How to Find the Area of a Triangle Using Basic Trigonometry?

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SUMMARY

The discussion focuses on calculating the area of a triangle formed by a circle's radius R, a distance D from the center, and an angle A1. The user employs the Law of Sines to derive necessary angles and heights, ultimately expressing the area as B*H/2, where B and H are derived from trigonometric relationships. The user questions the complexity of their solution and seeks a more straightforward method. The discussion highlights the importance of accurately identifying trigonometric laws in problem-solving.

PREREQUISITES
  • Understanding of basic trigonometric functions (sine, cosine)
  • Familiarity with the Law of Sines and Law of Cosines
  • Knowledge of triangle area calculation methods
  • Ability to manipulate angles and distances in trigonometric contexts
NEXT STEPS
  • Research the Law of Sines and its applications in triangle area calculations
  • Explore alternative methods for calculating triangle areas using trigonometry
  • Learn about the relationship between angles and side lengths in non-right triangles
  • Investigate geometric interpretations of trigonometric identities
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Mathematicians, physics students, and anyone interested in applying trigonometry to solve geometric problems, particularly those involving triangles and circles.

MechatronO
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We got a circle with a radius R.

From a distance D from the centerpoint a line is inserted at an offset angle A1 from a line drawn though the centerpoint C of the circle, see the picture below.

http://cdn.imghack.se/medium/0861cdab13b8957018f8e167342f2b8e.png

I would like the are of the red triangle, provided D, R and A1.

I drew another triangle with one of its corner in the circles center for help, extracted the angle A2, got H and could then solve the problem but. However I wonder if there is a neater way than this.

What I did:

A3 = 180°-A1

Law of cosines give

sin (A3) / R = (sin A4) /D

A4 = arcsin( sin(A3) * D/R ) = arcsin( sin(180°-A1) * D/R)

A2 = 180° - A3 - A4 = 180° - (180°-A1) - arcsin( sin(180°-A1) * D/R) =

= A1 - arcsin( sin(A1)* D/R)

sin(A1) = H/R

H= R*sin(A1)

cos(A2) = (B+D)/R

B= R*cos(A2) -D

The red area = B*H/2 = (R*cos(A2)-D)*R*sin(A2)/2

And so forth. However, I get the feeling that this solution is more complicated than necessary?

EDIT: Btw the circle hasn't got much to do with the problem, but I'm just using it in a next step.
 
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What you call the Law of Cosines is actually the Law of Sines.
 
Typo.

More suggestions?
 

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