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Unit Circle Animation

Unit Circle Animation
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2019-04-15 09:15
CircleVectorTrigonometryCoordinate Geometry
This example demonstrates the dynamic relationship between a point, vector, and trigonometric values (sine, cosine) on a unit circle. It involves core mathematical concepts such as the unit circle, trigonometric functions, vectors, and analytical geometry. Key GeoGebra commands and objects used include If, Intersect, Point, Segment, CircleArc, OrthogonalLine, Circle, Polygon, CircleSector, LaTeX, Numerator, Denominator. Suitable for high school mathematics teaching to visually understand the connection between trigonometric functions and geometric figures, enhancing spatial reasoning skills.

GeoGebra Commands & Parameters

  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