Question

A triangle has two corners with angles of `pi / 6` and `(5 pi )/ 8`. If one side of the triangle has a length of `15`, what is the largest possible area of the triangle?

Answer 1

Largest possible area = 126.5489

Explanation:

Three angles are `pi/6, (5pi)/8, (5pi)/24`
Least angle is `pi/6` which will correspond to side of length 15 to get maximum area.
`C/ sin (/_C ) = A / sin( /_A ) = B / sin ( /_B)`
`C/ sin ((5pi)/8) = A / sin ((5pi) / 24)= 15 / sin (pi/6)`

`C= (15 * sin ((5pi)/8))/sin (pi/6) = 27.7164`

`A = (5 * sin ((5pi)/24))/sin (pi/6) = 18.2628`

Semi Perimeter of `Delta ABC s = (15 + 27.7154 + 18.2628) / 2 = 30.49`

`s - a = 30.49 - 27.7164 = 2.7732`
`s - b = 30.49 - 15 = 15.49`
`s - c = 30.49 - 18.2628 = 12.2272`

Area of `Delta ABC = sqrt(s(s-a)(s-b)(s-c))`

Area of `Delta ABC = sqrt (30.49 * 2.7732 * 15.49 * 12.2272) = **126.5489**`