# 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/12. If side B has a length of 25, what is the area of the triangle?

May 3, 2018

color(maroon)("Area of Triangle " A_t = 57.2 " sq units"

#### Explanation:

color(blue)("Given " hat A = pi/12, hat C = pi/6, hat B = pi - pi/12 - pi/6 = (3pi)/4, b = 25

"As per " color(red)("Law of Sines", color(green)(a / sin A = b / sin B = c / sin C

$\frac{a}{\sin} \left(\frac{\pi}{12}\right) = \frac{25}{\sin} \left(\frac{3 \pi}{4}\right) = \frac{c}{\sin} \left(\frac{\pi}{6}\right)$

$c = \frac{25 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{3 \pi}{4}\right) = 17.68$

$\text{Area of } \Delta = {A}_{t} = \left(\frac{1}{2}\right) \cdot b \cdot \sin A$

${A}_{t} = \left(\frac{1}{2}\right) \cdot 25 \cdot 17.68 \cdot \sin \left(\frac{\pi}{12}\right) = 57.2 \text{ sq units}$