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Wrapping Real Numbers Around the Unit Circle

Wrapping Real Numbers Around the Unit Circle
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2019-09-09 07:35
CircleAngleFunction
This example demonstrates how to wrap real numbers around the unit circle. Key mathematical concepts include real numbers, unit circle, and angles. Using GeoGebra commands such as If, Point, Textfield, and Angle, the construction shows the correspondence between real numbers and the unit circle. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. If(X_1 ≥ 0, Arc[c, (1, 0), (cos(X_1), sin(X_1))], Arc[c, (cos(X_1), sin(X_1)), (1, 0)]) → X
  2. If(X_1 > 0, If[((-1)^(floor(X_1 / π)) * x) < ((-1)^(floor(X_1 / π)) * cos(X_1)), (((-cos(X_1))) / sin(X_1) * (x - cos(X_1))) + sin(X_1)]) → f
  3. If(X_1 ≟ 0, x = 1) → g
  4. If(X_1 > 0, Segment[(1, 0), (1, -10)], X_1 < 0, Segment[(1, 0), (1, 10)]) → h
  5. If(X_1 < 0, If[((-1)^(floor(X_1 / π)) * x) > ((-1)^(floor(X_1 / π)) * cos(X_1)), (((-cos(X_1))) / sin(X_1) * (x - cos(X_1))) + sin(X_1)]) → p
  6. If(0 < X_1 < 1, Point[{cos(X_1) - ((1 - X_1) * sin(X_1)), sin(X_1) + ((1 - X_1) * cos(X_1))}]) → B
  7. If(X_1 ≥ 1, Point[{cos(1), sin(1)}]) → C
  8. If(0 < X_1 < 2, Point[{cos(X_1) - ((2 - X_1) * sin(X_1)), sin(X_1) + ((2 - X_1) * cos(X_1))}]) → D
  9. If(X_1 ≤ 0, Point[{1, 2}]) → E
  10. If(X_1 ≤ 0, Point[{1, 1}]) → A
  11. If(X_1 ≥ 2, Point[{cos(2), sin(2)}]) → F
  12. If(X_1 ≤ 0, Point[{1, 3}]) → G
  13. If(0 < X_1 < 3, Point[{cos(X_1) - ((3 - X_1) * sin(X_1)), sin(X_1) + ((3 - X_1) * cos(X_1))}]) → H
  14. If(X_1 ≥ 3, Point[{cos(3), sin(3)}]) → I
  15. If(X_1 ≥ 0, Point[{1, -1}]) → J
  16. If(-1 < X_1 < 0, Point[{cos(X_1) - ((-1 - X_1) * sin(X_1)), sin(X_1) + ((-1 - X_1) * cos(X_1))}]) → K
  17. If(X_1 ≤ -1, Point[{cos(-1), sin(-1)}]) → L
  18. If(X_1 ≥ 0, Point[{1, -2}]) → M
  19. If(-2 < X_1 < 0, Point[{cos(X_1) - ((-2 - X_1) * sin(X_1)), sin(X_1) + ((-2 - X_1) * cos(X_1))}]) → N
  20. If(X_1 ≤ -2, Point[{cos(-2), sin(-2)}]) → O
  21. If(X_1 ≥ 0, Point[{1, -3}]) → P
  22. If(-3 < X_1 < 0, Point[{cos(X_1) - ((-3 - X_1) * sin(X_1)), sin(X_1) + ((-3 - X_1) * cos(X_1))}]) → Q
  23. If(X_1 ≤ -3, Point[{cos(-3), sin(-3)}]) → R
  24. If(X_1 ≥ 0, Point[{1, -4}]) → S
  25. If(-4 < X_1 < 0, Point[{cos(X_1) - ((-4 - X_1) * sin(X_1)), sin(X_1) + ((-4 - X_1) * cos(X_1))}]) → T
  26. If(X_1 ≤ -4, Point[{cos(-4), sin(-4)}]) → U
  27. If(0 < X_1 < 4, Point[{cos(X_1) - ((4 - X_1) * sin(X_1)), sin(X_1) + ((4 - X_1) * cos(X_1))}]) → W
  28. If(X_1 ≥ 4, Point[{cos(4), sin(4)}]) → Z
  29. Point({cos(X_1), sin(X_1)}) → A_1
  30. If(X_1 ≤ 0, Point[{1, 4}]) → V
  31. If(X_1 ≤ 0, Point[{1, 5}]) → B_1
  32. If(0 < X_1 < 5, Point[{cos(X_1) - ((5 - X_1) * sin(X_1)), sin(X_1) + ((5 - X_1) * cos(X_1))}]) → C_1
  33. If(X_1 ≥ 5, Point[{cos(5), sin(5)}]) → D_1
  34. If(X_1 ≤ 0, Point[{1, 6}]) → E_1
  35. If(0 < X_1 < 6, Point[{cos(X_1) - ((6 - X_1) * sin(X_1)), sin(X_1) + ((6 - X_1) * cos(X_1))}]) → F_1
  36. If(X_1 ≥ 6, Point[{cos(6), sin(6)}]) → G_1
  37. Point({1, 0}) → H_1
  38. Textfield(X_1) → InputBox1
  39. Point({cos(X_1), sin(X_1)}) → myPoint
  40. Angle(Vector[(1, 0)], victor) → θ
  41. Angle(victor, Vector[(1, 0)]) → theta