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

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

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Answer 1 · 2016-01-27T07:22:22

$A r e a = 5.70576$ $Area=5.70576$ 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: $A r e a = \frac{1}{2} \cdot b \cdot h$ $Area = 1/2*b*h$
Compute height $h$ $h$ first. The altitude from angle B to side b:

Given angle $A = \frac{π}{12}$ $A=pi/12$ and angle $C = \frac{2 π}{3}$ $C=(2pi)/3$ and side $b = 6$ $b=6$

$h = \frac{b}{cot A + cot C}$ $h=b/(cot A+cot C)$

$h = \frac{6}{cot (\frac{π}{12}) + cot (\frac{2 π}{3})}$ $h=6/(cot (pi/12)+cot ((2pi)/3))$

$h = 1.90192$ $h=1.90192$

Compute Area:

$A r e a = \frac{1}{2} \cdot b \cdot h = (\frac{1}{2}) (6) (1.90192)$ $Area=1/2*b*h=(1/2)(6)(1.90192)$

$A r e a = 5.70576$ $Area=5.70576$ square units

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

Compute angle $B$ $B$ :

$B = π - A - C = π - \frac{π}{12} - \frac{2 π}{3} = \frac{π}{4}$ $B=pi-A-C=pi-pi/12-(2pi)/3=pi/4$

Compute side $c$ $c$ using Sine Law:

$c = \frac{b \cdot sin C}{sin B} = \frac{6 \cdot sin (\frac{2 π}{3})}{sin (\frac{π}{4})}$ $c=(b*sin C)/sin B=(6*sin ((2pi)/3))/sin (pi/4)$

$c = 7.34847$ $c=7.34847$

Compute the $A r e a$ $Area$ :

$A r e a = \frac{1}{2} \cdot b \cdot c \cdot sin A = \frac{1}{2} (6) (7.34847) sin (\frac{π}{12})$ $Area = 1/2*b*c*sin A=1/2(6)(7.34847)sin (pi/12)$

$A r e a = 5.70576$ $Area=5.70576$ square units

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

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

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Explanation:

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A triangle has sides A, B, and C. The angle between sides A and B is 2π3(2pi)/3 and the angle between sides B and C is π12pi/12. If side B has a length of 6, what is the area of the triangle?

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Explanation:

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