Question

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?

Answer 1

Area of triangle `A_t = ~~ 27.05` sq units

Explanation:

Given `hat C = pi / 4, hat A = pi / 12, hat B = pi - pi /4 - pi / 12 = (2pi)/3, b = 16`

`c = (b sin C) / sin B = (16 * sin (pi/4)) / sin ((2pi) / 3) = (16 * 2) / (sqrt 2 * sqrt3) = 16 sqrt(2/3)`

Area of the triangle `A_t = (1/2) b c sin A = (1/2) 16 * 16 (sqrt (2/3)) * sin (pi/12)`

`A_t = 128 sqrt(2/3) sin (pi/12) ~~ 27.05` sq units