# A triangle has sides A, B, and C. The angle between sides A and B is (5pi)/6 and the angle between sides B and C is pi/12. If side B has a length of 7, what is the area of the triangle?

The Area is $12.25$ square units
The angle between sides A and B $\angle C = 5 \cdot \frac{180}{6} = {150}^{0}$
The angle between sides B and C $\angle A = \frac{180}{12} = {15}^{0}$
The angle between sides C and A $\angle B = 180 - \left(150 + 15\right) = {15}^{0}$
By sine law , we know $\frac{A}{\sin} A = \frac{B}{\sin} B = \frac{C}{\sin} C \therefore A = S \in \frac{A}{\sin} B \cdot B = 7 \cdot \sin \frac{15}{\sin} 15 = 7$ Now  A=7 ; B=7 and their included angle $\angle C = {150}^{0} \therefore$The Area $= A \cdot B \cdot \sin \frac{C}{2} = 7 \cdot 7 \cdot \sin \frac{150}{2} = 12.25$ square units[Ans}