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30°倍数的三角函数单位圆演示

30°倍数的三角函数单位圆演示
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2023-04-14 00:59
圆角函数三角函数几何作图
①本范例通过单位圆演示30°倍数的角度对应的三角函数值；②涉及核心数学概念包括角度、单位圆、三角函数（正弦、余弦）、特殊角；③作图要点运用了Mirror（镜像）、Intersect（交点）、Dilate（缩放）、Segment（线段）、Polygon（多边形）、Midpoint（中点）、Rotate（旋转）、Circle（圆）、Line（直线）、CircleSector（扇形）、Vector（向量）、Translate（平移）等GeoGebra命令；④适合高中数学学生学习和理解三角函数与单位圆的关系，可用于课堂教学或自主探究。

GeoGebra 命令及参数

  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 绘图提示语参考

在单位圆上绘制30°倍数的三角函数演示图。以圆心为原点，画出单位圆及水平、垂直直径。标记圆上30°、60°、90°等特殊角对应的点，并连接这些点形成三角形。通过镜像、旋转和缩放操作，逐步展示不同角度下正弦、余弦值对应的线段长度变化。图形包含多个动态三角形，其顶点随角度参数移动，边线颜色区分不同函数值。最终呈现一个交互式单位圆，清晰显示各特殊角度的三角函数关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。