Question

A triangle has two corners with angles of `pi / 4` and `pi / 6`. If one side of the triangle has a length of `16`, what is the largest possible area of the triangle?

Answer 1

largest possible area of the triangle `color(purple)(A_t ~~ 699.405`

Explanation:

Third angle `C = pi - pi/4 - pi/6 = (7pi)/12`

To get the largest possible area, length 16 should correspond to least angle `pi/6`

Other sides are

`a / sin(pi/4) = b / sin (7pi)/12 = c / sin (pi/6) = 16 / sin (pi/6) = 32` as `sin(pi/6) = sin 30 = 1/2`

`a = 32 sin (pi/4) = 32 sqrt2 = 45.2548`

Area of triangle `A_t = (1/2) a c sin B = cancel(1/2) cancel(32)^color(red)(16) sqrt2 * 32 * sin ((7pi)/12)`

`A_t = 512 sqrt2 sin ((7pi)/12) = color(purple)(699.405` sq units