# A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/6, the angle between sides B and C is pi/3, and the length of B is 3, what is the area of the triangle?

Jul 9, 2018

$A = \frac{9}{2} \cdot \sqrt{3}$

#### Explanation:

Since the third angle is given by

$\pi - \frac{\pi}{6} - \frac{\pi}{3} = \frac{6 \pi - \pi - 2 \pi}{6} = \frac{3 \pi}{6} = \frac{\pi}{2}$
so we get

$\tan \left(\frac{\pi}{6}\right) = \frac{3}{a}$ so

$a = \frac{3}{\tan} \left(\frac{\pi}{6}\right) = \frac{3}{\frac{\sqrt{3}}{3}} = \frac{9}{\sqrt{3}} = 9 \cdot \frac{\sqrt{3}}{3} = 3 \sqrt{3}$

So
$A = \frac{1}{2} \cdot 3 \cdot 3 \cdot \sqrt{3} = \frac{9}{2} \cdot \sqrt{3}$