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Trig Identity via Areas (2): Dynamic Illustrator

Trig Identity via Areas (2): Dynamic Illustrator
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2020-06-01 19:24
CircleTrigonometryTransformationGeometric ConstructionSlider Animation
This GeoGebra example dynamically illustrates a trigonometric identity through area relationships in the unit circle. It demonstrates how the two sides of the identity correspond to equal areas of constructed polygons, using reflection and rotation to map regions. Core mathematical concepts include trigonometric functions (sine, cosine), unit circle, area, reflection, and transformation. The construction employs commands such as Point, Circle, Polygon, Line, Intersect, OrthogonalLine, Distance, Segment, Dilate, Vector, Translate, Angle, and Rotate to build the unit circle, reflected lines, perpendiculars, polygons, and auxiliary segments. The interactive sliders and draggable points allow users to explore the dynamic equality of areas. Suitable for high school or early college students learning geometric proofs of trigonometric identities, and for teachers creating dynamic visual demonstrations.

GeoGebra Commands & Parameters

  1. Point(Segment[(0, 0), (1, 0)]) → A
  2. Polygon(C, A, 4) → poly1, h, i, j, k, E, F
  3. Polygon(D, A, 4) → poly2, l, m, n, p, G, H
  4. Line(C, F) → q
  5. Intersect(q, n) → I
  6. Line(D, H) → r
  7. OrthogonalLine(F, r) → s
  8. Intersect(r, s) → J
  9. Distance(E, G) → b
  10. Line(C, F) → t
  11. Circle(C, 1) → c
  12. Intersect(c, t, 2) → K
  13. Intersect(c, r) → L, M
  14. Polygon(E, G, I, F) → q1, e, g_1, i_1, f_1
  15. Polygon(C, I, H, D) → q2, c_1, i_2, h_1, d
  16. Segment(C, F) → j_1
  17. Segment(F, E) → k_1
  18. Segment(E, G) → l_1
  19. Segment(G, H) → m_1
  20. Segment(C, D) → n_1
  21. Segment(A, E) → p_1
  22. Segment(A, C) → q_1
  23. Dilate(D, If[0 < a ≤ 1, 1 - a, 0], L) → D'
  24. Vector(D, D') → u
  25. Translate(H, u) → H'
  26. Polygon(D', C, I, H') → q3, d', c_2, i_3, h'
  27. Line(I, H') → b_1
  28. Line(C, L) → d_1
  29. OrthogonalLine(L, d_1) → e_1
  30. Intersect(b_1, e_1) → N
  31. Dilate(H', If[2 < a ≤ 3, 3 - a, 0], N) → H''
  32. Vector(H', H'') → v
  33. Translate(I, v) → I'
  34. Polygon(L, C, I', H'') → q4, l_2, c_3, i', h''
  35. Dilate(N, If[4 < a ≤ 5, 5 - a, 0], O) → N'
  36. Vector(N, N') → w
  37. Translate(L, w) → L'
  38. Translate(C, w) → C'
  39. Translate(I', w) → I''
  40. Translate(H'', w) → H'''
  41. Polygon(L', C', I'', H''') → q4', l', c', i'', h'''
  42. Point(r_1) → P
  43. Angle(I, N, P) → α
  44. Rotate(L', If[4 < a ≤ 5, ((a - 4) * α), α], H''') → L''
  45. Rotate(C', If[4 < a ≤ 5, ((a - 4) * α), α], H''') → C''
  46. Rotate(I'', If[4 < a ≤ 5, ((a - 4) * α), α], H''') → I'''
  47. Rotate(H''', If[4 < a ≤ 5, ((a - 4) * α), α], H''') → Q
  48. Polygon(L'', C'', I''', Q) → q4'', l'', c'', i''', q_2
  49. Line(C, F) → t_1
  50. Circle(G, 1) → s_1
  51. Line(F, E) → a_1
  52. Intersect(s_1, a_1) → R, S
  53. Dilate(E, If[6 < a ≤ 7, 7 - a, 0], S) → F'
  54. Vector(E, F') → u_1
  55. Translate(E, u_1) → E'
  56. Translate(F, u_1) → F'_1
  57. Polygon(I, F'_1, F', G) → q5, i_4, f'_1, f', g_2
  58. Distance(G, F') → o
  59. Line(I, F'_1) → f_2
  60. OrthogonalLine(S, f_2) → h_2
  61. Intersect(f_2, h_2) → T
  62. Dilate(F'_1, If[9 < a ≤ 10, 10 - a, 0], T) → U
  63. Vector(F'_1, U) → v_1
  64. Translate(A, v_1) → A'
  65. Translate(I, v_1) → I'_1
  66. Polygon(I'_1, U, S, G) → q6, i'_1, u_2, s_2, g_3
  67. Dilate(T, If[11 < a ≤ 12, 12 - a, 0], L'') → T'
  68. Vector(T, T') → w_1
  69. Translate(I'_1, w_1) → V
  70. Translate(U, w_1) → U'
  71. Translate(S, w_1) → S'
  72. Translate(G, w_1) → G'
  73. Polygon(V, U', S', G') → q6', v_2, u', s', g'
  74. Point(f_2) → W
  75. Point(p_2) → Z
  76. Angle(Z, T, W) → β
  77. Rotate(V, If[11 < a ≤ 12, ((-(a - 11)) * β), (-β)], U') → V'
  78. Rotate(U', If[11 < a ≤ 12, ((-(a - 11)) * β), (-β)], U') → U''
  79. Rotate(S', If[11 < a ≤ 12, ((-(a - 11)) * β), (-β)], U') → S''
  80. Rotate(G', If[11 < a ≤ 12, ((-(a - 11)) * β), (-β)], U') → G''
  81. Polygon(V', U'', S'', G'') → q6'', v', u'', s'', g''
  82. Line(I, F'_1) → j_2
  83. Line(G, F') → k_2
  84. Segment(A, G) → m_2
  85. Segment(G, H) → n_2
  86. Segment(C, D) → r_2
  87. Segment(D, H) → t_2
  88. Segment((0, 0), A) → a_2

AI Drawing Prompt Reference

In the unit circle, draw two rays from the center intersecting the circle at two points, and place a point on each ray to form two squares. By reflecting and rotating, divide these squares into several polygonal regions with equal areas, visually demonstrating a trigonometric identity. Include auxiliary lines, perpendiculars, intersection points, and draggable sliders to dynamically adjust angles and positions, observing how the area relationships change.
The prompt above was compiled from the original drawing commands and is for reference only.