A triangle has two corners with angles of # pi / 6 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #8 #, what is the largest possible area of the triangle?

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Answer 1 · 2018-02-19T05:01:41

Area of triangle #A_t = (1/2) a b sin C ~~ color (red)(36#

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Given #hatA = pi/6, hat B (5pi)/8#, one side = 8#

To find the largest possible area of the triangle.

Third angle #hatC = pi - pi/6 - (5pi)/8 = (5pi)/24#

To get the largest possible area, side8 should correspond to to the least angle.

#a/ sin ((5pi)/24) = b / sin ((5pi)/8) = c / sin (pi)/6# using, law of sines.

#a = (8 * sin ((5pi)/24)) / sin (pi/6) = 9.7402#

#b = (8 * sin ((5pi)/8)) / sin (pi/6) = 14.7821#

Area of triangle #A_t = (1/2) a b sin C = (1/2) * 9.7402 * 14.7821 * sin (pi/6) ~~ color (red)(36#