Question

A triangle has sides A, B, and C. The angle between sides A and B is `(2pi)/3` and the angle between sides B and C is `pi/12`. If side B has a length of 16, what is the area of the triangle?

Answer 1

Two angles of the triangle are`(2pi/3,pi/12)`
So the third angle between sides A and C `=pi-2pi/3-pi/12=(12pi-8pi-pi)/12=3*pi/12=pi/4`
Pl consider the fig. below

From Properties of triangle we know the sides of a triangle are proportional to the `sine` of opposite angle
`:.` `A/sin(pi/12)=16/sin(pi/4)`
`=>A =16sin(pi/12)/sin(pi/4)=16sqrt2sin(pi/12)`

Now area of the triangle `=(1/2)*A*Bsin(2pi/3)`
`=(1/2)*16sqrt2sin(pi/12)*16*sin(2pi/3)`
`=128sqrt2sqrt3/2*sin(pi/12)`
`=64sqrt6*sin(pi/12 )=40.6`squnit