Two corners of a triangle have angles of #(3 pi ) / 8 # and # pi / 3 #. If one side of the triangle has a length of #1 #, what is the longest possible perimeter of the triangle?

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Answer 1 · 2018-06-02T16:04:46

Longest possible perimeter # color(crimson)(P = 3.25#

#hat A = (3pi)/8, hat B = pi/3, hat C = (7pi)/24#

Least angle #hat C = (7pi)/24 should correspond to the side of length 1 to get the longest possible perimeter.

Applying the law of Sines,

#a / sin A = b / sin B = c / sin C = 1 / sin ((7pi)/24)#

#a = sin ((3pi)/8)* (1 / sin ((7pi)/24) )= 1.16#

#b = sin (pi/3) * (1/sin ((7pi)/24) )= 1.09#

Longest possible perimeter # color(crimson)(P = 1.16 + 1.09 + 1 = 3.25#