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Unit Circle and Trigonometric Functions by Craig Cullen

Unit Circle and Trigonometric Functions by Craig Cullen
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2016-12-08 02:16
CircleFunctionTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the geometric interpretation of six trigonometric functions on a unit circle. It involves core concepts like the unit circle, trigonometric functions, geometric objects, angles, and arcs. The construction uses GeoGebra commands such as Vector, Translate, Ray, Point, Segment, Circle, CircleArc, Line, OrthogonalLine, Intersect, Rotate, and Mirror. It is suitable for high school mathematics education and learning, providing an intuitive understanding of the relationship between trigonometric functions and the unit circle.

GeoGebra Commands & Parameters

  1. Vector((0, 0), (1, 0)) → v
  2. Translate(A, v) → A'
  3. Ray(A, A') → a
  4. Point(a) → B
  5. Segment(A, B) → b
  6. Vector(A, B) → Radius
  7. Translate(C, Radius) → C'
  8. Circle(C, C') → c
  9. Point(c) → D
  10. CircleArc(C, C', D) → d
  11. Line(C, C') → e
  12. OrthogonalLine(D, e) → f
  13. Intersect(f, e) → E
  14. Segment(D, E) → g
  15. Point(Circle[F, abs(d)]) → G
  16. Segment(F, G) → h
  17. Vector(E, D) → Chord_1
  18. OrthogonalLine(F, Radius) → i
  19. Translate(i, Radius) → i'
  20. Translate(i', Radius) → i''
  21. Translate(i'', Radius) → i'''
  22. Translate(i''', Radius) → j
  23. Translate(j, Radius) → j'
  24. Translate(j', Radius) → j''
  25. Translate(G, Chord_1) → G'
  26. Segment(G', G) → k
  27. Intersect(i', h) → H
  28. Intersect(i'', h) → I
  29. Intersect(i''', h) → J
  30. Intersect(j, h) → K
  31. Intersect(j', h) → L
  32. Intersect(j'', h) → M
  33. Rotate(B, 90°, A) → B'
  34. Translate(A, Radius) → A'_1
  35. Vector(A, A'_1) → u
  36. Vector(A, B') → w
  37. Ray(F, G) → m
  38. Translate(F, w) → F'
  39. Translate(G, w) → G'_1
  40. Ray(F', G'_1) → m'
  41. Translate(F', w) → F''
  42. Translate(G'_1, w) → N
  43. Ray(F'', N) → m''
  44. Translate(F'', w) → F'''
  45. Translate(N, w) → N'
  46. Ray(F''', N') → m'''
  47. Mirror(F''', m) → O
  48. Mirror(N', m) → N''
  49. Ray(O, N'') → n
  50. Mirror(F'', m) → F'''_1
  51. Mirror(N, m) → N'_1
  52. Ray(F'''_1, N'_1) → m'''_1
  53. Mirror(F', m) → F''_1
  54. Mirror(G'_1, m) → P
  55. Ray(F''_1, P) → m''_1
  56. OrthogonalLine(C', m') → l
  57. Line(C, D) → p
  58. Intersect(l, p) → Q
  59. Segment(Q, C') → r
  60. Vector(C', Q) → Tangent
  61. Translate(G, Tangent) → G'_2
  62. Segment(G'_2, G) → t
  63. Intersect(l, p) → R
  64. Segment(R, C) → a_1
  65. Point(a_1) → S
  66. Vector(C, R) → Secant
  67. Line(C, C') → b_1
  68. OrthogonalLine(D, Secant) → c_1
  69. Intersect(b_1, c_1) → T
  70. Vector(C, T) → Secant_2
  71. Translate(G, Secant_2) → G'_3
  72. Rotate(G'_3, 90°, G) → U
  73. Segment(G, U) → d_1
  74. Segment(C, T) → e_1
  75. Ray(C, C') → g_1
  76. Rotate(C, 57.29577951308232°, C) → C'_1
  77. Rotate(C', 57.29577951308232°, C) → C''
  78. Ray(C'_1, C'') → h_1
  79. Intersect(d, h_1) → V,
  80. Rotate(C'_1, 57.29577951308232°, C) → W
  81. Rotate(C'', 57.29577951308232°, C) → C'''
  82. Ray(W, C''') → i_1
  83. Rotate(W, 57.29577951308232°, C) → W'
  84. Rotate(C''', 57.29577951308232°, C) → Z
  85. Ray(W', Z) → j_1
  86. Rotate(W', 57.29577951308232°, C) → W''
  87. Rotate(Z, 57.29577951308232°, C) → Z'
  88. Ray(W'', Z') → k_1
  89. Rotate(W'', 57.29577951308232°, C) → W'''
  90. Rotate(Z', 57.29577951308232°, C) → Z''
  91. Ray(W''', Z'') → l_1
  92. Rotate(W''', 57.29577951308232°, C) → A_1
  93. Rotate(Z'', 57.29577951308232°, C) → Z'''
  94. Ray(A_1, Z''') → m_1
  95. Intersect(d, i_1, 1) → B_1
  96. Intersect(d, j_1, 1) → C_1
  97. Intersect(d, k_1, 1) → D_1
  98. Intersect(d, l_1, 1) → E_1
  99. Intersect(d, m_1, 1) → F_1
  100. Line(D, b) → p_1
  101. OrthogonalLine(C, p_1) → q_1
  102. Intersect(p_1, q_1) → G_1
  103. Segment(D, G_1) → r_1
  104. Vector(G_1, D) → Cochord
  105. Translate(G, Cochord) → G'_4
  106. Rotate(G'_4, 90°, G) → H_1
  107. Segment(H_1, G) → s_1
  108. Rotate(C', 90°, C) → C''_1
  109. OrthogonalLine(C''_1, l) → t_1
  110. Intersect(p, t_1) → I_1
  111. Segment(C''_1, I_1) → a_2
  112. Vector(C''_1, I_1) → Cotangent
  113. Translate(G, Cotangent) → G'_5
  114. Rotate(G'_5, 90°, G) → J_1
  115. Segment(J_1, G) → b_2
  116. Line(C, C') → c_2
  117. Intersect(p, t_1) → K_1
  118. Segment(C, K_1) → d_2
  119. Line(C, C''_1) → e_2
  120. OrthogonalLine(D, d_2) → f_2
  121. Intersect(e_2, f_2) → L_1
  122. Segment(C, L_1) → g_2
  123. Vector(C, L_1) → Cosecant
  124. Translate(G, Cosecant) → G'_6
  125. Segment(G, G'_6) → h_2

AI Drawing Prompt Reference

Draw a unit circle and demonstrate the geometric interpretation of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. Start by drawing the unit circle, then draw a radius from point A to the center C. Next, take point B on the circle and draw a segment AB. Draw a ray from B to the positive x-axis, and take point D on the circle, drawing a circular arc CD. Draw a line perpendicular to the ray at point D, intersecting the x-axis at point E, and draw a segment DE. Construct the sine line, cosine line, tangent line, cotangent line, secant line, and cosecant line, and finally use reflection to obtain the secant and cosecant lines.
The prompt above was compiled from the original drawing commands and is for reference only.