Question

A triangle has sides A, B, and C. The angle between sides A and B is `(2pi)/3`. If side C has a length of `1` and the angle between sides B and C is `pi/12`, what is the length of side A?

Answer 1

`a=sqrt(6)/3`

Explanation:

At first we compute the third angle:
`pi-pi/12-2/3*pi=(12pi-pi-8pi)/12=3/12pi=pi/4`
So we get by the Theorem of sines:

`a=sin(pi/4)/sin(2/3pi)`
Not that

`sin(pi/4)=sqrt(2)/2`
`sin(2/3*pi)=sqrt(3)/2`

so we get

`a=(sqrt(2)/2)/(sqrt(3)/2)=sqrt(2)/sqrt(3)=(sqrt(2)*sqrt(3))/(sqrt(3)*sqrt(3))=sqrt(6)/3`