MathPlotAI Logo MathPlotAI

Radians in the Unit Circle by Vladimir Ortega Garcia

Radians in the Unit Circle by Vladimir Ortega Garcia
Loading the math board and drawing, please wait…
2017-07-27 09:10
CircleAngleVectorTrigonometryGeometric Construction
This interactive example provides a visual demonstration of radians within the unit circle, illustrating the fundamental relationship between the central angle and the corresponding arc length. The core mathematical concepts include radian measure, the constant $\pi$, and the geometric definition of angles. The construction utilizes GeoGebra commands such as Circle, CircleArc, Rotate, Angle, and Vector to create a dynamic system of points and segments that trace the unit circle. It is designed for high school or early college students studying trigonometry and geometry, serving as an effective tool for conceptualizing how radians are derived from the unit circle's properties.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(C, D) → b
  3. Circle(E, 2) → c
  4. OrthogonalLine(E, a) → d
  5. Intersect(d, a) → F
  6. Segment(A, F) → e
  7. Rotate(F, (-(180 * e / 6.28))°, E) → F'
  8. Angle(F', E, F) → α
  9. CircleArc(E, F, F') → f
  10. Vector(E, F') → u
  11. Rotate(F', 3.141592653589793, E) → F''_1
  12. Vector(P, Q) → v
  13. Translate(Q, v) → Q'
  14. Vector(Q, Q') → w
  15. Translate(Q', w) → Q''
  16. Translate(Q'', w) → Q'''
  17. Translate(Q''', w) → R
  18. Translate(R, w) → R'
  19. Vector(Q', Q'') → z
  20. Vector(Q'', Q''') → m
  21. Vector(Q''', R) → n
  22. Vector(R, R') → p
  23. Vector(P, R') → q
  24. Rotate(F', 0.7853981633974483, E) → F''
  25. Angle(F', E, F'') → β
  26. Rotate(F', 0.2617993877991494, E) → F''_2
  27. Angle(F', E, F''_2) → γ
  28. Rotate(F', 0.5235987755982988, E) → F''_3
  29. Angle(F', E, F''_3) → δ
  30. Rotate(F', 1.0471975511965976, E) → F''_4
  31. Angle(F', E, F''_4) → ε
  32. Rotate(F', 1.3089969389957472, E) → F''_5
  33. Angle(F', E, F''_5) → ζ
  34. Rotate(F', 1.5707963267948966, E) → F''_6
  35. Angle(F', E, F''_6) → η
  36. Rotate(F', 1.8325957145940461, E) → F''_7
  37. Angle(F', E, F''_7) → θ
  38. Rotate(F', 2.0943951023931953, E) → F''_8
  39. Angle(F', E, F''_8) → ι
  40. Rotate(F''_8, 0.2617993877991494, E) → U
  41. Angle(F''_8, E, U) → κ
  42. Rotate(U, 0.2617993877991494, E) → U'
  43. Angle(U, E, U') → λ
  44. Rotate(U', 0.2617993877991494, E) → U''
  45. Angle(U', E, U'') → μ
  46. Rotate(U'', 0.2617993877991494, E) → U'''
  47. Angle(U'', E, U''') → ν
  48. Rotate(U''', 0.2617993877991494, E) → V
  49. Angle(U''', E, V) → ξ
  50. Rotate(V, 0.2617993877991494, E) → V'
  51. Angle(V, E, V') → ο
  52. Rotate(V', 0.2617993877991494, E) → V''
  53. Angle(V', E, V'') → ρ
  54. Rotate(V'', 0.2617993877991494, E) → V'''
  55. Angle(V'', E, V''') → σ
  56. Rotate(V''', 0.2617993877991494, E) → W
  57. Angle(V''', E, W) → τ
  58. Rotate(W, 0.2617993877991494, E) → W'
  59. Angle(W, E, W') → υ
  60. Rotate(W', 0.2617993877991494, E) → W''
  61. Angle(W', E, W'') → ϕ
  62. Rotate(W'', 0.2617993877991494, E) → W'''
  63. Angle(W'', E, W''') → χ
  64. Rotate(W''', 0.2617993877991494, E) → Z
  65. Angle(W''', E, Z) → ψ
  66. Rotate(Z, 0.2617993877991494, E) → Z'
  67. Angle(Z, E, Z') → ω
  68. Rotate(Z', 0.2617993877991494, E) → Z''
  69. Angle(Z', E, Z'') → α_1
  70. Mirror(A_1, J) → B_1
  71. Line(B_1, a) → g
  72. OrthogonalLine(T, g) → h
  73. Intersect(g, h) → C_1
  74. Segment(J, B_1) → i
  75. Segment(B_1, C_1) → j
  76. Segment(C_1, T) → k
  77. Segment(T, J) → l
  78. Point(l) → D_1
  79. OrthogonalLine(D_1, j) → r
  80. Intersect(j, r) → E_1
  81. Segment(D_1, E_1) → s
  82. OrthogonalLine(G, a) → t
  83. Intersect(t, j) → F_1
  84. Intersect(t, l) → G_1
  85. Segment(F_1, G_1) → a_1
  86. Polygon(B_1, J, T, C_1) → poly1, b_1, j_1, t_1, c_1
  87. OrthogonalLine(E, a) → d_1
  88. Point(d_1) → H_1
  89. Segment(H_1, F''_2) → e_1
  90. Segment(H_1, F''_3) → f_1
  91. Segment(H_1, F'') → g_1
  92. Segment(H_1, F''_4) → h_1
  93. Segment(H_1, F''_5) → i_1
  94. Segment(H_1, F''_6) → k_1
  95. Segment(H_1, F''_7) → l_1
  96. Segment(H_1, F''_8) → m_1
  97. Segment(H_1, U) → n_1
  98. Segment(H_1, U') → p_1
  99. Segment(H_1, U'') → q_1
  100. Point(d_1) → I_1
  101. Segment(H_1, I_1) → r_1
  102. Segment(H_1, V) → s_1
  103. Segment(H_1, V') → a_2
  104. Segment(H_1, V'') → b_2
  105. Segment(H_1, V''') → c_2
  106. Segment(H_1, W) → d_2
  107. Segment(H_1, W') → e_2
  108. Segment(H_1, W'') → f_2
  109. Segment(H_1, W''') → g_2
  110. Segment(H_1, Z) → h_2
  111. Segment(H_1, Z') → i_2
  112. Segment(H_1, Z'') → j_2
  113. Point(d_1) → J_1
  114. Ray(H_1, J_1) → k_2
  115. Point(k_2) → K_1
  116. Segment(H_1, K_1) → l_2
  117. Segment(F', H_1) → m_2
  118. Segment(H_1, U''') → n_2
  119. Ray(H_1, K_1) → p_2
  120. Intersect(e_1, f_1) → O
  121. Point(m_2) → L_1
  122. Point(k_1) → M_1
  123. Point(n_2) → N_1
  124. Point(e_2) → P_1
  125. Point(g_1) → Q_1
  126. Point(n_1) → R_1
  127. Point(b_2) → S_1
  128. Point(h_2) → T_1
  129. Point(e_1) → U_1
  130. Point(f_1) → V_1
  131. Point(h_1) → W_1
  132. Point(i_1) → Z_1
  133. Point(l_1) → A_2
  134. Point(m_1) → B_2
  135. Point(p_1) → C_2
  136. Point(q_1) → D_2
  137. Point(s_1) → E_2
  138. Point(c_2) → F_2
  139. Point(d_2) → G_2
  140. Point(f_2) → H_2
  141. Point(g_2) → I_2
  142. Point(i_2) → J_2
  143. Point(j_2) → K_2
  144. Point(a_2) → L_2

AI Drawing Prompt Reference

Draw a unit circle and mark the arc lengths and angle values for several special angles (0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, 2π). Start by drawing a circle with a radius of 2, then use rotation and angle measurement to mark these special angles and connect them with arcs.
The prompt above was compiled from the original drawing commands and is for reference only.