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Demonstration of Sine, Cosine, and Tangent

Demonstration of Sine, Cosine, and Tangent
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2016-06-11 02:41
VectorTrigonometryCoordinate GeometryGeometric ConstructionGraphing
This example dynamically demonstrates the relationship between sine, cosine, and tangent functions in a right triangle, visually showcasing the definition and geometric meaning of trigonometric functions. Core math concepts include trigonometry, right triangles, unit circle, vectors, and coordinate geometry. Construction highlights GeoGebra commands like Intersect, Circle, Point, Rotate, Vector, Midpoint, Segment, CurveCartesian, Polygon, OrthogonalLine, and If. Suitable for high school students learning trigonometric geometry and university foundational mathematics education.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 3) → c
  3. Point(c) → B
  4. Rotate(B, β, A) → B'_1
  5. Vector(A, B'_1) → v
  6. Point(v) → E
  7. Midpoint(I, F) → J
  8. Midpoint(F, J) → K
  9. Midpoint(J, I) → L
  10. Midpoint(B, G) → M
  11. Midpoint(B, H) → N
  12. Segment(J, B) → k
  13. Segment(K, M) → l
  14. Midpoint(I, H) → O
  15. Midpoint(I, O) → P
  16. Midpoint(O, H) → Q
  17. Midpoint(F, G) → R
  18. Midpoint(F, R) → S
  19. Midpoint(R, G) → T
  20. Segment(P, S) → m
  21. Segment(Q, T) → p
  22. Point(g) → U
  23. Point(g) → V
  24. Point(g) → W
  25. Point(g) → Z
  26. Segment(R, O) → a_1
  27. CurveCartesian(((a * α) * cos(α)), ((a * α) * sin(α)), α, 0, β) → d
  28. CurveCartesian(((a * α) * cos(α)), (((-a) * α) * sin(α)), α, 0, (-β)) → e
  29. Segment(L, N) → t
  30. Vector(C_3, C_2) → u
  31. Vector(C_4, C_1) → w
  32. Point(g) → C
  33. Point(g) → D
  34. Point(g) → A_1
  35. Point(g) → B_1
  36. Point(g) → D_1
  37. Point(g) → E_1
  38. Point(j) → F_1
  39. Point(g) → G_1
  40. Point(g) → H_1
  41. Point(g) → I_1
  42. Point(g) → J_1
  43. Point(g) → K_1
  44. Intersect(g, yAxis, (0, -6)) → L_1
  45. Vector(C_6, C_5) → u_1
  46. Polygon(M_1, N_1, O_1, P_1) → poly2, m_1, n_1, o_1, p_3
  47. Vector(J_1, W_1) → v_1
  48. Vector(J_1, Z_1) → w_1
  49. Point(yAxis) → A_2
  50. Vector(J_1, A_2) → c_1
  51. Point(xAxis) → B_2
  52. Point(xAxis) → D_2
  53. Vector(B_2, D_2) → d_1
  54. Point(yAxis) → E_2
  55. Vector(A_2, E_2) → e_1
  56. OrthogonalLine(B'_1, xAxis) → i_1
  57. Intersect(i_1, xAxis) → F_2
  58. Vector(F_2, B'_1) → j_1
  59. Segment(B'_1, F_2) → k_1
  60. Point(e_1) → G_2
  61. Midpoint(F_2, B'_1) → H_2
  62. Segment(G_2, F_2) → l_1
  63. Midpoint(G_2, F_2) → I_2
  64. OrthogonalLine(B'_1, a_1) → a_2
  65. Intersect(a_1, yAxis) → J_2
  66. Intersect(a_2, yAxis) → K_2
  67. Vector(G_2, F_2) → u_2
  68. Polygon(L_2, M_2, N_2, O_2) → poly1, l_2, m_2, n_2, o_2
  69. Segment(A, B'_1) → b
  70. Midpoint(A, B'_1) → P_2
  71. Point(b) → Q_2
  72. Polygon(R_2, S_2, T_2, U_2) → poly3, r_2, s_2, t_2, u_3
  73. If(1000 < yxtan < -1000, text45) → text46
  74. Point(c_1) → V_2
  75. Point(c) → W_2
  76. Point(c) → Z_2
  77. Point(c) → A_3
  78. Point(c) → B_3
  79. Point(c) → D_3
  80. Point(c) → E_3
  81. Point(c) → F_3
  82. Point(c) → G_3
  83. Point(c) → H_3
  84. Point(c) → I_3
  85. Point(c) → J_3
  86. Point(c) → K_3
  87. Point(c) → L_3
  88. Point(c) → M_3
  89. Point(c) → N_3
  90. Point(c) → O_3