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三角恒等式的几何探究

三角恒等式的几何探究
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2019-09-09 21:41
圆角向量三角函数几何作图
本范例通过动态几何构造，直观地演示了三角函数恒等式的几何意义及其内在联系。核心数学概念涵盖了正弦（Sine）、余弦（Cosine）以及三角比率的几何定义。在作图实现上，利用了 GeoGebra 的 Intersect（交点）、Semicircle（半圆）、Rotate（旋转）、OrthogonalLine（垂线）以及 Dilate（位似/伸缩）等高级命令，构建了包含圆弧、向量和多边形的复杂几何网络，通过点 A、B、C 等关键对象的联动来验证三角关系。该资源非常适合高中数学教师用于讲解三角函数定义，或作为学生探究几何证明与代数恒等式之间关系的教学辅助工具。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(Ray[(0, 0), (1, 0)]) → B
  3. Semicircle(A, B) → c
  4. Midpoint(A, B) → C
  5. Rotate(B, π / 2, C) → B'
  6. Center(c) → D
  7. CircleArc(C, B, B') → d
  8. Point(d) → E
  9. OrthogonalLine(E, xAxis) → f
  10. OrthogonalLine(E, f) → g
  11. Intersect(c, g, 2) → F
  12. OrthogonalLine(F, g) → h
  13. Intersect(h, xAxis) → G
  14. Intersect(f, xAxis) → H
  15. Polygon(G, H, E, F) → q1, g_1, h_1, e, f_1
  16. Angle(E, H, C) → α
  17. Angle(H, G, F) → β
  18. Angle(G, F, E) → γ
  19. Angle(B, C, E) → δ
  20. Dilate(D, If[0 < a ≤ 1, 1 - a, 0], E) → D'
  21. Segment(E, H) → j
  22. Segment(G, F) → l
  23. Rotate(E, δ, D) → E'
  24. Segment(C, H) → k
  25. Segment(C, E) → i
  26. Circle(C, (0.4 * Distance[D, H])) → p
  27. Intersect(p, i, 1) → I
  28. Intersect(p, k, 1) → J
  29. Dilate(D, If[1 < a ≤ 2, 2 - a, 0], E') → D'_1
  30. Segment(C, D'_1) → m
  31. OrthogonalLine(E', xAxis) → n
  32. Intersect(n, xAxis) → K
  33. Segment(F, E) → q
  34. Rotate(J, If[0 < a ≤ 1, (a * δ), δ], D) → J'
  35. CircleSector(C, J, J') → r
  36. Rotate(r, If[2 < a ≤ 3, ((a - 2) * δ), δ], D) → r'
  37. Angle(E, C, E') → ε
  38. Dilate(E', If[4 < a ≤ 5, 5 - a, 0], K) → E''
  39. Segment(E', E'') → s
  40. Angle(E', K, C) → ζ
  41. Segment(C, A) → t
  42. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E') → C'
  43. Vector(C, C') → u
  44. Translate(C, u) → C'_1
  45. Translate(E, u) → E'_1
  46. Segment(C'_1, E'_1) → i'
  47. Rotate(C'_1, If[6 < a ≤ 7, ((a - 6) * δ), (-δ)], C'_1) → L
  48. Rotate(E'_1, If[6 < a ≤ 7, ((-(a - 6)) * δ), (-δ)], C'_1) → M
  49. Segment(L, M) → i''
  50. OrthogonalLine(M, xAxis) → b
  51. Intersect(b, xAxis) → N
  52. Ray(G, A) → j_1
  53. Line(M, m) → k_1
  54. Dilate(M, If[8 < a ≤ 9, 9 - a, 0], N) → M'
  55. Angle(E', M, N) → η
  56. Angle(M, N, K) → θ_1
  57. Dilate(K, If[7 < a ≤ 8, 8 - a, 0], N) → K'
  58. Vector(K, K') → v
  59. Translate(E', v) → E''_1
  60. Translate(E'', v) → E'''
  61. Segment(E''_1, E''') → s'
  62. Line(F, E) → l_1
  63. Circle(H, Distance[C, A]) → c_1
  64. Intersect(c_1, l_1, 2) → O
  65. Line(H, N) → i_1
  66. Dilate(E, If[9 < a ≤ 10, 10 - a, 0], O) → E'_2
  67. Vector(E, E'_2) → w
  68. Translate(F, w) → F'
  69. Polygon(G, F', E'_2, H) → q2, g_2, f', e'_2, h_2
  70. Line(G, F') → m_1
  71. OrthogonalLine(O, m_1) → n_1
  72. Intersect(m_1, n_1) → P
  73. Dilate(F', If[11 < a ≤ 12, 12 - a, 0], P) → F''
  74. Vector(F', F'') → u_1
  75. Translate(G, u_1) → G'
  76. Polygon(G', F'', O, H) → q3, g', f'', o, h_3
  77. Dilate(H, If[13 < a ≤ 14, 14 - a, 0], K) → H'
  78. Vector(H, H') → v_1
  79. Translate(G', v_1) → G''
  80. Translate(F'', v_1) → F'''
  81. Translate(O, v_1) → O'
  82. Translate(H, v_1) → H'_1
  83. Polygon(G'', F''', O', H'_1) → q3', g'', f''', o', h'_1
  84. Line(F'', G') → p_1
  85. Line(O, H) → q_1
  86. Line(F, E) → r_1
  87. Angle(N, H, O) → θ
  88. Rotate(G'', If[14 < a ≤ 15, ((-(a - 14)) * θ), (-θ)], K) → G'''
  89. Rotate(F''', If[14 < a ≤ 15, ((-(a - 14)) * θ), (-θ)], K) → Q
  90. Rotate(O', If[14 < a ≤ 15, ((-(a - 14)) * θ), (-θ)], K) → O''
  91. Rotate(H'_1, If[14 < a ≤ 15, ((-(a - 14)) * θ), (-θ)], K) → R
  92. Polygon(G''', Q, O'', R) → q3'', g''', q_2, o'', r_2
  93. Angle(B, E'', O') → κ
  94. Rotate(G'', If[14 < a ≤ 15, ((-(a - 14)) * κ), (-κ)], H') → G'''_1
  95. Rotate(F''', If[14 < a ≤ 15, ((-(a - 14)) * κ), (-κ)], H') → S
  96. Rotate(O', If[14 < a ≤ 15, ((-(a - 14)) * κ), (-κ)], H') → O''_1
  97. Rotate(H'_1, If[14 < a ≤ 15, ((-(a - 14)) * κ), (-κ)], H') → T
  98. Polygon(G'''_1, S, O''_1, T) → q3''_1, g'''_1, s_1, o''_1, t_1