# A triangle has sides A, B, and C. The angle between sides A and B is pi/4 and the angle between sides B and C is pi/12. If side B has a length of 16, what is the area of the triangle?

Mar 10, 2018

Area of triangle ${A}_{t} = \approx 27.05$ sq units

#### Explanation:

Given $\hat{C} = \frac{\pi}{4} , \hat{A} = \frac{\pi}{12} , \hat{B} = \pi - \frac{\pi}{4} - \frac{\pi}{12} = \frac{2 \pi}{3} , b = 16$

$c = \frac{b \sin C}{\sin} B = \frac{16 \cdot \sin \left(\frac{\pi}{4}\right)}{\sin} \left(\frac{2 \pi}{3}\right) = \frac{16 \cdot 2}{\sqrt{2} \cdot \sqrt{3}} = 16 \sqrt{\frac{2}{3}}$

Area of the triangle ${A}_{t} = \left(\frac{1}{2}\right) b c \sin A = \left(\frac{1}{2}\right) 16 \cdot 16 \left(\sqrt{\frac{2}{3}}\right) \cdot \sin \left(\frac{\pi}{12}\right)$

${A}_{t} = 128 \sqrt{\frac{2}{3}} \sin \left(\frac{\pi}{12}\right) \approx 27.05$ sq units