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Animated Trigonometric Ratios in a Right Triangle (sin, cos, tan)

Animated Trigonometric Ratios in a Right Triangle (sin, cos, tan)
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2021-01-14 17:33
TriangleAngleTrigonometryGeometric ConstructionSlider Animation
This example uses an animation to demonstrate how the trigonometric ratios of a right triangle—sine, cosine, and tangent—relate to changes in an angle α. As the triangle’s configuration evolves, the displayed ratios correspond to the geometric proportions of the triangle’s sides. The key concepts are right-triangle trig ratios, angle–side relationships, and constructing auxiliary curves/lines (circles, arcs, tangents) to make the functional meanings visible in the diagram. From the construction summary, it is built with core GeoGebra commands such as Point and Angle (to define key vertices and α), Segment and Polygon (to form the right triangle and its edges), Circle/CircleArc/CircumcircleArc with Center (to generate relevant circles and arcs), Tangent plus Intersect (to create tangent lines and their intersection points), and Geometry tools like Midpoint, Centroid, Vector, Translate, and Rotate for coordinated transformations. Animation drives the interactive exploration. Suitable for secondary-school instruction on trigonometry, geometry-construction practice, and classroom demonstrations.

GeoGebra Commands & Parameters

  1. Midpoint(A, C) → L_1
  2. Midpoint(C, A) → H_1
  3. Circle(C, A) → c
  4. Tangent(A, c) → a,
  5. Point(a) → B
  6. Polygon(C, A, B) → poly1, c_1, a_1, b
  7. Centroid(poly1) → Q
  8. Midpoint(C, Q) → R
  9. Midpoint(C, R) → S
  10. Angle(B, A, C) → A_2
  11. Angle(A, C, B) → α
  12. Translate(B, Vector[(Animation / 10 * Vector[B, D])]) → B'
  13. Translate(A, Vector[(Animation / 10 * Vector[A, F])]) → A'
  14. Segment(B', A') → f
  15. Translate(C, Vector[(Animation / 10 * Vector[C, G])]) → C'
  16. Translate(B, Vector[(Animation / 10 * Vector[B, H])]) → B'_{1}
  17. Segment(C', B'_{1}) → g
  18. Translate(C, Vector[(Animation / 10 * Vector[C, I])]) → C'_{1}
  19. Translate(A, Vector[(Animation / 10 * Vector[A, J])]) → A'_{1}
  20. Segment(C'_{1}, A'_{1}) → h
  21. Translate(C, Vector[(Animation / 10 * Vector[C, K])]) → C'_{2}
  22. Translate(B, Vector[(Animation / 10 * Vector[B, L])]) → B'_{2}
  23. Segment(C'_{2}, B'_{2}) → i
  24. Translate(B, Vector[(Animation / 10 * Vector[B, N])]) → B'_{3}
  25. Translate(A, Vector[(Animation / 10 * Vector[A, M])]) → A'_{2}
  26. Segment(A'_{2}, B'_{3}) → j
  27. Translate(C, Vector[(Animation / 10 * Vector[C, O])]) → C'_{3}
  28. Translate(A, Vector[(Animation / 10 * Vector[A, P])]) → A'_{3}
  29. Segment(C'_{3}, A'_{3}) → k
  30. Centroid(poly1) → T
  31. Centroid(poly1) → U
  32. Centroid(poly1) → V
  33. Midpoint(B, A) → W
  34. Midpoint(Q, W) → Z
  35. Midpoint(Z, W) → A_1
  36. Midpoint(S, R) → B_1
  37. Midpoint(B_1, S) → C_1
  38. Translate(C_1, Vector[(Animation / 10 * Vector[C_1, W])]) → D_1
  39. Midpoint(C, B) → F_1
  40. Midpoint(Q, F_1) → E_1
  41. Midpoint(E_1, F_1) → G_1
  42. Translate(A, Vector[(Animation / 10 * Vector[A, F_1])]) → A'_{4}
  43. Vector(A, A'_{4}) → u
  44. Vector(C_1, D_1) → v
  45. Midpoint(C, H_1) → I_1
  46. Rotate(C, If[Animation < (2 * π), (((-Animation) * π)) / 10, (-π)], I_1) → C'_{4}
  47. Midpoint(I_1, H_1) → J_1
  48. CircumcircleArc(S, R, H_1) → d
  49. Circle(S, R, H_1) → e
  50. Center(e) → K_1
  51. Angle(L_1, K_1, S) → β
  52. Rotate(S, If[Animation < 10, ((-Animation) * β / 10), (-β)], K_1) → S'
  53. CircleArc(K_1, S', S) → p
  54. Midpoint(C, S) → M_1
  55. Midpoint(M_1, S) → N_1
  56. Midpoint(N_1, S) → O_1
  57. Intersect(e, c_1) → Q_1, R_1
  58. Midpoint(S, Q_1) → S_1
  59. Midpoint(S, S_1) → T_1
  60. Midpoint(S, S_1) → U_1
  61. Rotate(U_1, If[Animation < 10, ((-Animation) * β / 10), (-β)], K_1) → P_1
  62. Vector(P_1, S') → w
  63. Midpoint(C, A) → V_1
  64. Midpoint(V_1, A) → W_1

AI Drawing Prompt Reference

Draw a right triangle ABC with angle A as the right angle and point C as the right-angle vertex. Use a slider to control the angle BAC, and animate to show how the sine, cosine, and tangent ratios change with the angle. Construct auxiliary circles, tangents, and midpoints to illustrate the relationships between the triangle's sides and angles, ultimately forming a dynamic trigonometric graph.
The prompt above was compiled from the original drawing commands and is for reference only.