A triangle has two corners with angles of # pi / 12 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #3 #, what is the largest possible area of the triangle?

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Answer 1 · 2018-05-12T03:58:50

#color(maroon)("Largest possible Area of triangle " A_t = 12.5 " sq units"#

#hat A = pi/12, hat B = (5pi) / 8, hat C = pi - pi/12 - (5pi) / 8 = (7pi)/24#

To get the largest area, side 3 should correspond to least angle #hatA#

Applying the Law of Sines,

#a / sin A = b / sin B = c / sin C#

#3 / sin (pi/12) = b / sin ((5pi)/8)#

#b = (3 * sin ((5pi)/8)) / sin (pi/12) = 10.71#

#"Area of Triangle " A_t = (1/2) a b sin C#

#A_t = (1/2) * 3 * 10.71 * sin ((7pi)/24) = 12.5 " sq units"#