Question

A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is `(pi)/3` and the angle between B and C is `(pi)/4`. What is the area of the triangle?

Answer 1

`A~~6.54`

Explanation:

The area of a triangle is given by `1/2*Base*height`
The height `h` in this example can be found from
`sin(pi/3) =h/A = h/3`
`h = 3sin(pi/3)`

The base `C = x +y` and `x` and `y` can also be found from trigonometric functions.
`x/A = cos(pi/3)` so `x = 3cos(pi/3)`
`y/B = cos(pi/4)` so `y = 5cos(pi/4)`

Area `A = 1/2*(3cos(pi/3)+5cos(pi/4))*3sin(pi/3)`

`A =1/2*(3*0.5 +5*0.7071)*3*0.866`
`A~~6.54`