A triangle has sides A,B, and C. If the angle between sides A and B is pi/6π6, the angle between sides B and C is pi/12π12, and the length of B is 3, what is the area of the triangle?

1 Answer
Sonnhard
May 21, 2018

The third angle of the given triangle is given by alpha=3/4*piα=34⋅π and the area is given by A=a*b/2*sin(pi/6)A=a⋅b2⋅sin(π6)
The side length of BCBC can be calculated with the theorem of sines.

Explanation:

alpha=pi-pi/6-pi/12=3/4*piα=π−π6−π12=34⋅π
A=a*b/2*sin(pi/6)A=a⋅b2⋅sin(π6)
sin(3/4*pi)/sin(pi/12)=a/3sin(34⋅π)sin(π12)=a3
so we get
A=1/2*3*sin(3/4*pi)/sin(pi/12)*sin(pi/6)A=12⋅3⋅sin(34⋅π)sin(π12)⋅sin(π6)
Further you can use:
sin(3/4*pi)=sqrt(2)/2sin(34⋅π)=√22
sin(pi/12)=1/4*sqrt(2)*(sqrt(3)-1)sin(π12)=14⋅√2⋅(√3−1)
sin(pi/6)=1/2sin(π6)=12

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