# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/6 and the angle between sides B and C is pi/6. If side B has a length of 2, what is the area of the triangle?

Mar 27, 2018

color(indigo)(Delta " " A_t = (1/2) ab sin C = 0.575 " sq units"

#### Explanation:

$\hat{A} = \frac{\pi}{6} , \hat{C} = \frac{\pi}{6} , b = 2$

To find the area of the isosceles triangle.

$\hat{B} = \pi - \frac{\pi}{6} - \frac{\pi}{6} = \frac{2 \pi}{3}$

Applying the law of sines,

$a = \frac{b \cdot \sin A}{\sin} B = \frac{2 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{2 \pi}{3}\right) = 1.15 \text{ units}$

$\Delta \text{ } {A}_{t} = \left(\frac{1}{2}\right) a b \sin C = \left(\frac{1}{2}\right) \cdot 1.15 \cdot 2 \cdot \sin \left(\frac{\pi}{6}\right) = 0.575$