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 #32 # and the angle between sides B and C is #pi/12#, what is the length of side A?

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Answer 1 · 2017-01-09T06:55:57

#A=16/3(3sqrt(2)-sqrt6)#

You would apply the Law of Sines:

#color(red)(A/sin hat(BC))=B/sin hat(AC)color(red)(=C/sin hat(AB))#

Then

#A/sin(pi/12)=32/sin((2pi)/3)#

and

#A=(32*sin(pi/12))/(sin((2pi)/3))=(cancel32^8*(sqrt(6)-sqrt(2))/cancel4)/(sqrt(3)/2)=16(sqrt(6)-sqrt(2))/sqrt(3)#

#=16*(sqrt(18)-sqrt(6))/3=16/3(3sqrt(2)-sqrt6)#