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Unit Circle to Sine and Cosine Functions Dynamic Illustrator by Megan

Unit Circle to Sine and Cosine Functions Dynamic Illustrator by Megan
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2020-04-10 01:01
CircleFunctionTrigonometryCoordinate GeometrySlider Animation
This example dynamically illustrates the relationship between the unit circle and sine/cosine functions, visually demonstrating the geometric meaning of trigonometric functions. Core math concepts include the unit circle, trigonometric functions, similarity transformation, tangent line, continuity, etc. Construction highlights involve GeoGebra commands like Rotate, OrthogonalLine, Intersect, Vector, Dilate, Translate, Angle, Segment, CircleArc, and Circle. Suitable for high school students and teachers to understand the origin and properties of trigonometric function graphs, as well as for teaching demonstrations in calculus and analytical geometry.

GeoGebra Commands & Parameters

  1. Rotate(A, ((a * 2) * π), (0, 0)) → A'
  2. OrthogonalLine(A', xAxis) → f
  3. Intersect(f, xAxis) → B
  4. Vector(B, A') → u
  5. Dilate(D, If[0 < a ≤ 1, 1 - a, 0], C) → D'
  6. Vector(D', E) → v
  7. Dilate(B, If[0 < b ≤ 1, 1 - b, 0], D') → B'
  8. Vector(B, B') → w
  9. Translate(A', w) → A''
  10. Vector(B', A'') → d
  11. Angle(A, F, A') → θ
  12. Segment(F, A') → k
  13. CircleArc(F, A, A') → q
  14. Dilate(A, If[1 < b ≤ 2, 2 - b, 0], D) → A'_1
  15. Vector(A, A'_1) → e
  16. Dilate(D, If[2 < b ≤ 3, 3 - b, 0], B') → D'_1
  17. Vector(D, D'_1) → m
  18. Circle(D'_1, 1) → r
  19. Dilate(D, If[3 < b ≤ 4, 4 - b, 0], D') → D'_2
  20. Vector(D, D'_2) → t
  21. Dilate(D, If[1 < b ≤ 2, 2 - b, 0], A) → D'_3
  22. Vector(D, D'_3) → n
  23. Translate(D, n) → D'_4
  24. Vector(H, A') → s
  25. Dilate(H, If[0 < b ≤ 1, 1 - b, 0], D') → H'
  26. Vector(H, H') → u_1
  27. Translate(s, u_1) → s'
  28. Translate(A', u_1) → A''_1
  29. Vector(H', A''_1) → v_1
  30. Rotate(A''_1, If[0 < b ≤ 1, (b * π / 2), π / 2], H') → I
  31. Vector(H', I) → w_1
  32. Rotate(A''_1, If[0 < b ≤ 1, ((-b) * π / 2), ((-π)) / 2], H') → J

AI Drawing Prompt Reference

Draw a unit circle and select a point A on it. Rotate A around the origin to get A'. Draw a perpendicular line f from A' to the x-axis, and let B be the intersection of f and the x-axis. Connect B and A' to form vector u. Scale B to get B' based on the value of b, then connect B' and A' to form vector w. Translate A' along w to get A''. Connect A' and A'' to form vector d. Calculate the angle θ between A' and F. Draw an arc q from F with A' as the starting point. Scale A' to get A'_1, then connect A' and A'_1 to form vector e. Repeat this process with D' based on the value of b, resulting in a series of circles and line segments that illustrate the relationship between the unit circle and sine and cosine functions.
The prompt above was compiled from the original drawing commands and is for reference only.