A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/4, the angle between sides B and C is (5pi)/12, and the length of B is 4, what is the area of the triangle?

1 Answer
Leland Adriano Alejandro
Jan 23, 2016

Area=4/3*(3+sqrt(3))

Area=6.3094 square units

Explanation:

Try drawing the triangle
Angle A=(5pi)/12 and angle C=pi/4 and side b=4
side b is the base of the triangle. There 's a need to solve for the height h from angle B to side b to compute the area.

h cot A+h cot C=b
h=b/(cot A+cot C)=4/(cot ((5pi)/12)+cot (pi/4)
From double angle formulas:

cot ((5pi)/12)=cos(pi/4+pi/6)/sin(pi/4+pi/6)=(cos (pi/4)*cos (pi/6)-sin (pi/4)*sin (pi/6))/(sin (pi/4)*cos (pi/6)+cos (pi/4)*sin (pi/6))

cot ((5pi)/12)=(sqrt3-1)/(sqrt3+1)
Solve h now
h=4/(cot ((5pi)/12)+cot (pi/4))=4/((sqrt3-1)/(sqrt3+1)+1)
h=2/3*(3+sqrt3)
Solve Area=1/2*b*h
Area=1/2*4*2/3*(3+sqrt3)

Area=4/3*(3+sqrt(3))

Area=6.3094 square units

Have a nice day !!! from the Philippines .

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