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单位圆动画

单位圆动画
几何画板及绘图内容加载中，请稍等……
2019-04-15 09:15
圆向量三角函数解析几何
①本范例演示单位圆上点、向量与三角函数值（正弦、余弦）的动态关系；②涉及核心数学概念包括单位圆、三角函数、向量、解析几何；③作图要点运用了If、Intersect、Point、Segment、CircleArc、OrthogonalLine、Circle、Polygon、CircleSector、LaTeX、Numerator、Denominator等GeoGebra命令与对象；④适合高中数学教学，用于直观理解三角函数与几何图形的联系，增强空间想象能力。

GeoGebra 命令及参数

  1. If(smooth ≟ 0, ((step1 + step2 + step3 + step4 + step5 + step6 + step7 + step8 + step9 + step10 + step11 + step12 + step13 + step14 + step15 + step16) * π / 6) + ((2 * π) * n), smoothplay) → argument
  2. Intersect(xAxis, yAxis) → C
  3. Point(snap) → D
  4. Point(p) → Z
  5. Point(snap) → B
  6. If(n ≥ 1, 359.99 + (0.01 * (n^(2) - 1) / n^(2)), 0.01 / (n^(2) + 1)) → close
  7. Intersect(xAxis, yAxis) → A
  8. If(Row[A2] / 2 ≟ ceil(Row[A2] / 2), LaTeX["\text{" + (Row[A3]) + "\pi}"], Row[A3] + 1) → A4
  9. If(Row[A3] / 2 ≟ ceil(Row[A3] / 2), LaTeX["\text{" + (Row[A4]) + "\pi}"], Row[A4] + 1) → A5
  10. If(Row[A4] / 2 ≟ ceil(Row[A4] / 2), LaTeX["\text{" + (Row[A5]) + "\pi}"], Row[A5] + 1) → A6
  11. If(Row[A5] / 2 ≟ ceil(Row[A5] / 2), LaTeX["\text{" + (Row[A6]) + "\pi}"], Row[A6] + 1) → A7
  12. If(Row[A6] / 2 ≟ ceil(Row[A6] / 2), LaTeX["\text{" + (Row[A7]) + "\pi}"], Row[A7] + 1) → A8
  13. If(Row[A7] / 2 ≟ ceil(Row[A7] / 2), LaTeX["\text{" + (Row[A8]) + "\pi}"], Row[A8] + 1) → A9
  14. If(Row[A8] / 2 ≟ ceil(Row[A8] / 2), LaTeX["\text{" + (Row[A9]) + "\pi}"], Row[A9] + 1) → A10
  15. If(Row[A9] / 2 ≟ ceil(Row[A9] / 2), LaTeX["\text{" + (Row[A10]) + "\pi}"], Row[A10] + 1) → A11
  16. If(Row[A10] / 2 ≟ ceil(Row[A10] / 2), LaTeX["\text{" + (Row[A11]) + "\pi}"], Row[A11] + 1) → A12
  17. If(Row[A11] / 2 ≟ ceil(Row[A11] / 2), LaTeX["\text{" + (Row[A12]) + "\pi}"], Row[A12] + 1) → A13
  18. If(Row[A12] / 2 ≟ ceil(Row[A12] / 2), LaTeX["\text{" + (Row[A13]) + "\pi}"], Row[A13] + 1) → A14
  19. If(Row[A13] / 2 ≟ ceil(Row[A13] / 2), LaTeX["\text{" + (Row[A14]) + "\pi}"], Row[A14] + 1) → A15
  20. If(Row[A14] / 2 ≟ ceil(Row[A14] / 2), LaTeX["\text{" + (Row[A15]) + "\pi}"], Row[A15] + 1) → A16
  21. If(Row[A15] / 2 ≟ ceil(Row[A15] / 2), LaTeX["\text{" + (Row[A16]) + "\pi}"], Row[A16] + 1) → A17
  22. If(Row[A16] / 2 ≟ ceil(Row[A16] / 2), LaTeX["\text{" + (Row[A17]) + "\pi}"], Row[A17] + 1) → A18
  23. If(cosine, LaTeX["\text{\cos}"], If[sine, LaTeX["\text{\sin}"], LaTeX["\text{\tan}"]]) → functiontext
  24. If(cosine, LaTeX["\fcolorbox{black}{white}{\text{The Cosine Function}}"], If[sine, LaTeX["\fcolorbox{black}{white}{\text{The Sine Function}}"], LaTeX["\fcolorbox{black}{white}{\text{The Tangent Function}}"]]) → Title
  25. If(cosine, LaTeX["\fcolorbox{black}{white}{\text{The Cosine Function}}"], If[sine, LaTeX["\fcolorbox{black}{white}{\text{The Sine Function}}"], LaTeX["\fcolorbox{black}{white}{\text{The Tangent Function}}"]]) → Titleguide
  26. Segment((1, 0), (-1, 0)) → b_1
  27. Segment((0, 1), (0, -1)) → h_2
  28. Segment(A, D_1) → f
  29. CircleArc(A, (1 / 3 * D), (1 / 3 * D_1)) → g
  30. CircleArc(A, (2 / 3 * D), (2 / 3 * D_1)) → h
  31. If(0 < argument ≤ (2 * π), (1 / 3 * D_1), If[argument > (2 * π), (2 / 3 * D_1)]) → E
  32. OrthogonalLine(E, f) → i
  33. Circle(E, arrowlength) → q
  34. Intersect(q, i, 1) → F
  35. OrthogonalLine(F, i) → j
  36. Circle(F, arrowwidth) → r
  37. Intersect(r, j, 1) → G
  38. Intersect(r, j, 2) → H
  39. Polygon(E, H, G) → t1, g_1, e_1, h_1
  40. CircleSector(A, (1 / 3, 0), (1 / 3 * D_1)) → c
  41. If(sine, If[0 ≤ x ≤ argument, sin(x)], If[cosine, If[0 ≤ x ≤ argument, cos(x)], If[0 ≤ x ≤ argument, tan(x)]]) → trig
  42. LaTeX("\text{\frac{\sqrt{3}}{2}}") → s32
  43. LaTeX("\text{\minus\frac{\sqrt{3}}{2}}") → ns32
  44. LaTeX("\text{\frac{\sqrt{2}}{2}}") → s22
  45. LaTeX("\text{\minus\frac{\sqrt{2}}{2}}") → ns22
  46. LaTeX("\text{\frac{1}{2}}") → oh
  47. LaTeX("\text{\minus\frac{1}{2}}") → noh
  48. LaTeX("\text{0}") → zero
  49. LaTeX("\text{1}") → one
  50. LaTeX("\text{\minus 1}") → none
  51. CircleSector(A, (2 / 3, 0), (2 / 3 * D_1)) → d
  52. Numerator(argument / π) → num
  53. Denominator(argument / π) → denom
  54. If(num ≠ 1 ∧ denom ≠ 1, LaTeX[(((("\text{\frac{" * LaTeX[num]) * "\pi}{") * LaTeX[denom]) * "}}")], If[num ≟ 1 ∧ denom ≠ 1, LaTeX[(("\text{\frac{\pi}{" * LaTeX[denom]) * "}}")], If[denom ≟ 1 ∧ num ≠ 1 ∧ num ≠ 0, LaTeX[(("\text{" * LaTeX[num]) * "\pi}")], If[num ≟ 0, LaTeX["\text{0}"], LaTeX["\text{\pi}"]]]]) → text34
  55. If(num ≠ 1 ∧ denom ≠ 1, LaTeX[(((("\text{\frac{" * LaTeX[num]) * "\pi}{") * LaTeX[denom]) * "}}")], If[num ≟ 1 ∧ denom ≠ 1, LaTeX[(("\text{\frac{\pi}{" * LaTeX[denom]) * "}}")], If[denom ≟ 1 ∧ num ≠ 1 ∧ num ≠ 0, LaTeX[(("\text{" * LaTeX[num]) * "\pi}")], If[num ≟ 0, LaTeX["\text{0}"], LaTeX["\text{\pi}"]]]]) → text34guide
  56. If(0 ≤ argument ∧ argument < (2 * π), argument, If[(2 * π) ≤ argument ∧ argument < (4 * π), argument - (2 * π), argument - (4 * π)]) → argforread
  57. If(smooth ≟ 1, LaTeX["y"], If[cosine, LaTeX[cosoutput], If[sine, LaTeX[sinoutput], LaTeX[tanoutput]]]) → output
  58. Segment(I, Tracepoint) → l
  59. Segment(Tracepoint, J) → m
  60. If(smooth ≟ 1, LaTeX["y"], If[cosine, LaTeX[cosoutput], If[sine, LaTeX[sinoutput], LaTeX[tanoutput]]]) → guideoutput
  61. If((¬(D_1 ∈ snap)) ∨ smooth ≟ 1, LaTeX["x"], If[showdegrees, LaTeX[degtext], LaTeX[text34]]) → guidextext
  62. If((¬(D_1 ∈ snap)) ∨ smooth ≟ 1, LaTeX["x"], If[showdegrees, LaTeX[degtext], LaTeX[text34]]) → graphxtext
  63. If(tangent, "tan", If[cosine, "cos", If[sine, "sin"]]) → functext