Question

A triangle has sides A, B, and C. If the angle between sides A and B is `(pi)/4`, the angle between sides B and C is `(2pi)/3`, and the length of side B is 5, what is the area of the triangle?

Answer 1

Let

1.the angle between sides A and B opposite to side C is `gamma=pi/4`

2.the angle between sides B and C opposite to side A is `alpha=(2pi)/3`

3.the angle between sides C and A opposite to side B is `beta=pi-(2pi)/3-pi/4=pi/12`

By sine law of triangle

`C/singamma=B/sinbeta`

`=>C=(Bsingamma)/sinbeta`

Area if the triangle

`Delta=1/2BxxCsinalpha`

`=1/2B^2(sinalphaxxsingamma)/sinbeta`

`=1/2xx5^2(sin((2pi)/3)sin(pi/4))/sin(pi/12)`

`~~29.57squnit`