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 #pi/3#, and the length of B is 12, what is the area of the triangle?

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Answer 1 · 2018-06-25T12:46:50

#A=(72*sqrt(3))/(1+sqrt(3))#

We use the Formula

#A=1/2*a*b*sin(gamma)#
where #b,gamma# is given. So we must compute the side length of #a#:

The third angle is given by #5pi/12#
so we can use the Theorem of sines:

#sin(5/12*pi)/sin(pi/3)=12/a#

From here we get

#a=12sin(pi/3)/sin(5*pi/12)#
putting Things together we get

#A=1/2*12*12*sin(pi/3)*sin(pi/4)/sin(5*pi/12)#

Note that

#sin(pi/3)=sqrt(3)/2#

#sin(5*pi/12)=(1+sqrt(3))/(2*sqrt(2))#

#sin(pi/4)=sqrt(2)/2#
thus

#A=(72*sqrt(3))/(1+sqrt(3))#