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Trigonometry and the Unit Circle by Thomas Lingefjärd

Trigonometry and the Unit Circle by Thomas Lingefjärd
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2022-03-04 22:46
CircleFunctionTrigonometry
This example demonstrates the relationship between trigonometric functions and the unit circle. Through the unit circle, it provides a visual understanding of the definitions and variations of sine, cosine, tangent, etc. The construction involves commands like Circle, Tangent, Segment, etc., suitable for high school mathematics education and learning.

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. Segment(A, E) → secα
  16. OrthogonalLine(D, d) → f
  17. Intersect(f, m) → H
  18. Segment(A, H) → cosα
  19. Angle(Q, A, D) → α
  20. Segment(A, D) → b
  21. Segment(F, D) → cotα
  22. Segment(D, E) → tanα
  23. Segment(A, F) → cscα
  24. Segment(D, H) → sinα
  25. OrthogonalLine(D, sinα) → i
  26. Function(1 / sin((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cosecantα
  27. Intersect(cosecantα, h, (2.176513564978313, 6.038860142308602)) → G
  28. Function(sin((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → sineα
  29. Intersect(sineα, h, (2.176513564978313, 0.1655941645334593)) → I
  30. Function(tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → tangentα
  31. Intersect(tangentα, h, (2.176513564978313, 0.16791236545719676)) → J
  32. Function(cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cosineα
  33. Intersect(cosineα, h, (2.176513564978313, 0.9861939832874999)) → K
  34. Function(1 / cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → secantα
  35. Intersect(secantα, h, (2.176513564978313, 1.0139992911602214)) → L
  36. Function(1 / tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cotangentα
  37. Intersect(cotangentα, h, (2.176513564978313, 5.955487538259439)) → M
  38. Segment(H, D) → j
  39. Segment(A, F) → k
  40. Intersect(j, m) → N
  41. Intersect(k, l) → P
  42. Function(1, 1.9, 5.5) → p
  43. Function(-1, 1.9, 5.5) → q
  44. Point(c) → R
  45. Segment(C, R) → a

AI Drawing Prompt Reference

Draw a coordinate plane with both x- and y-axes, then define a function curve and mark its intersection point with the constructed geometry. Use an x–y axis intersection as the center and another point on the y-axis as the radius reference to construct a circle, and place a point on the circle. Construct a tangent from that circle point, take its intersections with the y- and x-axes, and form segments connecting key axis intersections and circle points to set up the baseline geometry. From the resulting perpendiculars and intersections, draw the line segments that visually represent sine, cosine, tangent and their reciprocals, organizing them around the circle and the tangent to illustrate the unit-circle relationships.
The prompt above was compiled from the original drawing commands and is for reference only.