# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/4, the angle between sides B and C is (2pi)/3, and the length of side B is 5, what is the area of the triangle?

Dec 26, 2016

Let

1.the angle between sides A and B opposite to side C is $\gamma = \frac{\pi}{4}$

2.the angle between sides B and C opposite to side A is $\alpha = \frac{2 \pi}{3}$

3.the angle between sides C and A opposite to side B is $\beta = \pi - \frac{2 \pi}{3} - \frac{\pi}{4} = \frac{\pi}{12}$

By sine law of triangle

$\frac{C}{\sin} \gamma = \frac{B}{\sin} \beta$

$\implies C = \frac{B \sin \gamma}{\sin} \beta$

Area if the triangle

$\Delta = \frac{1}{2} B \times C \sin \alpha$

$= \frac{1}{2} {B}^{2} \frac{\sin \alpha \times \sin \gamma}{\sin} \beta$

$= \frac{1}{2} \times {5}^{2} \frac{\sin \left(\frac{2 \pi}{3}\right) \sin \left(\frac{\pi}{4}\right)}{\sin} \left(\frac{\pi}{12}\right)$

$\approx 29.57 s q u n i t$