# A triangle has sides A, B, and C. Sides A and B have lengths of 6 and 3, respectively. The angle between A and C is (13pi)/24 and the angle between B and C is  (3pi)/8. What is the area of the triangle?

Mar 30, 2017

The area of the triangle is $2.33 \left(2 \mathrm{dp}\right)$ sq.unit.

#### Explanation:

The angle between sides $A \mathmr{and} C$ is $\angle b = \frac{13 \pi}{24} = \frac{13 \cdot 180}{24} = {97.5}^{0}$
The angle between sides $B \mathmr{and} C$ is $\angle a = \frac{3 \pi}{8} = \frac{3 \cdot 180}{8} = {67.5}^{0}$

The angle between sides $A \mathmr{and} B$ is $\angle c = 180 - \left(97.5 + 67.5\right) = {15}^{0}$

We know sides $A = 6 , B = 3$ and their included angle $\angle c = {15}^{0}$

The area of the triangle is ${A}_{t} = \frac{1}{2} \cdot A \cdot B \cdot \sin c = \frac{1}{2} \cdot 6 \cdot 3 \cdot \sin 15 = 9 \cdot \sin 15 \approx 2.33 \left(2 \mathrm{dp}\right)$sq.unit [Ans]