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Trigonometry and the Unit Circle

Trigonometry and the Unit Circle
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2022-03-04 22:47
CircleFunctionTrigonometryGeometric Construction
This example dynamically demonstrates the relationship between trigonometric values and angles by showing the movement of a point on the unit circle. Core mathematical concepts include trigonometric functions (sin, cos, tan), the unit circle, angles, tangent lines, and sine values. It utilizes GeoGebra commands such as Intersect, Function, Point, Circle, Tangent, Segment, OrthogonalLine, and Angle. Suitable for high school mathematics teaching to help students visually understand the definitions and properties of trigonometric 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. 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, (2.572354239330577, 1.2223938567602033)) → G
  28. Function(sin((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → sineα
  29. Intersect(sineα, h, (2.572354239330577, 0.8180669384664372)) → I
  30. Function(tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → tangentα
  31. Intersect(tangentα, h, (2.572354239330577, 1.4224207950255017)) → J
  32. Function(cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cosineα
  33. Intersect(cosineα, h, (2.572354239330577, 0.5751230165696295)) → K
  34. Function(1 / cos((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → secantα
  35. Intersect(secantα, h, (2.572354239330577, 1.7387584415671375)) → L
  36. Function(1 / tan((2 * (x - (2 * 3.14 / 3)))), 2.09, 5.24) → cotangentα
  37. Intersect(cotangentα, h, (2.572354239330577, 0.7030268423361117)) → 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 unit circle with the intersection point of the origin and the x-axis, mark a point on the x-axis, and draw the unit circle through these two points. Mark a point on the unit circle, draw the tangent line at this point, and find its intersections with the x-axis and y-axis. Connect these points to form a right triangle, and based on this, draw sine, cosine, and tangent lines. Display the angle α and also draw secant, cosecant, and cotangent lines, adding auxiliary segments and angle markings to visually demonstrate the relationship between trigonometric functions and the unit circle.
The prompt above was compiled from the original drawing commands and is for reference only.