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探索单位圆 by Anthony DiLaura

探索单位圆 by Anthony DiLaura
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2016-03-08 21:02
圆三角函数解析几何几何作图
本范例演示单位圆及其与特殊直角三角形的关系。涉及的核心数学概念包括单位圆的定义、三角函数值、特殊直角三角形的边角关系。作图要点使用了GeoGebra的Point、Sequence、Segment、Min、If、Polygon、Element、Random、Angle等命令和对象，通过动态演示帮助理解。适合高中数学教学和学生学习三角函数与几何。

GeoGebra 命令及参数

  1. Point(c) → A
  2. Sequence(((π * k)) / 12, k, 0, 24) → Angles
  3. Sequence((cos(Element[Angles306090, k]), sin(Element[Angles306090, k])), k, 1, 9) → Pts306090
  4. Segment(O, B) → d
  5. Min(abs(Angles306090 - acos(x(A)))) → d306090
  6. Min(abs(Angles459045 - acos(x(A)))) → d459045
  7. Min(abs(AnglesVH - acos(x(A)))) → dVH
  8. If((d306090 ≤ d459045) ∧ (d306090 ≤ dVH), 1, 0) → is306090
  9. If((d459045 < d306090) ∧ (d459045 < dVH), 1, 0) → is459045
  10. If((dVH < d306090) ∧ (dVH < d459045), 1, 0) → isVH
  11. If(y(A) > 0, acos(x(A)), π + acos((-x(A)))) → theta
  12. Polygon(O, A, B) → Triangle306090, b, o, a
  13. Polygon(O, A, B) → Triangle459045, b_1, o_1, a_1
  14. Polygon(O, A, B) → TriangleVH, b_2, o_2, a_2
  15. Sequence((cos(Element[Angles459045, k]), sin(Element[Angles459045, k])), k, 1, 4) → Pts459045
  16. Sequence((cos(Element[AnglesVH, k]), sin(Element[AnglesVH, k])), k, 1, 4) → PtsVH
  17. Sequence((cos(Element[AnglesAll, k]), sin(Element[AnglesAll, k])), k, 1, Length[AnglesAll]) → PtsAll
  18. Element(AnglesAllLabels, 2) → Fail
  19. Random(1, 14) → randInt
  20. If((π / 2 < theta) ∧ (theta < ((3 * π)) / 2), "-", "") → negCos
  21. If(tan(theta) < 0, "-", "") → negTan
  22. Angle(O2, O, A) → alphaGreen
  23. If(theta ≟ π / 3 ∨ theta ≟ (2 * π / 3) ∨ theta ≟ (4 * π / 3) ∨ theta ≟ (5 * π / 3), 1) → isTall306090
  24. If(theta ≟ π / 6 ∨ theta ≟ (5 * π / 6) ∨ theta ≟ (7 * π / 6) ∨ theta ≟ (11 * π / 6), 1) → isSkinny306090
  25. If(theta ≟ π / 4 ∨ theta ≟ (3 * π / 4) ∨ theta ≟ (5 * π / 4) ∨ theta ≟ (7 * π / 4), 1) → isReal459045
  26. If(theta ≟ 0 ∨ theta ≟ π ∨ theta ≟ (2 * π), 1) → isH
  27. Point(xAxis) → C
  28. Point(xAxis) → D
  29. Point(xAxis) → E
  30. Point(xAxis) → F
  31. Point(yAxis) → G
  32. Point(yAxis) → H
  33. Point(yAxis) → I
  34. Point(yAxis) → J
  35. Point(c_1) → A_1
  36. Point(c_2) → A_2
  37. Angle(O2, O, A) → alphaRed
  38. Angle(O2, O, A) → alphaBlue
  39. Point(c_3) → A_3
  40. If(y(A_3) > 0, acos(x(A_3)), π + acos((-x(A_3)))) → theta_1
  41. If((π < theta) ∧ (theta < (2 * π)), "-", "") → negSin