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Special Angles and Reference Triangles

Special Angles and Reference Triangles
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2016-08-25 02:55
TriangleCircleAngleTrigonometryGeometric Construction
This example demonstrates how to use GeoGebra to draw special angles (30°, 45°, 60°, 90°, 180°, 270°) and their corresponding reference triangles. It involves core mathematical concepts such as angle measurement, trigonometric functions, geometric construction, and coordinate system applications. The construction highlights the use of GeoGebra commands and objects like Intersect, Circle, Point, Segment, Angle, CircleArc, Rotate, Textfield, and Line. Suitable for high school students learning trigonometric properties and geometric figure construction.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, P) → c
  3. Point(c) → Q
  4. Segment(O, P) → f
  5. Segment(O, Q) → g
  6. Angle(f, g) → α
  7. CircleArc(O, P, Q) → d
  8. Rotate(P, rad, O) → P'
  9. Angle(P, O, P') → β
  10. Rotate(P, (2 * rad), O) → P'_1
  11. Angle(P, O, P'_1) → γ
  12. Rotate(P, (3 * rad), O) → P'_2
  13. Angle(P, O, P'_2) → δ
  14. Rotate(P, (4 * rad), O) → P'_3
  15. Angle(P, O, P'_3) → ε
  16. Rotate(P, (5 * rad), O) → P'_4
  17. Angle(P, O, P'_4) → ζ
  18. Rotate(P, (6 * rad), O) → P'_5
  19. Angle(P, O, P'_5) → η
  20. Textfield(r) → InputBox1
  21. Segment((0, 0), ((r * cos(α)), 0)) → i
  22. Segment(Q, ((r * cos(α)), 0)) → h
  23. Rotate(P, (π / 6 * rad), O) → P_30
  24. Angle(P, O, P_30) → θ
  25. Rotate(P, (π / 4 * rad), O) → P_45
  26. Angle(P, O, P_45) → ι
  27. Rotate(P, (π / 3 * rad), O) → P_60
  28. Angle(P, O, P_60) → κ
  29. Rotate(P, (π / 2 * rad), O) → P_90
  30. Angle(P, O, P_90) → λ
  31. Rotate(P, (π * rad), O) → P_180
  32. Angle(P, O, P_180) → μ
  33. Rotate(P, ((3 * π / 2) * rad), O) → P_270
  34. Angle(P, O, P_270) → ν
  35. Rotate(P, ((3 * π / 4) * rad), O) → P_135
  36. Angle(P, O, P_135) → ξ
  37. Rotate(P, ((5 * π / 4) * rad), O) → P_225
  38. Angle(P, O, P_225) → ο
  39. Rotate(P, ((7 * π / 4) * rad), O) → P_315
  40. Angle(P, O, P_315) → ρ
  41. Rotate(P, ((4 * π / 3) * rad), O) → P_240
  42. Angle(P, O, P_240) → σ
  43. Rotate(P, ((2 * π / 3) * rad), O) → P_120
  44. Angle(P, O, P_120) → τ
  45. Rotate(P, ((5 * π / 3) * rad), O) → P_300
  46. Angle(P, O, P_300) → υ
  47. Rotate(P, ((5 * π / 6) * rad), O) → P_150
  48. Angle(P, O, P_150) → ϕ
  49. Rotate(P, ((7 * π / 6) * rad), O) → P_210
  50. Angle(P, O, P_210) → χ
  51. Rotate(P, ((11 * π / 6) * rad), O) → P_330
  52. Angle(P, O, P_330) → ψ
  53. Rotate(P, (0 * rad), O) → P_0
  54. Angle(P, O, P_0) → ω
  55. Line((0, 0), (1, 0)) → X
  56. Line((0, 0), (0, 1)) → Y
  57. Line(Q, X) → hor
  58. Line(Q, Y) → ver
  59. Intersect(h, i) → R
  60. Angle(Line[R, O], Line[R, Q]) → α_1
  61. Angle(Line[O, R], Line[O, Q]) → β_1