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Radians in Unit Circle

Radians in Unit Circle
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2020-11-22 08:27
CircleAngleTrigonometryGeometric ConstructionSlider Animation
① This example visually demonstrates the definition of radians using a unit circle, showing the relationship between angle, arc length, and radius. ② Core mathematical concepts include radian measure, unit circle, trigonometric functions (cosine), central angle and arc length, and conversion between degrees and radians. ③ Construction highlights: using GeoGebra commands such as Circle, Line, Segment, Rotate, Angle, CircleArc to build the unit circle and dynamic angle, with Vector, Translate, Mirror for transformations to illustrate that arc length equals radius for one radian, and Point, Polygon, Ray for annotations, plus text and images to explain the concept. ④ Suitable for high school or pre-university students learning trigonometry and angle measurement, and for teachers demonstrating or students exploring the geometric meaning of radians.

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 dynamically demonstrate the 1 radian angle where the arc length equals the radius. Start by drawing the diameter AB and CD, then draw a circle c with center E. Draw the perpendicular line d to AB and find the intersection point F. Connect AF and rotate F to form a 1 radian angle F'. Mark different radian angles on the circle and use rotation and translation to show angles like 1/4 radian, 1/2 radian, etc. Finally, label the degree measures of these angles on the circle.
The prompt above was compiled from the original drawing commands and is for reference only.