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三角恒等式几何探究

三角恒等式几何探究
几何画板及绘图内容加载中，请稍等……
2021-10-26 09:53
向量三角函数几何作图
①本范例通过动态几何演示三角恒等式，揭示余弦、正弦等函数与几何图形的内在联系；②核心概念包括三角函数、几何变换、向量运算；③构造中运用了点、旋转、圆、线段、正交线、交点、镜像、多边形、半圆、直线、缩放、角度、扇形、文本旋转、向量、平移等GeoGebra命令；④适合高中数学教学，用于加深对三角恒等式和几何直观的理解。

GeoGebra 命令及参数

  1. Point(Ray[(0, 0), (1, 0)]) → B
  2. Rotate(B, π / 4, A) → B'
  3. Circle(A, B) → c
  4. Point(CircleArc[A, B, B']) → D
  5. Rotate(D, Angle[B, A, D], A) → D'
  6. Segment(A, D') → f
  7. OrthogonalLine(D', xAxis) → g
  8. Intersect(g, xAxis) → E
  9. Intersect(h, yAxis) → F
  10. Intersect(g, h) → G
  11. Mirror(D, yAxis) → D'_{1}
  12. OrthogonalLine(D'_{1}, xAxis) → i
  13. Segment(A, D'_{1}) → j
  14. Intersect(i, xAxis) → H
  15. Polygon(A, H, 4) → poly1, k, l, m, n, I, J
  16. Polygon(H, D'_{1}, 4) → poly2, p, q, r, s, K, L
  17. Semicircle(M, B) → d
  18. Polygon(A, E, G, F) → q1, a_1, e, g_1, f_1
  19. Line(D'_{1}, K) → t
  20. Circle(H, Distance[H, I]) → c_1
  21. Intersect(c_1, t, 2) → N
  22. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D) → A'
  23. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D') → A'_{1}
  24. Segment(A, A'_{1}) → h_1
  25. Segment(A, A') → i_1
  26. Circle(A, 0.4) → d_1
  27. Intersect(d_1, i_1, 1) → O
  28. Angle(B, A, D) → α
  29. Angle(B, A, O) → β
  30. Angle(D, A, D') → γ
  31. Intersect(d_1, xAxis, 2) → P
  32. Rotate(P, If[2 < a ≤ 3, ((a - 2) * α), α], A) → P'
  33. CircleSector(A, P, P') → e_1
  34. Rotate(e_1, If[4 < a ≤ 5, ((a - 4) * α), α], A) → k_1
  35. Angle(D'_{1}, A, M) → δ
  36. Rotate(k_1, If[6 < a ≤ 7, ((a - 6) * (π - (2 * α))), π - (2 * α)], A) → p_1
  37. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], D'_{1}) → A'_{2}
  38. Segment(A, A'_{2}) → j_1
  39. Dilate(D'_{1}, If[9 < a ≤ 10, 10 - a, 0], H) → Q
  40. Dilate(D', If[9 < a ≤ 10, 10 - a, 0], E) → D''
  41. Segment(D', D'') → l_1
  42. Segment(D'_{1}, Q) → m_1
  43. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], E) → A'_{3}
  44. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], H) → A'_{4}
  45. Segment(A, A'_{3}) → n_1
  46. Segment(A, A'_{4}) → q_1
  47. RotateText(text1, π / 2) → text2
  48. Rotate(H, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], D'_{1}) → H'
  49. Dilate(D'_{1}, If[14 < a ≤ 15, 15 - a, 0], H) → R
  50. Vector(D'_{1}, R) → u
  51. Translate(D'_{1}, u) → S
  52. Translate(H', u) → H''
  53. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], L) → H'_{1}
  54. Vector(H, H'_{1}) → v
  55. Translate(D'_{1}, v) → T
  56. Translate(Q, v) → Q'
  57. Rotate(A, If[17 < a ≤ 18, ((-(a - 17)) * π / 2), ((-π)) / 2], H) → A'_{5}
  58. Segment(H, A'_{5}) → b_1
  59. Dilate(H, If[18 < a ≤ 19, 19 - a, 0], I) → H'_{2}
  60. Vector(H, H'_{2}) → w
  61. Translate(A, w) → A'_{6}
  62. Translate(A'_{4}, w) → U
  63. Segment(A'_{6}, U) → f_2
  64. Dilate(I, If[19 < a ≤ 20, 20 - a, 0], J) → I'
  65. Vector(I, I') → u_1
  66. Translate(H, u_1) → H'_{3}
  67. Translate(A'_{5}, u_1) → V
  68. Segment(H'_{3}, V) → g_2
  69. Angle(F, A, D') → ε
  70. Rotate(D', If[20 < a ≤ 21, ((-(a - 20)) * ε), (-ε)], A) → D''_{1}
  71. Segment(A, D''_{1}) → i_2
  72. Dilate(A, If[21 < a ≤ 22, 22 - a, 0], F) → A'_{7}
  73. Vector(A, A'_{7}) → v_1
  74. Translate(A, v_1) → A'_{8}
  75. Translate(A'_{3}, v_1) → W
  76. Segment(A'_{8}, W) → j_2
  77. Dilate(F, If[22 < a ≤ 23, 23 - a, 0], G) → F'
  78. Vector(F, F') → w_1
  79. Translate(A, w_1) → A'_{9}
  80. Translate(D''_{1}, w_1) → Z
  81. Segment(A'_{9}, Z) → k_2
  82. Polygon(A, E, G, F) → q3, a_2, e_2, g_3, f_3
  83. Dilate(D'_{1}, If[26 < a ≤ 27, 27 - a, 0], N) → A_1
  84. Vector(D'_{1}, A_1) → u_2
  85. Translate(H', u_2) → H''_{1}
  86. Line(H''_{1}, L) → n_2
  87. OrthogonalLine(H, n_2) → p_2
  88. Intersect(n_2, p_2) → B_1
  89. Dilate(L, If[28 < a ≤ 29, 29 - a, 0], B_1) → L'
  90. Vector(L, L') → v_2
  91. Translate(H''_{1}, v_2) → C_1
  92. Dilate(H, If[30 < a ≤ 31, 31 - a, 0], A) → H'_{4}
  93. Vector(H, H'_{4}) → w_2
  94. Angle(B_1, H, I) → ζ
  95. Translate(N, w_2) → N'
  96. Translate(C_1, w_2) → D_1
  97. Translate(L', w_2) → L''
  98. Translate(H, w_2) → H'_{5}
  99. Polygon(N', D_1, L'', H'_{5}) → q5', n', r_2, l'', h'_{5}
  100. Rotate(N', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → N''
  101. Rotate(D_1, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → E_1
  102. Rotate(L'', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → L'''
  103. Rotate(H'_{5}, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → F_1
  104. Line(A, E) → b_2
  105. Circle(A, Distance[I, J]) → c_2
  106. Segment(A, B) → k_4
  107. Segment(A, M) → l_4
  108. Dilate(V_1, If[9 < a ≤ 10, 10 - a, 0], A) → W_1
  109. Rotate(A, If[11 < a ≤ 12, ((a - 11) * (π / 2 - (2 * α))), π / 2 - (2 * α)], A) → A'_{11}
  110. Rotate(A'_{1}, If[11 < a ≤ 12, ((a - 11) * (π / 2 - (2 * α))), π / 2 - (2 * α)], A) → A_2
  111. Segment(A'_{11}, A_2) → n_4
  112. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], Z_1) → A'_{12}
  113. Vector(A, A'_{12}) → u_4
  114. Translate(A'_{11}, u_4) → B_2
  115. Translate(A_2, u_4) → C_2
  116. Segment(B_2, C_2) → m_4
  117. Rotate(B_2, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], C_2) → D_2
  118. Rotate(C_2, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], C_2) → E_2
  119. Segment(D_2, E_2) → p_4
  120. Dilate(C_2, If[14 < a ≤ 15, 15 - a, 0], Z_1) → F_2
  121. Vector(C_2, F_2) → v_4
  122. Translate(D_2, v_4) → G_2
  123. Translate(E_2, v_4) → H_2
  124. Segment(G_2, H_2) → q_4
  125. Dilate(C_2, If[15 < a ≤ 16, 16 - a, 0], D_2) → I_2
  126. Vector(C_2, I_2) → w_4
  127. Translate(B_2, w_4) → J_2
  128. Translate(C_2, w_4) → K_2
  129. Segment(J_2, K_2) → r_4
  130. Polygon(G_2, Z_1, C_2, D_2) → q8, s_4, z_1, t_4, a_4
  131. Rotate(U, If[23 < a ≤ 24, ((a - 23) * π), π], A'_{5}) → U'
  132. Rotate(H, If[23 < a ≤ 24, ((a - 23) * π / 2), π / 2], A'_{5}) → H'_{6}
  133. Rotate(A'_{5}, If[23 < a ≤ 24, ((a - 23) * π / 2), π / 2], A'_{5}) → M_2
  134. Segment(H'_{6}, M_2) → s_1
  135. Dilate(A'_{5}, If[24 < a ≤ 25, 25 - a, 0], H'_{6}) → L_2
  136. Vector(A'_{5}, L_2) → r_1
  137. Translate(H, r_1) → H'_{7}
  138. Translate(A'_{5}, r_1) → N_2
  139. Segment(H'_{7}, N_2) → t_1
  140. RotateText(text3, π / 2) → text5
  141. Circle(A, (2 * f_2)) → h_2
  142. Line(A'_{8}, W) → l_2
  143. Intersect(h_2, l_2, 1) → O_2
  144. Dilate(A'_{8}, If[28 < a ≤ 29, 29 - a, 0], O_2) → P_2
  145. Vector(A'_{8}, P_2) → m_2
  146. Translate(W, m_2) → W'
  147. Polygon(A, A'_{3}, W', P_2) → q2, a_5, a'_{3}, w', q_2
  148. Line(O_2, a'_{3}) → h_3
  149. OrthogonalLine(A'_{3}, h_3) → l_3
  150. Intersect(h_3, l_3) → Q_2
  151. Dilate(A, If[30 < a ≤ 31, 31 - a, 0], Q_2) → A'_{13}
  152. Vector(A, A'_{13}) → s_2
  153. Translate(O_2, s_2) → R_2
  154. Polygon(W', R_2, A'_{13}, A'_{3}) → q4, t_2, n_3, a'_{13}, c_3
  155. Line(W', A'_{3}) → d_3
  156. Dilate(W', If[32 < a ≤ 33, 33 - a, 0], N_2) → W''
  157. Vector(W', W'') → e_3
  158. Translate(W', e_3) → W''_{1}
  159. Translate(R_2, e_3) → S_2
  160. Translate(A'_{13}, e_3) → T_2
  161. Translate(A'_{3}, e_3) → U_2
  162. Polygon(W''_{1}, S_2, T_2, U_2) → q4', w''_{1}, h_4, b_4, c_4
  163. Line(W', f_2) → d_4
  164. Point(d_4) → V_2
  165. Angle(V_2, W', A'_{3}) → θ
  166. Rotate(W''_{1}, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → W_2
  167. Rotate(S_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → Z_2
  168. Rotate(T_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → A_3
  169. Rotate(U_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → B_3
  170. Polygon(W_2, Z_2, A_3, B_3) → q4'', e_4, z_2, g_5, h_5
  171. Circle(Z_1, (2 * f_2)) → d_2
  172. Line(C_2, p_4) → i_3
  173. Intersect(d_2, i_3) → G_1, M_1
  174. Line(G_2, i_3) → j_3
  175. Dilate(C_2, If[34 < a ≤ 35, 35 - a, 0], M_1) → H_1
  176. Vector(C_2, H_1) → u_3
  177. Translate(D_2, u_3) → I_1
  178. Polygon(I_1, H_1, Z_1, G_2) → q5, k_3, m_3, p_3, q_3
  179. Line(M_1, Z_1) → r_3
  180. Line(I_1, G_2) → s_3
  181. OrthogonalLine(G_2, s_3) → t_3
  182. Intersect(t_3, r_3) → J_1
  183. Dilate(Z_1, If[36 < a ≤ 37, 37 - a, 0], J_1) → K_1
  184. Vector(Z_1, K_1) → v_3
  185. Translate(M_1, v_3) → L_1
  186. Polygon(I_1, L_1, K_1, G_2) → q6, a_3, b_3, f_4, g_4
  187. Dilate(J_1, If[38 < a ≤ 39, 39 - a, 0], Z_2) → O_1
  188. Vector(J_1, O_1) → w_3
  189. Translate(I_1, w_3) → P_1
  190. Translate(L_1, w_3) → Q_1
  191. Translate(K_1, w_3) → R_1
  192. Translate(G_2, w_3) → S_1
  193. Polygon(P_1, Q_1, R_1, S_1) → q6', i_4, j_4, f_5, i_5
  194. Angle(A'_{3}, Z_1, M_1) → η
  195. Rotate(P_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → T_1
  196. Rotate(Q_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → U_1
  197. Rotate(R_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → C_3
  198. Line(Z_2, s_1) → j_5
  199. Point(j_5) → D_3
  200. Rotate(P_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → E_3
  201. Rotate(Q_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → F_3
  202. Rotate(R_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → G_3
  203. Rotate(S_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → H_3
  204. Polygon(E_3, F_3, G_3, H_3) → q6'', k_5, l_5, m_5, n_5

AI 绘图提示语参考

在坐标平面上，从原点出发沿水平方向取一点，将其绕原点旋转45度得到另一点，并以原点为圆心过这两点画圆弧。在圆弧上任取一点，将其绕原点旋转至与第一点同角度，连接原点与该旋转点作线段，并过该点作水平线的垂线，交水平轴于一点。将圆弧上的点关于竖直轴镜像，过镜像点作水平线垂线，交水平轴于另一点。以原点、该交点及镜像点构造正方形，并在镜像点与交点间再构造正方形。以原点为圆心画小圆，在圆上取点并旋转生成扇形，同时用多个辅助点与线段展示角度关系，最后将文字标签旋转90度。
以上提示语为根据原始绘图命令整理而来，仅供参考。