A triangle has two corners with angles of # pi / 12 # and # pi / 3 #. If one side of the triangle has a length of #14 #, what is the largest possible area of the triangle?

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Answer 1 · 2017-06-30T09:35:10

The area of the triangle is #=669.4u^2#

The third angle of the triangle is

#=pi-(1/12pi+1/3pi)#

#=pi-5/12pi#

#=7/12pi#

The angles are

#1/12pi, 5/12pi, 7/12pi#

To have the greatest area , the side of length #14# is opposite the smallest angle

We apply the sine rule

#14/sin(1/12pi)=A/sin(5/12pi)=B/sin(7/12pi)#

#A=14sin(5/12pi)/sin(1/12pi)=52.2#

#B=14sin(7/12pi)/sin(1/12pi)=99.1#

The area of the triangle is

#=1/2*AB*sin(1/12pi)#

#=0.5*52.2*99.1*sin(1/12pi)#

#=669.4#