A triangle has two corners with angles of # pi / 12 # and # pi / 8 #. If one side of the triangle has a length of #5 #, what is the largest possible area of the triangle?

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Answer 1 · 2017-10-18T03:48:13

Area of triangle #A = sqrt(s (s-a) (s-b) (s - c))#

#**A = 145.1551**#

Three angles are #pi/12, pi/8, (19pi)/24#
#a/sin a = b / sin b = c / sin c#
#5/sin (pi/12) = b/sin (pi/8) = c /sin ((19pi)/24)#

#b = (5* sin (pi/8))/sin (pi/12) = 7.3929#

#c = (5* sin ((19pi)/24)/sin (pi/12)=11.7604#

#s = (a + b + c) /2 = (5+7.3928+11.7604)/2= 18.273#
#s-a = 13.273#
#s-b = 10.8801#
#s-c = 6.5126#

Area of triangle #A = sqrt(s (s-a) (s-b) (s - c))#

#A = sqrt(18.273 * 13.273 * 10.8801 * 6.5126) = 145.1551#