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Geometric Investigation of Trigonometric Identities

Geometric Investigation of Trigonometric Identities
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2019-09-09 21:41
CircleAngleVectorTrigonometryGeometric Construction
This interactive example provides a visual exploration of trigonometric identities through geometric construction. It demonstrates the fundamental relationships between sine, cosine, and ratios using a dynamic geometric model. The construction is built using a variety of GeoGebra tools, including Intersect, Semicircle, Rotate, OrthogonalLine, and Dilate, incorporating elements such as vectors, circle sectors, and polygons to map out the trigonometric properties. This resource is ideal for high school students and educators seeking to bridge the gap between algebraic trigonometric formulas and their spatial geometric representations.

GeoGebra Commands & Parameters

  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