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单位圆三角函数比值

单位圆三角函数比值
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2016-09-14 01:57
圆函数三角函数解析几何几何作图
①本范例演示单位圆上任意角度α对应的三角函数sinα, cosα, tanα的几何意义；②涉及核心数学概念包括单位圆、三角函数定义、直角三角形边长比值、角度与弧度制转换；③作图要点提炼自GeoGebra命令：使用Intersect求交点、Function绘制函数图像、Point创建点、Circle画圆、Tangent作切线、Segment画线段、OrthogonalLine作垂线、Angle测量角度、Ray画射线；④适合高中数学学生理解三角函数与单位圆关系，也可用于微积分中三角函数极限的教学演示。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Function(0, 1.9, 5.5) → g
  3. Point(g) → Q
  4. Point(yAxis) → B
  5. Circle(A, B) → c
  6. Point(c) → D
  7. Tangent(D, c) → e,
  8. Intersect(e, yAxis) → F
  9. Intersect(xAxis, e) → E
  10. Segment(F, E) → l
  11. Segment(E, A) → m
  12. Intersect(m, l) → O
  13. Intersect(c, xAxis, 2) → C
  14. Segment(A, C) → d
  15. OrthogonalLine(D, d) → f
  16. Intersect(f, m) → H
  17. Segment(A, H) → cosα
  18. Angle(Q, A, D) → α
  19. Segment(A, D) → b
  20. Segment(D, E) → tanα
  21. Segment(D, H) → sinα
  22. OrthogonalLine(D, sinα) → i
  23. Function(sin((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → sineα
  24. Intersect(sineα, h, (2.612829828523947, 0.8618940213928555)) → I
  25. Function(tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → tangentα
  26. Intersect(tangentα, h, (2.612829828523947, 1.6996916845347747)) → J
  27. Function(cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cosineα
  28. Intersect(cosineα, h, (2.612829828523947, 0.5070884497671505)) → K
  29. Segment(H, D) → j
  30. Segment(A, F) → k
  31. Intersect(j, m) → N
  32. Intersect(k, l) → P
  33. Function(1, 1.9, 5.5) → p
  34. Function(-1, 1.9, 5.5) → q
  35. Point(c) → R
  36. Segment(C, R) → a
  37. Segment(A, H) → g_1
  38. Segment(D, H) → h_1
  39. Segment(A, E) → j_1
  40. Angle(A, D, E) → β
  41. Ray(D, F) → i_1
  42. Ray(D, E) → l_1
  43. Intersect(c, a) → G, L
  44. OrthogonalLine(I, xAxis) → d_1
  45. Intersect(g, xAxis, 1) → M
  46. Intersect(g, xAxis, 1) → S
  47. Intersect(g, xAxis, 1) → T
  48. Intersect(g, d_1) → U
  49. Segment(I, U) → e_1
  50. OrthogonalLine(K, xAxis) → k_1
  51. Intersect(k_1, xAxis) → V
  52. Segment(K, V) → m_1
  53. OrthogonalLine(J, xAxis) → n_1
  54. Intersect(n_1, xAxis) → W
  55. Segment(J, W) → p_1
  56. Point(yAxis) → Z
  57. Point(yAxis) → A_1
  58. Segment(Z, A_1) → q_1
  59. Point(xAxis) → B_1
  60. Point(xAxis) → C_1
  61. Segment(B_1, C_1) → r_1
  62. OrthogonalLine(D, r_1) → c_1
  63. Intersect(g, r_1, 1) → D_1
  64. Intersect(c_1, r_1) → E_1
  65. Segment(D, E_1) → s_1
  66. Angle(D, E_1, A) → γ