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Trigonometric Factorisation Subtraction Identity Geometric Demonstration

Trigonometric Factorisation Subtraction Identity Geometric Demonstration
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2017-11-09 12:04
PolygonAngleTrigonometryTransformationGeometric Construction
This GeoGebra example dynamically demonstrates a subtraction identity in trigonometric factorisation, such as the factored form of sin(A-B) or cos(A-B), through geometric construction. The core mathematical concepts include trigonometric sum/difference formulas, factorisation, and angle relationships in geometric figures. Construction highlights: using commands like Rotate, Line, Intersect, Segment, and Angle, the construction builds quadrilaterals, equilateral triangles, rectangles, and other shapes, with auxiliary points (A', B', C', etc.) and lines (h, m, j, etc.) linking angles α, β, γ, δ, etc. The identity is verified through logical reasoning and the concepts of difference and slope. Suitable for high school or early college students to grasp the geometric meaning of trigonometric identities, and useful for teachers creating dynamic lessons or students exploring mathematics independently.

GeoGebra Commands & Parameters

  1. Rotate(B, a°, A) → B'
  2. Line(A, B') → h
  3. Intersect(h, i) → C
  4. Rotate(A, (-(90°)), C) → A'
  5. Line(C, A') → m
  6. Rotate(C, b°, A) → C'
  7. Line(A, C') → j
  8. Intersect(j, m) → F
  9. Intersect(i, k) → D
  10. Segment(D, F) → c
  11. Segment(C, D) → t
  12. Angle(B, A, B') → α
  13. Segment(A, B) → f
  14. Line(A, B) → g
  15. Angle(C, A, C') → β
  16. Angle(A', C, A) → γ
  17. Line(A, E) → l
  18. Segment(A, F) → p
  19. Segment(A, C) → q
  20. Segment(C, F) → r
  21. Segment(B, C) → s
  22. Segment(F, E) → d
  23. Segment(E, A) → e
  24. Segment(C, G) → f_1
  25. Segment(G, H) → g_1
  26. Segment(H, A) → h_1
  27. Angle(E, F, A) → δ
  28. Rotate(C, 90°, A) → C'_1
  29. Angle(C, A, C'_1) → ζ
  30. Line(C'_1, A) → i_1
  31. Intersect(i_1, g_1) → I
  32. Segment(I, A) → j_1
  33. Segment(I, C) → k_1
  34. Angle(A, H, I) → ε
  35. Angle(I, G, C) → η
  36. Angle(A, E, F) → θ
  37. Angle(F, D, C) → ι
  38. Angle(C, B, A) → κ
  39. Angle(I, C, A) → λ
  40. Angle(H, I, A) → μ
  41. Angle(G, C, I) → ξ
  42. Angle(D, C, F) → ν
  43. Segment(G, I) → l_1
  44. Segment(I, H) → m_1
  45. Line(G, H) → n_1
  46. Intersect(k, n_1) → J
  47. Segment(J, E) → p_1
  48. Segment(J, G) → q_1
  49. Angle(G, J, E) → ο