Question

A triangle has sides A,B, and C. If the angle between sides A and B is `pi/6`, the angle between sides B and C is `pi/12`, and the length of B is 3, what is the area of the triangle?

Answer 1

The third angle of the given triangle is given by `alpha=3/4*pi` and the area is given by `A=a*b/2*sin(pi/6)`
The side length of `BC` can be calculated with the theorem of sines.

Explanation:

`alpha=pi-pi/6-pi/12=3/4*pi`
`A=a*b/2*sin(pi/6)`
`sin(3/4*pi)/sin(pi/12)=a/3`
so we get
`A=1/2*3*sin(3/4*pi)/sin(pi/12)*sin(pi/6)`
Further you can use:
`sin(3/4*pi)=sqrt(2)/2`
`sin(pi/12)=1/4*sqrt(2)*(sqrt(3)-1)`
`sin(pi/6)=1/2`