A triangle has sides A, B, and C. The angle between sides A and B is #(2pi)/3# and the angle between sides B and C is #pi/12#. If side B has a length of 6, what is the area of the triangle?

1 Answer

#Area=5.70576# square units

Explanation:

To compute for the Area by using the given, there are several ways to do it.
I will present 2 solutions.
1st solution: #Area = 1/2*b*h#
Compute height #h# first. The altitude from angle B to side b:

Given angle #A=pi/12# and angle #C=(2pi)/3# and side #b=6#

#h=b/(cot A+cot C)#

#h=6/(cot (pi/12)+cot ((2pi)/3))#

#h=1.90192#

Compute Area:

#Area=1/2*b*h=(1/2)(6)(1.90192)#

#Area=5.70576# square units

2nd solution:
If there are 2 sides and an included angle then, the area is determined.
Compute Angle #B# then apply sine law to compute side #c#
So that , sides #b#, #c#, and angle #A# area available.

Compute angle #B#:

#B=pi-A-C=pi-pi/12-(2pi)/3=pi/4#

Compute side #c# using Sine Law:

#c=(b*sin C)/sin B=(6*sin ((2pi)/3))/sin (pi/4)#

#c=7.34847#

Compute the #Area#:

#Area = 1/2*b*c*sin A=1/2(6)(7.34847)sin (pi/12)#

#Area=5.70576# square units

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