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Unit Circle Beyond 0 and 2π: Sine and Cosine of Any Angle

Unit Circle Beyond 0 and 2π: Sine and Cosine of Any Angle
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2020-05-08 16:01
CircleAngleFunctionTrigonometrySlider Animation
① This example demonstrates the sine and cosine values for any given angle, including those beyond the standard 0 to 2π range, using a dynamic unit circle visualization. ② Core mathematical concepts include the unit circle, trigonometric functions for arbitrary angles, sine and cosine definitions, angle periodicity, and related identities. ③ Construction highlights: The unit circle is drawn using the CurveCartesian command; Intersect and Point locate the terminal side's intersection with the circle; CircleArc, OrthogonalLine, and Polygon construct the angle arc, perpendiculars, and reference triangles; If and Textfield enable dynamic input and conditional display; a slider (increment) controls angle variation. ④ Suitable for high school or pre-calculus trigonometry lessons, ideal for teachers demonstrating the geometric meaning of trigonometric values for any angle and reinforcing the concept of periodicity.

GeoGebra Commands & Parameters

  1. CurveCartesian(cos(a), sin(a), a, 0, 6.283185307179586) → c
  2. CurveCartesian(((1 / 20 + a / 200) * cos(a)), ((-(1 / 20 + a / 200)) * sin(a)), a, 0, u) → b
  3. Intersect(xAxis, yAxis) → D
  4. Point(c) → E
  5. CircleArc(D, E, C) → e
  6. OrthogonalLine(C, xAxis) → f
  7. Intersect(f, xAxis) → F
  8. Polygon(C, D, F) → poly1, f_1, c_1, d_1
  9. If(π / 2 < e < (2 * π), Polygon[D, (abs(x(C)), abs(y(C))), (abs(x(F)), abs(y(F)))]) → poly2
  10. Textfield(w) → increment
  11. Intersect(e, i, 2) → G
  12. Intersect(e, g, 2) → H
  13. Intersect(e, h, 2) → I
  14. Intersect(e, k, 1) → J
  15. Intersect(e, m, 1) → K
  16. Intersect(e, l, 1) → L
  17. Intersect(e, n, 1) → M
  18. Intersect(e, j, 1) → N
  19. Intersect(e, i, 1) → O
  20. Intersect(e, g, 1) → P
  21. Intersect(e, h, 1) → Q
  22. Intersect(e, k, 2) → R
  23. Intersect(e, m, 2) → S
  24. Intersect(e, l, 2) → T
  25. Intersect(e, n, 2) → U
  26. Intersect(e, j, 2) → V
  27. CurveCartesian(((1 / 20 + a / 200) * cos(a)), ((1 / 20 + a / 200) * sin(a)), a, 0, t) → a
  28. CircleArc(D, A_1, Z) → q
  29. Intersect(q, l, 1) → W
  30. Intersect(q, m, 1) → B_1
  31. Intersect(q, n, 1) → C_1
  32. Intersect(q, j, 1) → D_1
  33. Intersect(q, i, 1) → E_1
  34. Intersect(q, g, 1) → F_1
  35. Intersect(q, h, 1) → G_1

AI Drawing Prompt Reference

Draw a dynamic unit circle centered at the origin with radius 1. Place a movable point on the circle corresponding to any angle, including angles beyond 360°. Draw a ray from the center to this point, and display the sine and cosine values for that angle. Use a spiral curve starting from the origin that expands outward as the angle increases, representing continuous angle variation. Inside the circle, construct a reference triangle with legs parallel to the axes, and show the arc corresponding to the angle. Add a slider to control the angle and display its current value. When the angle exceeds 90°, draw a dashed reference triangle to emphasize periodicity and symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.