Question

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?

Answer 1

`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

Have a nice day!!! from the Philippines..