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Exploring the Unit Circle

Exploring the Unit Circle
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2020-02-05 11:31
CircleVectorTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the unit circle and its connection with special right triangles. It involves core math concepts like the definition of the unit circle, trigonometric values, and geometric representations of special angles (30°, 45°, 60°). Construction highlights include using GeoGebra commands such as Point, Sequence, Segment, Min, If, Polygon, Element, Random, Angle to build the unit circle, special triangles, and angle relationships. Suitable for high school mathematics education to understand the relationship between trigonometric functions and geometric shapes.

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 - acosd(x(A)))) → d306090
  6. Min(abs(Angles459045 - acosd(x(A)))) → d459045
  7. Min(abs(AnglesVH - acosd(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, acosd(x(A)), π + acosd((-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, acosd(x(A_3)), π + acosd((-x(A_3)))) → theta_1
  41. If(π < theta ∧ theta < (2 * π), "-", "") → negSin

AI Drawing Prompt Reference

Draw a unit circle and mark the points corresponding to special angles like 30°, 45°, and 60° on it. Connect these points to the origin to form three special right triangles. Also, mark the projection points of the unit circle on the coordinate axes and display the trigonometric values of these angles. Use different colors to distinguish the triangles for different angles and label the angle values and trigonometric values.
The prompt above was compiled from the original drawing commands and is for reference only.