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Arcsine Function on Unit Circle

Arcsine Function on Unit Circle
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2017-02-05 09:06
CircleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the definition of the arcsine function on the unit circle, showing how the sine of any angle α corresponds to the y-coordinate, while the arcsine value is the x-coordinate of the intersection point of the terminal side with the unit circle. Core concepts include the unit circle, trigonometric functions, inverse trigonometric functions, and coordinate geometry. Construction involves commands like Point, Intersect, Circle, Segment, Angle, OrthogonalLine, Sequence, Vector, Line, Ray, CircleArc. Suitable for high school mathematics to understand the geometric meaning and domain/range of inverse trigonometric functions.

GeoGebra Commands & Parameters

  1. Point(yAxis) → D
  2. Intersect(xAxis, yAxis) → A
  3. Circle(A, B) → c
  4. Segment(A, C) → a
  5. Segment(B, A) → b
  6. Angle(B, A, C) → α
  7. OrthogonalLine(C, yAxis) → d
  8. Segment(D, A) → h
  9. Segment(H, B) → i
  10. Sequence((-1.5, t), t, -87.96459430051421, 87.96459430051421, 0.2617993877991494) → tval
  11. Vector(E, J) → z
  12. Vector(C, K) → j
  13. Vector(A, L) → k
  14. Segment(B, L) → l
  15. Vector(E, M) → m
  16. Line(L, B) → n
  17. Segment(T_0, t_2) → p
  18. Ray(A, B) → q
  19. Ray(A, C) → r
  20. CircleArc(A, B, C) → e
  21. Vector(C, t_2) → w_2
  22. Vector(D, C) → u_3
  23. CircleArc(A, C, B) → s
  24. Angle(C, A, B) → β
  25. Vector(F, E) → v_2
  26. Point(xAxis) → N
  27. OrthogonalLine(N, i) → c_1
  28. Intersect(e, c, 1) → O
  29. Vector(N, O) → u
  30. Intersect(c, c_1, 2) → P
  31. Vector(N, P) → v
  32. CircleArc(A, B, P) → d_1
  33. Vector(P, Q) → w
  34. Angle(B, A, P) → γ
  35. Ray(A, P) → e_1
  36. Point(f_1) → R
  37. Segment(R, A) → g_1
  38. Intersect(c, d_1) → S, , ,
  39. Intersect(c, g_1) → T,
  40. CircleArc(A, B, T) → h_1
  41. Vector(R, T) → u_1
  42. Angle(B, A, T) → δ
  43. Angle(B, A, T) → ε
  44. Vector(T, U) → v_1
  45. Segment(A, P) → j_1
  46. Segment(T_0, Q) → k_1
  47. CircleArc(A, B, T) → p_1