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Basic Trig Functions from the Unit Circle (Degrees)

Basic Trig Functions from the Unit Circle (Degrees)
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2017-02-22 22:41
CircleAngleFunctionTrigonometrySlider Animation
This GeoGebra example visually demonstrates the definitions and geometric meanings of the basic trigonometric functions (sine, cosine, and tangent) using the unit circle in degree mode. It illustrates how the coordinates of a point moving on the unit circle correspond to the values of sinα, cosα, and tanα, and how these values are projected onto their respective function graphs. Key mathematical concepts include the unit circle, trigonometric ratios, angle measurement in degrees, and the relationship between circular motion and periodic functions. The construction employs GeoGebra commands such as Circle, Point, Intersect, OrthogonalLine, Segment, Ray, Angle, Function, and Slider to create an interactive diagram where dragging the point or adjusting the angle slider updates the trigonometric values and graphs in real time. This resource is ideal for high school students learning trigonometry, teachers demonstrating the unit circle approach, and anyone seeking an intuitive visual foundation for understanding sine, cosine, and tangent functions.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Function(0, 1.9, 5.5) → g
  3. Point(g) → Q
  4. Point(yAxis) → B
  5. Circle(A, B) → c
  6. Point(c) → D
  7. Tangent(D, c) → e,
  8. Intersect(e, yAxis) → F
  9. Intersect(xAxis, e) → E
  10. Segment(F, E) → l
  11. Segment(E, A) → m
  12. Intersect(m, l) → O
  13. Intersect(c, xAxis, 2) → C
  14. Segment(A, C) → d
  15. OrthogonalLine(D, d) → f
  16. Intersect(f, m) → H
  17. Segment(A, H) → cosα
  18. Angle(Q, A, D) → α
  19. Segment(A, D) → b
  20. Segment(D, E) → tanα
  21. Segment(D, H) → sinα
  22. OrthogonalLine(D, sinα) → i
  23. Function(sin((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → sineα
  24. Intersect(sineα, h, (2.439250960476847, 0.6379515420040821)) → I
  25. Function(tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → tangentα
  26. Intersect(tangentα, h, (2.439250960476847, 0.8284261815090247)) → J
  27. Function(cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cosineα
  28. Intersect(cosineα, h, (2.439250960476847, 0.770076509221398)) → K
  29. Segment(H, D) → j
  30. Segment(A, F) → k
  31. Intersect(j, m) → N
  32. Intersect(k, l) → P
  33. Function(1, 1.9, 5.5) → p
  34. Function(-1, 1.9, 5.5) → q
  35. Point(c) → R
  36. Segment(C, R) → a
  37. Segment(A, H) → g_1
  38. Segment(D, H) → h_1
  39. Segment(A, E) → j_1
  40. Angle(A, D, E) → β
  41. Ray(D, F) → i_1
  42. Ray(D, E) → l_1
  43. Intersect(c, a) → G, L
  44. OrthogonalLine(I, xAxis) → d_1
  45. Intersect(g, xAxis, 1) → M
  46. Intersect(g, xAxis, 1) → S
  47. Intersect(g, xAxis, 1) → T
  48. Intersect(g, d_1) → U
  49. Segment(I, U) → e_1
  50. OrthogonalLine(K, xAxis) → k_1
  51. Intersect(k_1, xAxis) → V
  52. Segment(K, V) → m_1
  53. OrthogonalLine(J, xAxis) → n_1
  54. Intersect(n_1, xAxis) → W
  55. Segment(J, W) → p_1
  56. Point(yAxis) → Z
  57. Point(yAxis) → A_1
  58. Segment(Z, A_1) → q_1
  59. Point(xAxis) → B_1
  60. Point(xAxis) → C_1
  61. Segment(B_1, C_1) → r_1
  62. OrthogonalLine(D, r_1) → c_1
  63. Intersect(g, r_1, 1) → D_1
  64. Intersect(c_1, r_1) → E_1
  65. Segment(D, E_1) → s_1
  66. Angle(D, E_1, A) → γ

AI Drawing Prompt Reference

Draw a unit circle with a point on it representing angle α. Show the line from the origin to the point, and its projections onto the x-axis and y-axis. Illustrate the sine, cosine, and tangent lines, and project them onto their respective sine, cosine, and tangent curves. Include an angle slider that updates all visuals and values in real time when moved.
The prompt above was compiled from the original drawing commands and is for reference only.