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Trig Identity via Areas: Dynamic Illustrator

Trig Identity via Areas: Dynamic Illustrator
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2020-03-09 23:42
CircleAngleTrigonometryGeometric ConstructionSlider Animation
①This example dynamically illustrates trigonometric identities (such as those related to the Pythagorean theorem) by comparing areas within a unit circle and right triangles, making the algebraic identity visually evident. ②Core mathematical concepts include trigonometric functions, Pythagorean theorem, unit circle, area equivalence, similar triangles, and geometric transformations. ③Construction highlights: using GeoGebra commands such as Line, Intersect, Polygon, Circle, Segment, Dilate, Translate, OrthogonalLine, Vector, Point, Angle, and Rotate, the applet builds interactive figures with circles, lines, polygons, vectors, angles, and multiple auxiliary points; dragging sliders or points changes the angle and updates area relationships in real time. ④Suitable for high school or early college students learning trigonometric identities, and for teachers creating dynamic geometry demonstrations that connect algebra and geometry.

GeoGebra Commands & Parameters

  1. Line(A, B) → h
  2. Intersect(h, xAxis) → C
  3. Polygon(A, C, 4) → poly1, i, j, k, l, D, E
  4. Intersect(h, p) → F
  5. Polygon(B, F, 4) → poly2, m, n, q, r, G, H
  6. Intersect(xAxis, yAxis) → I
  7. Intersect(f, yAxis, (0, 1)) → J
  8. Polygon(I, J, 4) → poly3, s, t, c, d, K, L
  9. Circle(C, 1) → c_1
  10. Intersect(c_1, f_1) → M, N
  11. Segment(C, N) → g_1
  12. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], N) → A'
  13. Translate(E, Vector[A, A']) → E'
  14. Polygon(C, A', E', D) → q1, c_2, a', e', d_2
  15. OrthogonalLine(C, e') → h_1
  16. Line(E', D) → i_1
  17. Intersect(h_1, i_1) → O
  18. Dilate(D, If[2 < e ≤ 3, 3 - e, 0], O) → D'
  19. Translate(E', Vector[D, D']) → E''
  20. Polygon(A', E'', D', C) → q2, j_1, e'', d', c_3
  21. Segment(I, J) → k_1
  22. Segment(J, K) → l_1
  23. Segment(K, L) → m_1
  24. Segment(L, I) → n_1
  25. Segment(A, E) → p_1
  26. Segment(E, D) → q_1
  27. Segment(D, C) → r_1
  28. Segment(C, A) → s_1
  29. Segment(A', E') → t_1
  30. Segment(E', D) → a_1
  31. Segment(D, C) → b_1
  32. Segment(C, A') → d_1
  33. Segment(A', E'') → e_1
  34. Segment(E', D') → f_2
  35. Segment(C, D') → g_2
  36. Dilate(C, If[4 < e ≤ 5, 5 - e, 0], K) → C'
  37. Vector(C, C') → u
  38. Translate(A', u) → A''
  39. Translate(E'', u) → E'''
  40. Translate(D', u) → D''
  41. Translate(C, u) → C'_1
  42. Polygon(A'', E''', D'', C'_1) → q2', a'', e''', d'', c'_1
  43. Line(C', m_1) → h_2
  44. Point(h_2) → P
  45. Angle(P, C', D'') → α
  46. Rotate(A'', If[4 < e ≤ 5, ((-(e - 4)) * α), (-α)], C'_1) → A'''
  47. Rotate(E''', If[4 < e ≤ 5, ((-(e - 4)) * α), (-α)], C'_1) → Q
  48. Rotate(D'', If[4 < e ≤ 5, ((-(e - 4)) * α), (-α)], C'_1) → D'''
  49. Rotate(C'_1, If[4 < e ≤ 5, ((-(e - 4)) * α), (-α)], C'_1) → R
  50. Polygon(A''', Q, D''', R) → q2'', a''', q_2, d''', r_2
  51. Line(C, A') → i_2
  52. Line(D, E') → j_2
  53. Line(B, H) → k_2
  54. Segment(B, H) → m_2
  55. Segment(H, G) → n_2
  56. Segment(G, F) → s_2
  57. Circle(F, 1) → e_2
  58. Intersect(e_2, k_2) → S, T
  59. Dilate(B, If[6 < e ≤ 7, 7 - e, 0], T) → B'
  60. Vector(B, B') → v
  61. Translate(H, v) → H'
  62. Polygon(F, B', H', G) → q3, f_3, b', h', g_3
  63. OrthogonalLine(F, h') → t_2
  64. Line(H', G) → a_2
  65. Intersect(t_2, a_2) → U
  66. Dilate(G, If[8 < e ≤ 9, 9 - e, 0], U) → G'
  67. Vector(G, G') → w
  68. Translate(H', w) → H''
  69. Polygon(F, B', H'', G') → q4, f_4, b_2, h'', g'
  70. Line(B', F) → h_3
  71. Dilate(U, If[10 < e ≤ 11, 11 - e, 0], L) → F'
  72. Vector(U, F') → u_1
  73. Translate(F, u_1) → F'_1
  74. Translate(B', u_1) → B''
  75. Translate(H'', u_1) → H'''
  76. Translate(G', u_1) → G''
  77. Polygon(F'_1, B'', H''', G'') → q4', f'_1, b'', h''', g''
  78. Rotate(F'_1, If[10 < e ≤ 11, ((-(e - 10)) * (π / 2 - α)), α - π / 2], G'') → V
  79. Rotate(B'', If[10 < e ≤ 11, ((-(e - 10)) * (π / 2 - α)), α - π / 2], G'') → B'''
  80. Rotate(H''', If[10 < e ≤ 11, ((-(e - 10)) * (π / 2 - α)), α - π / 2], G'') → W
  81. Rotate(G'', If[10 < e ≤ 11, ((-(e - 10)) * (π / 2 - α)), α - π / 2], G'') → G'''
  82. Polygon(V, B''', W, G''') → q4'', v_1, b''', w_1, g'''
  83. Segment(A, Z) → i_3
  84. Segment(B, F) → l_2

AI Drawing Prompt Reference

Draw a unit circle with a triangle inside and a right triangle, comparing their areas to illustrate trigonometric identities. Start by drawing a unit circle and two intersecting lines, then construct four polygons to represent area relationships. Use similar triangles and geometric transformations (such as translation, rotation, and scaling) to dynamically show how angle changes affect area. Finally, rotate and scale the elements to ensure they fit the geometric interpretation of trigonometric identities.
The prompt above was compiled from the original drawing commands and is for reference only.