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Using the Unit Circle

Using the Unit Circle
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2015-09-29 09:45
CircleVectorTrigonometry
This example demonstrates how points on the unit circle represent angles and trigonometric values. Key mathematical concepts include angles, trigonometry, and vectors. Using GeoGebra commands such as Sequence, Intersect, and Circle, the relationship between angles and trigonometric values is dynamically illustrated. Suitable for high school mathematics education and learning.

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[(k * π / 2, 0), (k * π / 2, 1)], k, -100, 100) → list25