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Another Popular Trig Identity: Geometric Illustration via Areas

Another Popular Trig Identity: Geometric Illustration via Areas
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2021-09-09 09:48
PolygonAngleFunctionTrigonometryGeometric Construction
① This example demonstrates a popular trigonometric identity through a geometric area-based illustration, turning an algebraic identity into a visual equality of areas. ② Core mathematical concepts include trigonometric functions (sine, cosine, tangent), trigonometric identities, geometric area, and elements of function graphs, lines, and secants. ③ Construction highlights: Based on the construction summary, the example uses GeoGebra commands such as Point, Intersect, Polygon, Dilate, Segment, Vector, Translate, Rotate, Line, Ray, Circle, OrthogonalLine, and Angle, and creates numerous auxiliary objects (e.g., h, f, A, p, eq1, B, C, q, D, eq2, E, q1, a, e, d, c, b, g, F, G, H, I, q2, h_1, g_1, f_1, i, j, A', k, C', u, A'_1, A'', k', A'_2, l, B', m, D') to dynamically show that the areas on both sides of the identity are equal. ④ It is suitable for high school or early college students learning trigonometry, geometry, and introductory calculus, and also for teachers creating interactive lessons to help students grasp the geometric essence of trigonometric identities.

GeoGebra Commands & Parameters

  1. Point(Segment[(0, 0), (π / 2, 0)]) → A
  2. Intersect(f, eq1, (0.8228536275037948, 0.6801320145437539)) → B
  3. Intersect(h, eq1, (0.8228536275037948, 1.0778637671534934)) → C
  4. Intersect(xAxis, eq2) → E
  5. Polygon(A, E, D, C) → q1, a, e, d, c
  6. Intersect(p, eq1, (0.8228536275037948, -1.2669103446421748)) → F
  7. Intersect(eq1, g) → G
  8. Polygon(H, G, F, I) → q2, h_1, g_1, f_1, i
  9. Dilate(A, If[0 < j ≤ 1, 1 - j, 0], C) → A'
  10. Segment(A, A') → k
  11. Dilate(C, If[2 < j ≤ 3, 3 - j, 0], D) → C'
  12. Vector(C, C') → u
  13. Translate(A, u) → A'_{1}
  14. Translate(A', u) → A''
  15. Segment(A'_{1}, A'') → k'
  16. Dilate(A, If[3 < j ≤ 4, 4 - j, 0], B) → A'_{2}
  17. Segment(A, A'_{2}) → l
  18. Rotate(B, If[5 < j ≤ 6, ((-(j - 5)) * π / 2), ((-π)) / 2], A) → B'
  19. Segment(A, B') → m
  20. Dilate(D, If[6 < j ≤ 7, 7 - j, 0], C) → D'
  21. Vector(D, D') → v
  22. Translate(A'_{1}, v) → J
  23. Translate(A'', v) → A'''
  24. Segment(J, A''') → k''
  25. Dilate(A, If[7 < j ≤ 8, 8 - j, 0], C) → A'_{3}
  26. Vector(A, A'_{3}) → w
  27. Translate(A, w) → A'_{4}
  28. Translate(B', w) → B''
  29. Segment(A'_{4}, B'') → m'
  30. Polygon(A, E, D, C) → q3, a_1, e_1, d_1, c_1
  31. Intersect(h, yAxis, (0, 0)) → K
  32. Line(A, C) → n
  33. Ray(K, A) → r
  34. Ray(M, G) → s
  35. Dilate(G, If[9 < j ≤ 10, 10 - j, 0], F) → G'
  36. Segment(G, G') → t
  37. Dilate(K, If[11 < j ≤ 12, 12 - j, 0], L) → K'
  38. Segment(K, K') → i_1
  39. Dilate(K, If[13 < j ≤ 14, 14 - j, 0], F) → K'_{1}
  40. Vector(K, K'_{1}) → u_1
  41. Translate(K, u_1) → K'_{2}
  42. Translate(K', u_1) → K''
  43. Segment(K'_{2}, K'') → j_1
  44. Rotate(K'_{2}, If[14 < j ≤ 15, ((-(j - 14)) * π / 2), ((-π)) / 2], F) → N
  45. Rotate(K'', If[14 < j ≤ 15, ((-(j - 14)) * π / 2), ((-π)) / 2], F) → K'''
  46. Segment(N, K''') → k_1
  47. Dilate(I, If[15 < j ≤ 16, 16 - j, 0], H) → I'
  48. Vector(I, I') → v_1
  49. Translate(N, v_1) → N'
  50. Translate(K''', v_1) → O
  51. Segment(N', O) → l_1
  52. Dilate(F, If[16 < j ≤ 17, 17 - j, 0], I) → F'
  53. Vector(F, F') → w_1
  54. Translate(G, w_1) → G'_{1}
  55. Translate(G', w_1) → G''
  56. Segment(G'_{1}, G'') → t'
  57. Circle(D, 1) → p_1
  58. Intersect(p_1, n, 1) → P
  59. Dilate(C, If[19 < j ≤ 20, 20 - j, 0], P) → C'_{1}
  60. Vector(C, C'_{1}) → b_1
  61. Translate(C, b_1) → C'_{2}
  62. Translate(A, b_1) → A'_{5}
  63. Polygon(D, C'_{1}, A'_{5}, E) → q4, d_2, c'_{1}, a'_{5}, e_2
  64. Line(A'_{5}, E) → m_1
  65. OrthogonalLine(D, m_1) → n_1
  66. Intersect(m_1, n_1) → Q
  67. Line(P, D) → q_1
  68. Dilate(E, If[21 < j ≤ 22, 22 - j, 0], Q) → E'
  69. Vector(E, E') → r_1
  70. Translate(A'_{5}, r_1) → R
  71. Polygon(E', D, P, R) → q5, e', d_3, p_2, r_2
  72. Line(P, k_1) → s_1
  73. Point(s_1) → S
  74. Line(P, n_1) → t_1
  75. Point(t_1) → T
  76. Angle(A, P, T) → α
  77. Dilate(P, If[23 < j ≤ 24, 24 - j, 0], F) → P'
  78. Vector(P, P') → u_2
  79. Translate(E', u_2) → E''
  80. Translate(D, u_2) → D'_{1}
  81. Translate(P, u_2) → P'_{1}
  82. Translate(R, u_2) → R'
  83. Polygon(E'', D'_{1}, P'_{1}, R') → q5', e'', d'_{1}, p'_{1}, r'
  84. Rotate(E'', If[23 < j ≤ 24, ((-(j - 23)) * α), (-α)], P'_{1}) → E'''
  85. Rotate(D'_{1}, If[23 < j ≤ 24, ((-(j - 23)) * α), (-α)], P'_{1}) → U
  86. Rotate(P'_{1}, If[23 < j ≤ 24, ((-(j - 23)) * α), (-α)], P'_{1}) → V
  87. Rotate(R', If[23 < j ≤ 24, ((-(j - 23)) * α), (-α)], P'_{1}) → R''
  88. Polygon(E''', U, V, R'') → q5'', e''', u_3, v_2, r''
  89. Segment(K, A) → f_2
  90. Segment(M, W) → g_2

AI Drawing Prompt Reference

The diagram illustrates a trigonometric identity proven through area equality. It includes key points such as A, E, D, and C, forming polygons q1 and q2. The figure uses scaling, translation, and rotation to create symmetric auxiliary shapes, demonstrating the principle of equal areas. This design clearly shows the relationship between trigonometric functions and geometric areas, suitable for educational use.
The prompt above was compiled from the original drawing commands and is for reference only.