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Introduction to Special Angles in Standard Position

Introduction to Special Angles in Standard Position
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2017-01-26 01:42
AngleTransformationGeometric Construction
In collections Similarity & Right Triangle Trigonometry Similarity, Right Triangles, and Trigonometry (SRT) Cluster Trigonometric Functions of Angles Special Right Triangles Special Right Triangles
This example demonstrates special angles in standard position, involving angle concepts and transformations. Using commands like Mirror, Intersect, Dilate, Segment, and Polygon, it dynamically illustrates the properties and changes of angles. Suitable for high school math education and learning.

GeoGebra Commands & Parameters

  1. Mirror(G, f) → G'
  2. Mirror(F, f) → F'
  3. Mirror(E, g) → E'
  4. Mirror(F, g) → F'_1
  5. Mirror(G, g) → G'_1
  6. Mirror(F', g) → F''
  7. Mirror(G', g) → G''
  8. Mirror(E, f) → E'_1
  9. Mirror(E'_1, g) → H
  10. Intersect(g, j) → I
  11. Intersect(g, i) → J
  12. Intersect(g, h) → K
  13. Dilate(L, If[0 < a ≤ 1, 1 - a, 0], K) → L'
  14. Segment(L, L') → k
  15. Dilate(K, If[1 < a ≤ 2, 2 - a, 0], E) → K'
  16. Segment(K, K') → l
  17. Polygon(L, K, E) → poly1, e, l_1, k_1
  18. Midpoint(L, E) → M
  19. Dilate(E, If[2 < a ≤ 3, 3 - a, 0], M) → E'_2
  20. Segment(E, E'_2) → m
  21. Dilate(M, If[3 < a ≤ 4, 4 - a, 0], L) → M'
  22. Segment(M, M') → n
  23. Rotate(K, If[4 < a < 5, ((-(a - 4)) * π / 3), ((-π)) / 3], E) → K'_1
  24. Rotate(K', If[4 < a < 5, ((-(a - 4)) * π / 3), ((-π)) / 3], E) → K''
  25. Segment(K'_1, K'') → l'
  26. Rotate(K'_1, If[5 < a < 6, ((-(a - 5)) * π), (-π)], M) → N
  27. Rotate(K'', If[5 < a < 6, ((-(a - 5)) * π), (-π)], M) → K'''
  28. Segment(N, K''') → l''
  29. Rotate(L, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → L'_1
  30. Rotate(K, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → K'_2
  31. Rotate(E, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → E'_3
  32. Polygon(L'_1, K'_2, E'_3) → poly1', e'_3, l'_1, k'_2
  33. Rotate(K, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → K'_3
  34. Rotate(K', If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → K''_1
  35. Segment(K'_3, K''_1) → l'_2
  36. Rotate(L, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → L'_2
  37. Rotate(L', If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → L''
  38. Segment(L'_2, L'') → k'
  39. Rotate(K'_1, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → O
  40. Rotate(K'', If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → K'''_1
  41. Segment(O, K'''_1) → l''_1
  42. Rotate(N, If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → N'
  43. Rotate(K''', If[6 < a < 7, ((a - 6) * π / 2), π / 2], L) → P
  44. Segment(N', P) → l'''
  45. Rotate(L'_1, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → Q
  46. Rotate(K'_2, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → R
  47. Rotate(E'_3, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → S
  48. Polygon(Q, R, S) → poly1'', s, q, r
  49. Rotate(K'_3, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → T
  50. Rotate(K''_1, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → U
  51. Segment(T, U) → p
  52. Rotate(L'_2, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → V
  53. Rotate(L'', If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → L'''
  54. Segment(V, L''') → k''
  55. Rotate(O, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → O'
  56. Rotate(K'''_1, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → W
  57. Segment(O', W) → t
  58. Rotate(N', If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → N''
  59. Rotate(P, If[7 < a < 8, ((a - 7) * π / 2), π / 2], L) → P'
  60. Segment(N'', P') → b
  61. Rotate(Q, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → Q'
  62. Rotate(R, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → R'
  63. Rotate(S, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → S'
  64. Polygon(Q', R', S') → poly1''', s', q', r'
  65. Rotate(V, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → V'
  66. Rotate(L''', If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → Z
  67. Segment(V', Z) → k'''
  68. Rotate(N'', If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → N'''
  69. Rotate(P', If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → P''
  70. Segment(N''', P'') → b'
  71. Rotate(O', If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → O''
  72. Rotate(W, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → W'
  73. Segment(O'', W') → t'
  74. Rotate(T, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → T'
  75. Rotate(U, If[8 < a < 9, ((a - 8) * π / 2), π / 2], L) → U'
  76. Segment(T', U') → p'
  77. Intersect(d, f) → A_1
  78. Dilate(A_1, If[9 < a ≤ 10, 10 - a, 0], G) → B_1
  79. Segment(A_1, B_1) → f_1
  80. Rotate(K'_3, If[10 < a < 11, ((a - 10) * π), π], A_1) → C_1
  81. Rotate(K''_1, If[10 < a < 11, ((a - 10) * π), π], A_1) → D_1
  82. Segment(C_1, D_1) → g_1
  83. Rotate(K'_1, If[11 < a < 12, ((a - 11) * π / 6), π / 6], L) → E_1
  84. Rotate(K'', If[11 < a < 12, ((a - 11) * π / 6), π / 6], L) → K'''_2
  85. Segment(E_1, K'''_2) → l''_2
  86. Rotate(M, If[11 < a < 12, ((a - 11) * π / 6), π / 6], L) → M'_1
  87. Rotate(N, If[11 < a < 12, ((a - 11) * π / 6), π / 6], L) → N'_1
  88. Rotate(K''', If[11 < a < 12, ((a - 11) * π / 6), π / 6], L) → F_1
  89. Segment(N'_1, F_1) → l'''_1
  90. Rotate(A_1, π / 2) → G_1
  91. Rotate(G_1, π / 2) → H_1
  92. Dilate(G_1, If[9 < a ≤ 10, 10 - a, 0], E'_1) → I_1
  93. Dilate(H_1, If[9 < a ≤ 10, 10 - a, 0], G'') → J_1
  94. Dilate(K, If[9 < a ≤ 10, 10 - a, 0], E') → K'_4
  95. Segment(G_1, I_1) → h_1
  96. Segment(H_1, J_1) → i_1
  97. Segment(K, K'_4) → j_1
  98. Rotate(C_1, π / 2, L) → K_1
  99. Rotate(D_1, π / 2, L) → L_1
  100. Segment(K_1, L_1) → m_1
  101. Rotate(C_1, π, L) → M_1
  102. Rotate(D_1, π, L) → N_1
  103. Segment(M_1, N_1) → n_1
  104. Rotate(C_1, (3 * π / 2), L) → O_1
  105. Rotate(D_1, (3 * π / 2), L) → P_1
  106. Segment(O_1, P_1) → p_1
  107. Polygon(L, G, A_1) → poly2, a_1, l_2, g_2
  108. Polygon(L, E'_1, G_1) → poly3, g_3, l_3, e'_1
  109. Polygon(L, G'', H_1) → poly4, h_2, l_4, g''
  110. Polygon(L, K, E') → poly5, e', l_5, k_2
  111. Rotate(M'_1, π, L) → Q_1
  112. Rotate(M'_1, π / 2, L) → R_1
  113. Rotate(M'_1, (3 * π / 2), L) → S_1
  114. Rotate(E_1, π / 2, L) → T_1
  115. Rotate(K'''_2, π / 2, L) → U_1
  116. Segment(T_1, U_1) → q_1
  117. Rotate(E_1, π, L) → V_1
  118. Rotate(K'''_2, π, L) → W_1
  119. Segment(V_1, W_1) → r_1
  120. Rotate(E_1, (3 * π / 2), L) → Z_1
  121. Rotate(K'''_2, (3 * π / 2), L) → A_2
  122. Segment(Z_1, A_2) → s_1
  123. Rotate(N'_1, π / 2, L) → B_2
  124. Rotate(F_1, π / 2, L) → C_2
  125. Segment(B_2, C_2) → t_1
  126. Rotate(N'_1, π, L) → D_2
  127. Rotate(F_1, π, L) → E_2
  128. Segment(D_2, E_2) → b_1
  129. Rotate(N'_1, (3 * π / 2), L) → F_2
  130. Rotate(F_1, (3 * π / 2), L) → G_2
  131. Segment(F_2, G_2) → c_1
  132. Circle(L, (0, 0.5)) → d_1
  133. Intersect(d_1, x = 0) → H_2, I_2
  134. Line(G, L) → e_1
  135. Intersect(d_1, e_1) → J_2, K_2
  136. CircleSector(L, J_2, H_2) → c_2
  137. Rotate(c_2, If[13.5 < a ≤ 14.5, ((-(a - 13.5)) * π / 6), ((-π)) / 6], L) → d_2
  138. Rotate(d_2, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * π / 6), ((-π)) / 6], L) → e_2
  139. Segment(L', A) → f_2
  140. Segment(L'', B) → i_2
  141. Segment(L''', C) → j_2
  142. Segment(Z, D) → m_2

AI Drawing Prompt Reference

Draw a standard-position angle and demonstrate special angles through reflections, rotations, and dilations. First, reflect several points, then form new segments by intersecting lines. Next, scale and connect these segments to create polygons. Finally, rotate and reflect these polygons to dynamically show the changes and properties of angles.
The prompt above was compiled from the original drawing commands and is for reference only.