A triangle has sides A, B, and C. Sides A and B have lengths of 9 and 3, respectively. The angle between A and C is #(pi)/3# and the angle between B and C is # (pi)/4#. What is the area of the triangle?

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Answer 1 · 2015-12-27T07:23:26

#frac{27(1+sqrt(3))}{4sqrt{2}}#

Since the interior angles of a triangle add up to #pi#, the angle between #A# and #B#, #theta#, is

#theta = pi - pi/4 - pi/3 = frac{5pi}{12}#.

Area of triangle is calculated using

#1/2 xx A xx B xx sin(theta) #

#= 1/2 xx 9 xx 3 xx sin(frac{5pi}{12})#

#= frac{27(1+sqrt(3))}{4sqrt{2}}#

Note:

#sin(frac{5pi}{12}) = sin(pi - pi/4 - pi/3)#

#= sin(pi/4 + pi/3)#

#= sin(pi/4)cos(pi/3) + cos(pi/4)sin(pi/3)#

#= (1/sqrt{2})(1/2) + (1/sqrt{2})(sqrt{3}/2)#

#= frac{1 + sqrt{3}}{2sqrt{2}}#