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Trigonometric Ratios Visualized on the Unit Circle

Trigonometric Ratios Visualized on the Unit Circle
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2012-02-25 08:09
CircleAngleTrigonometryGeometric ConstructionSlider Animation
This GeoGebra example dynamically visualizes the six fundamental trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) as geometric lengths on the unit circle, allowing students to prove and understand the relationships between these ratios and the circle's segments, tangents, and perpendiculars. Core mathematical concepts include trigonometric definitions, the unit circle, similar triangles, tangent properties, and angle-radian correspondence. Construction highlights: using GeoGebra commands such as Intersect, Function, Point, Circle, Tangent, Segment, OrthogonalLine, and Angle, the applet builds a unit circle, an angle α, and perpendicular/tangent lines from the center and tangent points, generating labeled segments for sinα, cosα, tanα, cotα, secα, and cscα (objects like cosα, sinα, tanα, etc.), with dynamic updates via a slider or angle object. Suitable for high school or pre-university trigonometry instruction, it also supports exploration of trigonometric identities and periodicity.

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, (2.520357270523917, 1.3263553394304515)) → G
  28. Function(sin(2 * (x - 2 * 3.14 / 3)), 2.09, 5.24) → sineα
  29. Intersect(sineα, h, (2.520357270523917, 0.7539457717487749)) → I
  30. Function(tan(2 * (x - 2 * 3.14 / 3)), 2.09, 5.24) → tangentα
  31. Intersect(tangentα, h, (2.520357270523917, 1.1476688992921722)) → J
  32. Function(cos(2 * (x - 2 * 3.14 / 3)), 2.09, 5.24) → cosineα
  33. Intersect(cosineα, h, (2.520357270523917, 0.6569366584855378)) → K
  34. Function(1 / cos(2 * (x - 2 * 3.14 / 3)), 2.09, 5.24) → secantα
  35. Intersect(secantα, h, (2.520357270523917, 1.5222167724744418)) → L
  36. Function(1 / tan(2 * (x - 2 * 3.14 / 3)), 2.09, 5.24) → cotangentα
  37. Intersect(cotangentα, h, (2.520357270523917, 0.8713314446498922)) → 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

Visualize the six fundamental trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) as geometric lengths on a unit circle. Draw a unit circle and an angle α, then construct perpendiculars and tangents from the center and tangent points to create labeled segments for sinα, cosα, tanα, cotα, secα, and cscα. Use a slider or angle object to dynamically update the lengths, showing the relationships between these ratios and the circle's segments, tangents, and perpendiculars. Ideal for high school or pre-university trigonometry instruction, exploring trigonometric identities and periodicity.
The prompt above was compiled from the original drawing commands and is for reference only.