# A triangle has sides A, B, and C. Sides A and B have lengths of 12 and 7, respectively. The angle between A and C is (pi)/4 and the angle between B and C is  (2pi)/3. What is the area of the triangle?

Area of the triangle is $10.87 \left(2 \mathrm{dp}\right) u n i {t}^{2}$
The angle between sides A and C is $\angle b = \frac{\pi}{4} = \frac{180}{4} = {45}^{0}$
The angle between sides B and C is $\angle a = \frac{2 \pi}{3} = \frac{2 \cdot 180}{3} = {120}^{0}$
The angle between sides A and B is /_c=180-((45+120)=15^0
Now we know sides A=12 ; B=7 and their included angle $\angle c = {15}^{0}$
So area of the triangle is ${A}_{t} = \frac{1}{2} \cdot A \cdot B \cdot \sin c = \frac{1}{2} \cdot 12 \cdot 7 \cdot \sin 15 = 10.87 \left(2 \mathrm{dp}\right) u n i {t}^{2}$ [Ans]