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Unit Circle Demonstration by Tom Owsiak

Unit Circle Demonstration by Tom Owsiak
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2013-04-21 11:03
CircleFunctionTrigonometryCoordinate GeometryParametric Equation
This example demonstrates the construction of the unit circle and its relationship with trigonometric functions. It involves core mathematical concepts including circle, trigonometry (sine, cosine), parametric equations, and coordinate geometry. Key GeoGebra commands used in the construction are Sequence, Intersect, Circle, Textfield, Segment, Simplify, Angle, TrigExpand, Mod, Point, CurveCartesian, TableText, and Text. It is suitable for high school and above students to learn and understand trigonometric functions, coordinate geometry, and parametric equations.

GeoGebra Commands & Parameters

  1. Sequence(FractionText[k / fractionsofpie], k, 1, advancedpointoncircle) → list5
  2. Sequence(FractionText[k * 180 / fractionsofpie], k, 1, advancedpointoncircle) → list7
  3. Sequence((r * cos(k * π / fractionsofpie), r * sin(k * π / fractionsofpie)), k, 1, advancedpointoncircle) → list3
  4. Sequence(Text[(Element[list5, t] * "π=" * Element[list7, t])°, (x(Element[list3, t]) + movlabelshorizontally, y(Element[list3, t]) + movelabelsvertically), false, true], t, 1, 2 * fractionsofpie) → list6
  5. Sequence(Text["cos(" * FractionText[k / fractionsofpie] * " π )=", (0, 0), false, true], k, 1, advancedpointoncircle) → list8
  6. Intersect(xAxis, yAxis) → A
  7. Circle(A, 1) → c
  8. Textfield(fractionsofpie) → inputBox1
  9. Circle((0, 0), r) → d
  10. Sequence(Text["$\frac{" * 2 * π * "}{" * fractionsofpie * "}$", Element[list3, t], false, true], t, 1, Length[list3]) → list4
  11. FractionText(2 / fractionsofpie * π) → text2
  12. Textfield(r) → inputBox2
  13. Textfield(advancetriangletonextangle) → inputBox3
  14. Segment((0, 0), C) → e
  15. Segment(C, B) → f
  16. Segment(A, B) → g
  17. Simplify(newforcosine) → h
  18. Segment(A, B) → i
  19. Angle(C, A, B) → α
  20. FractionText(2 / fractionsofpie) → text4
  21. Angle(E, A, B) → β
  22. TrigExpand(sin(2 * x)) → p
  23. Angle(C, A, B) → γ
  24. Mod(375, 360) → l
  25. Sequence((cos(k * π / fractionsofpie), sin(k * π / fractionsofpie)), k, 0, 2 * fractionsofpie) → list1
  26. Sequence(Segment[(0, 0), (cos(k * π / fractionsofpie), sin(k * π / fractionsofpie))], k, 1, 2 * fractionsofpie) → list2
  27. Dimension(list6) → m
  28. Dimension(list6) → n
  29. Point(i) → F
  30. Sequence(Text["\frac{\sqrt{2}}{2}", (k, 0), false, true], k, 1, 4) → list9
  31. Sequence(Text[(Element[list5, t] * "π=" * Element[list7, t])°, (x(Element[list3, t]) + movlabelshorizontally, y(Element[list3, t]) + movelabelsvertically), false, true], t, 1, advancedpointoncircle) → list12
  32. Sequence(cos(k * π / fractionsofpie), k, 1, advancedpointoncircle) → list13
  33. CurveCartesian((1 / 10 + t / 100) * cos(t * π / fractionsofpie), (1 / 10 + t / 100) * sin(t * π / fractionsofpie), t, 0, advancedpointoncircle) → q
  34. Sequence(Text["sin(" * FractionText[k / fractionsofpie] * " π )=", (0, 0), false, true], k, 1, advancedpointoncircle) → list14
  35. Sequence(sin(k * π / fractionsofpie), k, 1, advancedpointoncircle) → list15
  36. Sequence(Angle[Element[list1, t], (0, 0), Element[list1, t + 1]], t, 1, Length[list1]) → list17
  37. Sequence(Text["tan(" * FractionText[k / fractionsofpie] * " π )=", (0, 0), false, true], k, 1, advancedpointoncircle) → list18
  38. Sequence(tan(k * π / fractionsofpie), k, 1, advancedpointoncircle) → list19
  39. TableText({list12, list8, list13, list14, list15, list18, list19}, "v") → text5
  40. Sequence((r / 3 * cos(π / fractionsofpie * k), r / 3 * sin(π / fractionsofpie * k)), k, 0, 2 * fractionsofpie, 0.5) → list20
  41. Sequence(Element[list20, t], t, 0, Length[list20], 2) → list22
  42. Sequence(Text["\frac{π}{" * fractionsofpie * "}", (x(Element[list22, k]) + movlabelshorizontally, y(Element[list22, k]) + movelabelsvertically), false, true], k, 1, Length[list22], 1) → list21
  43. Sequence(Text[(Element[list5, t] * "π=" * Element[list7, t])°, (x(Element[list3, t]) + movlabelshorizontally, y(Element[list3, t]) + movelabelsvertically), false, true], t, Length[list6] + 1, advancedpointoncircle) → secondsetpiover4
  44. Segment((-999.9714194038698, 0), (1000.0285805961302, 0)) → a_1
  45. Segment((0, -100), (0, 100)) → b_1
  46. Sequence((k * π / fractionsofpie, 0), k, -100, 100) → list23
  47. Sequence(Text["\frac{" * k * "π}{" * fractionsofpie * "}", (k * π / fractionsofpie, 0), false, true], k, -100, 100) → list24
  48. Sequence(Line[(π / 2 + k * π, 0), (π / 2 + k * π, 1)], k, -100, 100) → list25
  49. Text("\frac{\sqrt{2}}{2}", (0, 0), false, true) → text3
  50. Segment(G, H) → i_1
  51. Segment(G, J) → j_1
  52. Sequence(Line[(k * π, 0), (k * π, 1)], k, -100, 100) → list31

AI Drawing Prompt Reference

Draw a unit circle centered at the origin, including lines from the center to points on the circumference, and labels for angles and their corresponding sine and cosine values. First, draw a circle with radius 1, then draw segments from the center to points on the circle, and label the angles and their sine and cosine values. Next, draw the x-axis and y-axis, and label the degrees of the points on the circumference.
The prompt above was compiled from the original drawing commands and is for reference only.