A triangle has sides A, B, and C. The angle between sides A and B is (2pi)/32π3 and the angle between sides B and C is pi/12π12. If side B has a length of 16, what is the area of the triangle?

1 Answer
P dilip_k
Mar 5, 2016

Two angles of the triangle are(2pi/3,pi/12)(2π3,π12)
So the third angle between sides A and C =pi-2pi/3-pi/12=(12pi-8pi-pi)/12=3*pi/12=pi/4=π−2π3−π12=12π−8π−π12=3⋅π12=π4
Pl consider the fig. below
enter image source here
From Properties of triangle we know the sides of a triangle are proportional to the sine 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 )=40.6squnit

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