MathPlotAI Logo MathPlotAI

Derivation of Sum Formulas for Sine and Cosine

Derivation of Sum Formulas for Sine and Cosine
Loading the math board and drawing, please wait…
2015-04-22 19:24
FunctionTrigonometryGeometric Construction
This example demonstrates the proof of sum formulas for sine and cosine functions using geometric constructions. Key concepts include trigonometry, geometric construction, and angle relationships. Construction involves using GeoGebra commands like Intersect, Point, OrthogonalLine, and Rotate. Suitable for middle school students learning trigonometric basics or geometry teaching assistance.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Point(xAxis) → T
  3. OrthogonalLine(T, xAxis) → a
  4. Rotate(T, α, O) → B'
  5. Angle(T, O, B') → γ
  6. Line(O, B') → b
  7. Intersect(b, a) → S
  8. OrthogonalLine(S, b) → c
  9. Rotate(S, β, O) → C'
  10. Angle(S, O, C') → δ
  11. Line(O, C') → d
  12. Intersect(d, c) → P
  13. OrthogonalLine(P, xAxis) → e
  14. OrthogonalLine(S, e) → f
  15. Intersect(e, f) → R
  16. Segment(O, T) → g
  17. Segment(T, S) → h
  18. Segment(S, O) → i
  19. Segment(S, P) → j
  20. Segment(P, O) → k
  21. Segment(S, R) → l
  22. Intersect(g, xAxis) → F
  23. Intersect(e, g) → Q
  24. Segment(P, Q) → m
  25. Angle(P, Q, O) → ε
  26. Angle(S, T, Q) → ζ
  27. Angle(P, S, O) → η
  28. Angle(S, R, P) → θ
  29. Angle(T, O, S) → ι
  30. Angle(S, O, P) → κ
  31. Polygon(O, T, S) → poly1, s, o, t
  32. Polygon(O, S, P) → poly2, p, o_1, s_1
  33. Polygon(O, Q, P) → poly3, p_1, o_2, q
  34. Polygon(S, R, P) → poly4, p_2, s_2, r
  35. Angle(R, P, S) → λ
  36. Angle(R, S, O) → μ
  37. Angle(P, S, R) → ν
  38. RotateText("\cos(\beta)", α) → text3
  39. RotateText("\sin(\alpha) \cos(\beta)", -90*°) → text5
  40. RotateText("\cos(\alpha) \sin(\beta)", -90*°) → text9
  41. RotateText("\sin(\alpha+\beta)", -90*°) → text7
  42. RotateText("1", α + β) → text1
  43. RotateText("\sin(\beta)", α - 90°) → text2
  44. Segment(O, Q) → a_1
  45. Point(a_1) → B
  46. Segment(B, T) → c_1
  47. Segment(P, Q) → d_1
  48. Segment(P, R) → e_1
  49. Segment(S, T) → b_1
  50. Segment(R, S) → f_1

AI Drawing Prompt Reference

Draw a triangle O, T, S with O as the intersection of the x-axis and y-axis, and T on the x-axis. Rotate T to get point S. Then draw triangle O, S, P, and triangle O, Q, P. Construct point R using perpendicular lines and rotations, forming triangle S, P, R. Finally, add angle labels for angles α, β, γ, δ, and indicate the angle relationships in triangles O, T, S, O, S, P, O, Q, P, S, R, P, Q, P, S, R.
The prompt above was compiled from the original drawing commands and is for reference only.