Question

A triangle has two corners with angles of `pi / 4` and `pi / 6`. If one side of the triangle has a length of `8`, what is the largest possible area of the triangle?

Answer 1

`color(maroon)("Largest possible Area of Triangle " A_t = 87.43 " sq units"`

Explanation:

`hat A = pi/4, hat B = pi/6, hat C = pi - pi/4 - pi/6 = (7pi)/12`

To get the largest area, side 8 should correspond to least angle `hat B` and hence is side b.

Applying the Law of Sines,

`a / sin A = b / sin B`

`a = (8 * sin (pi/4)) / sin (pi/6) = 16/sqrt2`

`"Largest possible Area of Triangle " A_t = (1/2) a b sin C`

`A_t = (1/2) * (16/sqrt2) * 8 * sin ((7pi)/12) = 87.43 " sq units"`