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Unit Circle with sin2x Dynamic Visualization

Unit Circle with sin2x Dynamic Visualization
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2017-01-26 07:59
CircleAngleFunctionTrigonometrySlider Animation
① This example visually demonstrates the sine function sin(2x) on the unit circle, connecting the angle's rotation to the function's value as a projection. ② Core mathematical concepts include the unit circle, trigonometric functions (sine), radian measure, function graphing, and the double-angle relationship. ③ Construction uses GeoGebra commands: Point, Rotate, Angle, Function, Segment, Circle, Vector, and Sequence, creating rotating points, angle sliders, function curves, vectors, and sequences, with text labels for key values. ④ Ideal for high school or introductory college students studying trigonometry and the unit circle, and for teachers demonstrating the geometric intuition behind the double-angle formula.

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
  27. Point(b) → C
  28. Point(list3) → D
  29. Segment(C, D) → h
  30. Function(cos((120° * x)), 0, x(Q)) → p
  31. Segment(E, (x(E), 0)) → i
  32. Segment(D, (x(P), 0)) → j
  33. Point(c) → G
  34. Segment(C, H) → l
  35. Segment(H, G) → m
  36. Function(tan((120° * x)), 0, (θ / ((360 * γ)) * 3)) → q
  37. Segment(I, (x(I), 0)) → n
  38. Segment(H, I) → r
  39. Segment(C, J) → s
  40. Segment(C, K) → t
  41. Segment(K, D) → b_1
  42. Segment(D, J) → d_1
  43. Function(csc((120° * x)), 0, x(Q)) → f_1
  44. Segment(J, L) → e_1
  45. Segment(L, (x(L), 0)) → g_1
  46. Function(sec((120° * x)), 0, (θ / ((360 * γ)) * 3)) → p_1
  47. Segment(M, (x(M), 0)) → i_1
  48. Segment(C, K) → j_1
  49. Point(list3) → N
  50. Segment(O, N) → l_1
  51. Segment(C, O) → m_1
  52. Function(cot((120° * x)), 0, x(Q)) → q_1
  53. Segment(R, (x(R), 0)) → n_1
  54. Segment(C, J) → r_1
  55. Rotate(A, (2 * θ), B) → A'_4
  56. Segment(A'_4, D) → s_1
  57. Angle(C, D, A'_4) → δ
  58. Segment(A'_4, S) → f_2
  59. Segment(A'_4, T) → g_2
  60. Segment(T, (x(T), 0)) → h_2

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with radius 1. Place a moving point on the circle, controlled by an angle slider. Draw a radius segment from the center to this point. Outside the circle, set up a coordinate system with the horizontal axis for angle and vertical axis for function values. Plot the sine function sin(2x) from 0 to the current angle, and project the point's vertical coordinate onto this curve using a perpendicular segment. Label the angle on the circle. Additionally, plot cosine, tangent, cosecant, and secant curves in different colors or line styles. Add tick marks below the circle for key angle values.
The prompt above was compiled from the original drawing commands and is for reference only.