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Unit Circle to Sine and Cosine Functions

Unit Circle to Sine and Cosine Functions
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2020-02-29 04:53
CircleFunctionTrigonometrySlider AnimationGraphing
① This interactive GeoGebra example dynamically links a moving point on the unit circle to the graphs of sine and cosine functions, visually demonstrating how the angle determines the function values. ② Core mathematical concepts include the unit circle definition of trigonometric functions, sine and cosine, angle measurement in radians, periodic behavior, and function graphing. ③ Construction highlights: using GeoGebra commands such as Point, Circle, Segment, Vector, Angle, Sequence, Element, List, and Text to build the circle, radius vector, angle display, and point lists; a slider controls the angle parameter to animate the correspondence between the circle and the function graphs. ④ Suitable for high school trigonometry classes, college precalculus review, and anyone seeking an intuitive grasp of how sine and cosine arise from circular motion.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C
  2. Text("2π", (6.28319, -0.2)) → text2
  3. Text("π", (3.141592653589793, -0.2)) → text1
  4. Circle(C, 1) → c
  5. Segment(C + (1.2, 0), C - (1.2, 0)) → a
  6. Sequence(Rotate[C + (1, 0), (i * π / n), C], i, 0, (2 * n)) → list4s
  7. Segment(C + (0, 1.2), C - (0, 1.2)) → b
  8. Sequence((i, 0), i, 0, 6.283185307179586, π / n) → list3s
  9. Sequence(Segment[Element[list4s, i], (x(Element[list4s, i]), 0)], i, 1, k - 1) → list5
  10. Sequence(Segment[Element[list4s, i], C], i, 1, k - 1) → list6
  11. Segment(Element[list4s, k], C) → d
  12. Segment(Element[list4s, k], (x(Element[list4s, k]), 0)) → e
  13. Element(list4s, k) → E
  14. Vector(E, D) → w
  15. Angle(Element[list4s, 1], C, E) → α
  16. Text("x", Element[list3s, k] + (0, -0.2)) → text3
  17. Element(list3s, k) → F
  18. Segment(D, F) → f
  19. Vector((-0.3, 0), (6.783185307179586, 0)) → u
  20. Vector((0, -1.6), (0, 1.6)) → v
  21. Sequence(Segment[(((i - 1) * π / n), 0), (((i - 1) * π / n), y(Element[list4s, i]))], i, 1, k - 1) → list7
  22. Sequence(Segment[((i * π / 2), -1.6), ((i * π / 2), 1.6)], i, 1, 4) → list10
  23. Point(list4s) → G
  24. Segment(Element[list4s, k], D) → j
  25. Sequence((((i - 1) * π / n), y(Element[list4s, i])), i, 1, k - 1) → list8s
  26. Sequence(Segment[((i * π / n), -1.6), ((i * π / n), 1.6)], i, 0, (2 * n)) → list2
  27. Sequence(Segment[(0, i), ((2 * π), i)], i, -1.6, 1.6, 0.2) → list1
  28. Sequence((i, -1.6), i, 0, 6.28319, π / n) → liste3
  29. Sequence(Text["" + (i - 1) + " π / " + n, (x(Element[liste3, i]) - 0.2, y(Element[liste3, i]) - 0.2)], i, 2, k) → liste4
  30. Take(liste4, k, k) → liste5
  31. Sequence((((i - 1) * π / n), -1.6), i, 1, k) → liste2
  32. Text("π/2", (1.5707963267948966, -0.2)) → texte2
  33. Text("3π/2", (4.71238898038469, -0.2)) → texte3
  34. Sequence(Text["" + (i - 1) + " π / " + n, (x(Element[liste3, i]) - 0.1, y(Element[liste3, i]) - 0.2)], i, 2, k) → liste6
  35. Sequence(Text["" + (k) + " π / " + n, C - (0.12, 0) + (1 + 0.25; (k * π / n))], k, 0, (2 * n) - 1, 2) → liste1b
  36. Sequence(Text["" + (i) + " π / " + n, C - (0.12, 0) + (1 + 0.25; (i * π / n))], i, 0, k - 1, 1) → liste1
  37. Element(list4s, k) → M
  38. Sequence((((i - 1) * π / n), x(Element[list4s, i]) + 1.68), i, 1, k - 1) → liste8c
  39. Segment(C, (x(Element[list4s, k]), 0)) → l
  40. Segment(Element[list4s, k], (x(Element[list4s, k]), 0)) → m
  41. Sequence(Segment[(((i - 1) * π / n), 0), (((i - 1) * π / n), y(Element[liste8c, i]))], i, 1, k - 1) → liste7
  42. Sequence(Segment[Element[list4s, i], (x(C), y(Element[list4s, i]))], i, 1, k) → l1
  43. Sequence((((i - 1) * π / n), 0), i, 1, k - 1) → l4
  44. Sequence(Text[If[i ≟ 0, "0", If[i ≟ 4, "π", "\frac{" + (i / GCD[i, 4]) + ((π * "}{") + (4 / GCD[i, 4]) + "}")]], Element[list4s, i], true, true], i, 0, k) → l3
  45. Sequence(Text[If[i ≟ 0, "0", If[i ≟ n, "π", "\frac{" + (i / GCD[i, n]) + ((π * "}{") + (n / GCD[i, n]) + "}")]], Element[l4, i + 1] - (0.15, 0.1), true, true], i, 0, (2 * n) + 1) → l5
  46. Sequence(Text[If[i ≟ 0, "0", If[i ≟ n, "π", "\frac{" + (i / GCD[i, n]) + ((π * "}{") + (n / GCD[i, n]) + "}")]], C - (0.12, -0.125) + (1 + 0.25; (i * π / n)), true, true], i, 0, (2 * n)) → l2

AI Drawing Prompt Reference

Draw a unit circle and move a point on it to show the angle. Dynamically plot the sine and cosine functions based on the position of this point on the circle. First, draw the unit circle, then place a movable point on it to determine the sine and cosine values based on the angle, and finally plot the graphs of these functions in the coordinate system.
The prompt above was compiled from the original drawing commands and is for reference only.