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Trig Circle Start: Multiples of 30°

Trig Circle Start: Multiples of 30°
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2023-04-14 00:59
CircleAngleFunctionTrigonometryGeometric Construction
This example demonstrates the trigonometric values for multiples of 30° on the unit circle. It involves core concepts like angles, the unit circle, sine, cosine, and special angles. GeoGebra commands used include Mirror, Intersect, Dilate, Segment, Polygon, Midpoint, Rotate, Circle, Line, CircleSector, Vector, and Translate. It is suitable for high school students to learn and understand the relationship between trigonometric functions and the unit circle, ideal for classroom instruction or self-study.

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)) * pi / 3), ((-pi)) / 3], E) → K'_1
  24. Rotate(K', If[4 < a < 5, ((-(a - 4)) * pi / 3), ((-pi)) / 3], E) → K''
  25. Segment(K'_1, K'') → l'
  26. Rotate(K'_1, If[5 < a < 6, ((-(a - 5)) * pi), (-pi)], M) → N
  27. Rotate(K'', If[5 < a < 6, ((-(a - 5)) * pi), (-pi)], M) → K'''
  28. Segment(N, K''') → l''
  29. Rotate(L, If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → L'_1
  30. Rotate(K, If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → K'_2
  31. Rotate(E, If[6 < a < 7, ((a - 6) * pi / 2), pi / 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) * pi / 2), pi / 2], L) → K'_3
  34. Rotate(K', If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → K''_1
  35. Segment(K'_3, K''_1) → l'_2
  36. Rotate(L, If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → L'_2
  37. Rotate(L', If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → L''
  38. Segment(L'_2, L'') → k'
  39. Rotate(K'_1, If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → O
  40. Rotate(K'', If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → K'''_1
  41. Segment(O, K'''_1) → l''_1
  42. Rotate(N, If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → N'
  43. Rotate(K''', If[6 < a < 7, ((a - 6) * pi / 2), pi / 2], L) → P
  44. Segment(N', P) → l'''
  45. Rotate(L'_1, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → Q
  46. Rotate(K'_2, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → R
  47. Rotate(E'_3, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → S
  48. Polygon(Q, R, S) → poly1'', s, q, r
  49. Rotate(K'_3, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → T
  50. Rotate(K''_1, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → U
  51. Segment(T, U) → p
  52. Rotate(L'_2, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → V
  53. Rotate(L'', If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → L'''
  54. Segment(V, L''') → k''
  55. Rotate(O, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → O'
  56. Rotate(K'''_1, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → W
  57. Segment(O', W) → t
  58. Rotate(N', If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → N''
  59. Rotate(P, If[7 < a < 8, ((a - 7) * pi / 2), pi / 2], L) → P'
  60. Segment(N'', P') → b
  61. Rotate(Q, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → Q'
  62. Rotate(R, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → R'
  63. Rotate(S, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → S'
  64. Polygon(Q', R', S') → poly1''', s', q', r'
  65. Rotate(V, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → V'
  66. Rotate(L''', If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → Z
  67. Segment(V', Z) → k'''
  68. Rotate(N'', If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → N'''
  69. Rotate(P', If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → P''
  70. Segment(N''', P'') → b'
  71. Rotate(O', If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → O''
  72. Rotate(W, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → W'
  73. Segment(O'', W') → t'
  74. Rotate(T, If[8 < a < 9, ((a - 8) * pi / 2), pi / 2], L) → T'
  75. Rotate(U, If[8 < a < 9, ((a - 8) * pi / 2), pi / 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) * pi), pi], A_1) → C_1
  81. Rotate(K''_1, If[10 < a < 11, ((a - 10) * pi), pi], A_1) → D_1
  82. Segment(C_1, D_1) → g_1
  83. Rotate(K'_1, If[11 < a < 12, ((a - 11) * pi / 6), pi / 6], L) → E_1
  84. Rotate(K'', If[11 < a < 12, ((a - 11) * pi / 6), pi / 6], L) → K'''_2
  85. Segment(E_1, K'''_2) → l''_2
  86. Rotate(M, If[11 < a < 12, ((a - 11) * pi / 6), pi / 6], L) → M'_1
  87. Rotate(N, If[11 < a < 12, ((a - 11) * pi / 6), pi / 6], L) → N'_1
  88. Rotate(K''', If[11 < a < 12, ((a - 11) * pi / 6), pi / 6], L) → F_1
  89. Segment(N'_1, F_1) → l'''_1
  90. Rotate(A_1, pi / 2) → G_1
  91. Rotate(G_1, pi / 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, pi / 2, L) → K_1
  99. Rotate(D_1, pi / 2, L) → L_1
  100. Segment(K_1, L_1) → m_1
  101. Rotate(C_1, pi, L) → M_1
  102. Rotate(D_1, pi, L) → N_1
  103. Segment(M_1, N_1) → n_1
  104. Rotate(C_1, (3 * pi / 2), L) → O_1
  105. Rotate(D_1, (3 * pi / 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, pi, L) → Q_1
  112. Rotate(M'_1, pi / 2, L) → R_1
  113. Rotate(M'_1, (3 * pi / 2), L) → S_1
  114. Rotate(E_1, pi / 2, L) → T_1
  115. Rotate(K'''_2, pi / 2, L) → U_1
  116. Segment(T_1, U_1) → q_1
  117. Rotate(E_1, pi, L) → V_1
  118. Rotate(K'''_2, pi, L) → W_1
  119. Segment(V_1, W_1) → r_1
  120. Rotate(E_1, (3 * pi / 2), L) → Z_1
  121. Rotate(K'''_2, (3 * pi / 2), L) → A_2
  122. Segment(Z_1, A_2) → s_1
  123. Rotate(N'_1, pi / 2, L) → B_2
  124. Rotate(F_1, pi / 2, L) → C_2
  125. Segment(B_2, C_2) → t_1
  126. Rotate(N'_1, pi, L) → D_2
  127. Rotate(F_1, pi, L) → E_2
  128. Segment(D_2, E_2) → b_1
  129. Rotate(N'_1, (3 * pi / 2), L) → F_2
  130. Rotate(F_1, (3 * pi / 2), L) → G_2
  131. Segment(F_2, G_2) → c_1
  132. Circle(L, (0, 1)) → 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)) * pi / 6), ((-pi)) / 6], L) → d_2
  138. Rotate(d_2, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * pi / 6), ((-pi)) / 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
  143. Dilate(A_1, If[17 < a ≤ 18, 18 - a, 0], L) → L_2
  144. Vector(A_1, L_2) → u
  145. Translate(C_1, u) → M_2
  146. Translate(D_1, u) → N_2
  147. Segment(M_2, N_2) → n_2
  148. Mirror(u, A_1) → u'
  149. Translate(K'_3, u) → O_2
  150. Translate(M_1, u') → P_2
  151. Translate(N_1, u') → Q_2
  152. Segment(P_2, Q_2) → p_2
  153. Dilate(A_1, If[18 < a ≤ 19, 19 - a, 0], G) → R_2
  154. Vector(A_1, R_2) → v
  155. Mirror(R_2, L) → S_2
  156. Mirror(A_1, L) → T_2
  157. Vector(T_2, S_2) → w
  158. Translate(L'_2, v) → U_2
  159. Translate(L'', v) → L'''_1
  160. Segment(U_2, L'''_1) → k''_1
  161. Translate(T', u') → T''
  162. Translate(U', u') → U''
  163. Segment(T'', U'') → p''
  164. Translate(K'_3, u) → V_2
  165. Translate(K''_1, u) → W_2
  166. Segment(V_2, W_2) → q_2
  167. Translate(L'_2, w) → Z_2
  168. Translate(L'', w) → L'''_2
  169. Segment(Z_2, L'''_2) → k''_2
  170. Translate(V', w) → V''
  171. Translate(Z, w) → Z'
  172. Segment(V'', Z') → r_2
  173. Translate(V', v) → V''_1
  174. Translate(Z, v) → Z'_1
  175. Segment(V''_1, Z'_1) → s_2
  176. Intersect(t_2, y = 0) → A_3
  177. Intersect(a_2, y = 0) → B_3
  178. Polygon(L, B_3, G) → poly6, g_4, l_6, b_3
  179. Polygon(L, G', A_3) → poly7, a_3, l_7, g'
  180. Polygon(L, A_3, G'') → poly8, g''_1, l_8, a_4
  181. Polygon(L, G'_1, B_3) → poly9, b_2, l_9, g'_1
  182. Rotate(A, 45°, L) → A'
  183. Angle(A, L, A') → α
  184. Mirror(A', L) → A''
  185. Mirror(A', y = 0) → A''_1
  186. Mirror(A', x = 0) → A''_2
  187. Intersect(f_3, y = 0) → C_3
  188. Mirror(C_3, L) → D_3
  189. Line(A', A'') → h_3
  190. Intersect(d_1, h_3) → E_3, F_3
  191. Intersect(d_1, y = 0) → G_3, H_3
  192. CircleSector(L, G_3, F_3) → c_3
  193. Dilate(L, If[20 < a ≤ 21, 21 - a, 0], C_3) → L'_3
  194. Polygon(L, C_3, A') → poly10, a', l_{10}, c_4
  195. Polygon(L, A''_2, D_3) → poly11, d_3, l_{11}, a''_2
  196. Polygon(L, D_3, A'') → poly12, a'', l_{12}, d_4
  197. Polygon(L, A''_1, C_3) → poly13, c_5, l_{13}, a''_1
  198. Segment(L, L'_3) → i_3
  199. Segment(C_3, A) → j_3
  200. Mirror(C_3, L) → I_3
  201. Mirror(A, L) → A'_1
  202. Segment(I_3, A'_1) → k_3
  203. Rotate(L, If[21 < a ≤ 22, ((a - 21) * pi), pi], L) → L'_4
  204. Rotate(L'_3, If[21 < a ≤ 22, ((a - 21) * pi), pi], L) → J_3
  205. Segment(L'_4, J_3) → m_3
  206. Rotate(L, If[22 < a ≤ 23, ((-(a - 22)) * pi / 2), ((-pi)) / 2], C_3) → L'_5
  207. Rotate(L'_3, If[22 < a ≤ 23, ((-(a - 22)) * pi / 2), ((-pi)) / 2], C_3) → K_3
  208. Segment(L'_5, K_3) → n_3
  209. Mirror(L'_5, x = 0) → L_3
  210. Mirror(K_3, x = 0) → M_3
  211. Segment(L_3, M_3) → p_3
  212. Mirror(L'_5, L) → N_3
  213. Mirror(K_3, L) → O_3
  214. Segment(N_3, O_3) → q_3
  215. Mirror(L'_5, y = 0) → P_3
  216. Mirror(K_3, y = 0) → Q_3
  217. Segment(P_3, Q_3) → r_3
  218. Dilate(L, If[23 < a ≤ 24, 24 - a, 0], A') → L'_6
  219. Segment(L, L'_6) → s_3
  220. Mirror(L, x = 0) → L'_7
  221. Mirror(L'_6, x = 0) → R_3
  222. Segment(L'_7, R_3) → t_3
  223. Mirror(L, L) → L'_8
  224. Mirror(L'_6, L) → S_3
  225. Segment(L'_8, S_3) → e_3
  226. Mirror(L, y = 0) → L'_9
  227. Mirror(L'_6, y = 0) → T_3
  228. Segment(L'_9, T_3) → f_4
  229. Dilate(L, If[24 < a ≤ 25, 25 - a, 0], D) → L'_{10}
  230. Segment(L, L'_{10}) → h_4
  231. Mirror(L, y = 0) → L'_{11}
  232. Mirror(L'_{10}, y = 0) → U_3
  233. Segment(L'_{11}, U_3) → i_4
  234. Rotate(c_3, If[26 < a ≤ 27, ((a - 26) * pi / 4), pi / 4], L) → e_4

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with horizontal and vertical diameters. Mark points on the circle at multiples of 30° (30°, 60°, 90°, etc.) and connect them to form triangles. Use mirroring, rotation, and scaling to show how sine and cosine segment lengths change with angle. Include several dynamic triangles whose vertices move with an angle parameter, with edges colored to distinguish different function values. The final figure is an interactive unit circle clearly displaying trigonometric relationships for special angles.
The prompt above was compiled from the original drawing commands and is for reference only.