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Basic Trig Functions from the Unit Circle

Basic Trig Functions from the Unit Circle
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2015-04-21 07:23
CircleFunctionTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the geometric interpretation of basic trigonometric functions (sine, cosine, tangent) using the unit circle. It covers core math concepts including the unit circle, trigonometric function definitions, angle and radian conversions, and right triangle side relationships. The construction utilizes GeoGebra commands/objects like Intersect, Function, Point, Circle, Tangent, Segment, OrthogonalLine, Angle, and Ray. Suitable for high school students learning about the relationship between trig functions and the unit circle, or for classroom demonstrations.

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 center A and point B. Mark the intersection points C and D of the circle with the x-axis. Draw the tangent line e at point D, intersecting the y-axis at F and the x-axis at E. Connect F and E to form line l, and connect E and A to form line m; their intersection is point O. Draw a perpendicular line f from D to line d, intersecting m at point H. Define angle alpha as the angle QAD, where Q is on function g. Draw segments for sin alpha, tan alpha, and cos alpha connecting D to H, E, and H respectively. Draw a perpendicular line i from H to sin alpha. Draw sine alpha, cosine alpha, and tangent alpha curves, marking their intersections with a line at points I, J, and K. Connect various points such as H, D, A, F to form intersections G, L, and P. Overall, this illustrates the geometric relationship between trigonometric functions and the unit circle, including tangents, perpendiculars, intersections, angles, and function curves.
The prompt above was compiled from the original drawing commands and is for reference only.