.