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Angle Addition Formula Visualization

Angle Addition Formula Visualization
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2015-07-20 03:17
AngleTrigonometryGeometric Construction
This example demonstrates the geometric interpretation of the angle addition formula sin(α+β)=sinαcosβ+cosαsinβ. Key mathematical concepts include trigonometric functions, angle operations, and geometric transformations. Construction involves GeoGebra commands like Circle, Point, Angle, and OrthogonalLine. Suitable for high school students learning trigonometry and geometric transformations.

GeoGebra Commands & Parameters

  1. Circle(O, 1) → c
  2. Point(c) → A
  3. Segment(O, A) → a
  4. Point(c) → B
  5. Segment(B, O) → b
  6. Angle(A, O, B) → β
  7. OrthogonalLine(B, a) → d
  8. Intersect(d, a) → C
  9. Angle(B, C, O) → γ
  10. OrthogonalLine(A, xAxis) → e
  11. Intersect(e, xAxis) → D
  12. OrthogonalLine(C, xAxis) → f
  13. Intersect(f, xAxis) → E
  14. Segment(C, E) → sinαcosβ
  15. Segment(A, D) → h
  16. Segment(C, O) → i
  17. OrthogonalLine(B, f) → j
  18. Intersect(f, j) → F
  19. Angle(F, C, B) → δ
  20. Segment(B, C) → k
  21. Segment(F, C) → cosαsinβ
  22. Intersect(j, yAxis) → G
  23. Segment(G, O) → sinαpβ
  24. Angle(C, E, O) → ε
  25. Angle(A, D, O) → ζ
  26. Angle(B, F, C) → η
  27. Segment(F, B) → sinαsinβ
  28. Angle(D, O, A) → α
  29. Segment(F, C) → p
  30. CircleArc(O, M, N) → q
  31. Segment(O, D) → cosα
  32. Segment(E, O) → cosαcosβ
  33. Segment(B, F) → g
  34. Segment(G, B) → cosαpβ
  35. Angle(O, G, B) → θ

AI Drawing Prompt Reference

Draw a unit circle centered at the origin. Place two movable points on the circle, each connected to the center by a radius, with the angle between the radii labeled β. From one point, draw a perpendicular to the other radius, intersecting it at a point; from this intersection, drop a perpendicular to the horizontal axis. From the other point, drop a perpendicular to the horizontal axis. Connect relevant points to form segments representing products of sine and cosine. Add right-angle marks and an arc to indicate angles. The final figure shows several right triangles and perpendiculars inside the unit circle, illustrating the sine addition formula.
The prompt above was compiled from the original drawing commands and is for reference only.