MathPlotAI Logo MathPlotAI

Explore the Unit Circle

Explore the Unit Circle
Loading the math board and drawing, please wait…
2016-03-08 21:02
CircleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the unit circle and its connection with special right triangles. It involves core mathematical concepts such as the definition of the unit circle, trigonometric values, and the side-angle relationships of special right triangles. The construction uses GeoGebra commands and objects like Point, Sequence, Segment, Min, If, Polygon, Element, Random, Angle, etc., to provide a dynamic visualization for better understanding. Suitable for high school mathematics teaching and learning trigonometry and geometry.

GeoGebra Commands & Parameters

  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