Question

Two corners of a triangle have angles of `(5 pi )/ 12` and `(3 pi ) / 8`. If one side of the triangle has a length of `1`, what is the longest possible perimeter of the triangle?

Answer 1

Longest possible perimeter of the triangle is 4.1043

Explanation:

Given are the two angles `(5pi)/12` and `(3pi)/8` and the length 1

The remaining angle:

`= pi - (((5pi)/12) + (3pi)/8) = (5pi)/24`

I am assuming that length AB (1) is opposite the smallest angle

`a / sin A = b / sin B = c / sin C`

`1 / sin ((5pi)/24) = b / sin ((3pi) /8) = c / ((5pi) / 12)`

`b = (1*sin((3pi)/8)) / sin ((5pi) /24) = 1.5176`

`c = (1*sin ((5pi)/12)) / sin ((5pi)/24) = 1.5867`

Longest possible perimeter of the triangle is =`(a+b+c) = (1+1.5176+1.5867) = 4.1043`