# A triangle has sides A,B, and C. If the angle between sides A and B is pi/3, the angle between sides B and C is pi/12, and the length of B is 1, what is the area of the triangle?

May 19, 2018

color(blue)(A_t = 0.4019 sq units

#### Explanation:

$\hat{A} = \frac{\pi}{12} , \hat{C} = \frac{\pi}{3} , \hat{B} = \pi - \frac{\pi}{12} - \frac{\pi}{3} = \frac{7 \pi}{12} , b = 1$

Applying Law of Sines,

$\frac{a}{\sin} A = \frac{b}{\sin} B$

$a = \frac{1 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{7 \pi}{12}\right)$

$a = 0.2679$

Area of Triangle ${A}_{t} = \left(\frac{1}{2}\right) a b \sin C$

${A}_{t} = \left(\frac{1}{2}\right) \cdot 0.2679 \cdot 1 \cdot \sin \left(\frac{\pi}{3}\right)$

color(blue)(A_t = 0.4019 sq units