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Unit Circle and Sine Graph Dynamic Generation

Unit Circle and Sine Graph Dynamic Generation
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2020-03-11 19:43
CircleFunctionTrigonometrySlider AnimationGraphing
This GeoGebra example dynamically generates the sine graph from the unit circle, showing how the y-coordinate of a point rotating on the unit circle maps to the sine function curve. It covers core concepts such as the unit circle, radian measure, the definition of sine, and its periodic nature. The construction uses commands like Point, Rotate, Angle, Function, Segment, Circle, Vector, and Sequence to build the circle, rotating angle, sine curve, and auxiliary segments with text labels. By animating the angle θ, learners can visually connect the point's vertical position on the circle to the corresponding point on the sine wave. Ideal for high school trigonometry lessons and introductory college mathematics, this interactive tool strengthens the link between geometric visualization and function graphing.

GeoGebra Commands & Parameters

  1. Point(Ray[(0, 0), (-1, 0)]) → B
  2. Rotate(A, θ, B) → P
  3. Rotate(A, θ, B) → A'
  4. Angle(A, B, A') → α
  5. Rotate(A, θ, B) → A'_1
  6. Angle(A, B, A'_1) → β
  7. Function(sin((120° * x)), 0, x(Q)) → f
  8. Segment(Q, (x(Q), 0)) → e
  9. Circle(B, A) → c
  10. Segment(B, P) → a
  11. Vector((x(B), y(Corner[1])), (x(B), y(Corner[4]))) → u
  12. Vector((0, y(Corner[1])), (0, y(Corner[4]))) → w
  13. Vector((0, 0), (x(Corner[2]), 0)) → z
  14. Sequence(Segment[((0.75 * i), 0), ((0.75 * i), -0.025)], i, 1, 4) → list1
  15. Segment(P, (x(P), 0)) → b
  16. Vector(B - (1.2, 0), B + (1.2, 0)) → v
  17. Sequence(Segment[(0, (0.5 * i) - 1), (-0.025, (0.5 * i) - 1)], i, 0, 4) → list2
  18. Sequence(Segment[(x(B), (0.5 * i) - 1), (x(B) - 0.025, (0.5 * i) - 1)], i, 0, 4) → list3
  19. Segment(P, Q) → d
  20. Sequence(Segment[B + ((0.5 * i) - 1, 0), B + ((0.5 * i) - 1, -0.025)], i, 0, 4) → list4
  21. Point(Ray[(0, 0), (-1, 0)]) → B_1
  22. Rotate(A_1, θ, B_1) → P_1
  23. Rotate(A_1, θ, B_1) → A'_2
  24. Rotate(A_1, θ, B_1) → A'_3
  25. Circle(B_1, A_1) → c_1
  26. Segment(B_1, P_1) → a_1

AI Drawing Prompt Reference

Create a dynamic illustration of a unit circle and a sine wave, showing how the y-coordinate of a point rotating on the unit circle maps to the sine function curve. Include an adjustable angle to visualize the relationship between the angle and the sine value, with auxiliary segments and text labels to mark key positions and values, helping to understand the geometric meaning and periodicity of the sine function.
The prompt above was compiled from the original drawing commands and is for reference only.