# A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/12, the angle between sides B and C is pi/3, and the length of B is 7, what is the area of the triangle?

Dec 6, 2016

Let

$\alpha \to \text{angle between B and C} = \frac{\pi}{3}$

$\beta \to \text{angle between A and C} = \pi - \left(\frac{\pi}{3} + \frac{\pi}{12}\right) = \frac{7 \pi}{12}$

$\gamma \to \text{angle between B and A} = \frac{\pi}{12}$

$\text{side } B = 7$

By properties of triangle

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

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

Now area of the triangle

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

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

$= \frac{1}{2} \times {7}^{2} \times \frac{\sin \left(\frac{\pi}{12}\right) \sin \left(\frac{\pi}{3}\right)}{\sin} \left(\frac{7 \pi}{12}\right)$

$\approx 5.685 \text{ squnit}$