# A triangle has sides A, B, and C. Sides A and B have lengths of 10 and 8, respectively. The angle between A and C is (13pi)/24 and the angle between B and C is  (pi)/24. What is the area of the triangle?

Jun 29, 2016

Area of $\Delta A B C = 38.636$ sq. unit.

#### Explanation:

Let us denote by $\angle \left(A , C\right)$ the angle btwn. sides A & C.

Then, by what is given, we find, $\angle \left(A , C\right) + \angle \left(B , C\right) = \frac{13 \pi}{24} + \frac{\pi}{24} = \frac{14 \pi}{24} = \frac{7 \pi}{12.}$

Hence, $\angle \left(A , B\right) = \pi - \frac{7 \pi}{12} = \frac{5 \pi}{12.}$

Now by Formula from Trigo.,
Area of $\Delta A B C = \frac{1}{2} \cdot A \cdot B \cdot \sin \angle \left(A , B\right) = \frac{1}{2} \cdot 10 \cdot 8 \cdot \sin \left(\frac{5 \pi}{12}\right) = 40 \cdot \sin {75}^{o} = 40 \left(0.9659\right) = 38.636$ sq. unit.