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反三角函数直观探索

反三角函数直观探索
几何画板及绘图内容加载中，请稍等……
2019-01-29 22:52
函数三角函数几何作图函数图像
①本范例通过动态演示反三角函数图像与性质，直观展示其定义域、值域及几何意义；②涉及核心概念包括反三角函数、三角函数图像、几何变换、连续性等；③作图要点运用了 Rotate（旋转）、OrthogonalLine（垂线）、Intersect（交点）、Circle（圆）、Ray（射线）等 GeoGebra 命令及对象；④适合高中及以上数学学习者，用于理解反三角函数的直观概念与性质。

GeoGebra 命令及参数

  1. Rotate(B, β, A) → B'_4
  2. OrthogonalLine(B'_4, xAxis) → j_1
  3. Intersect(j_1, xAxis) → J
  4. Circle(A, 1) → c
  5. Rotate(B, α, A) → B'
  6. OrthogonalLine(B', xAxis) → i
  7. OrthogonalLine(B, xAxis) → j
  8. Ray(A, B') → k
  9. Segment(A, B') → l
  10. Intersect(i, xAxis) → C
  11. Rotate(C, If[0 < m ≤ 1, (m * π / 2), π / 2], A) → C'
  12. Dilate(A, If[1 < m ≤ 2, 2 - m, 0], D) → A'
  13. Vector(A, A') → u
  14. Translate(C', u) → C''
  15. Rotate(C'', If[2 < m ≤ 3, ((-(m - 2)) * π / 2), ((-π)) / 2], D) → C'''
  16. Mirror(B, A) → B'_1
  17. Semicircle(B'_1, B) → p
  18. CircleArc(A, B, B') → q
  19. Dilate(B, If[4 < m ≤ 5, 5 - m, 0], E) → B'_2
  20. Vector(B, B'_2) → v
  21. Translate(c, v) → c'
  22. Translate(B', v) → B''
  23. Center(c') → F
  24. Rotate(B'', If[4 < m ≤ 5, ((-(m - 4)) * α), (-α)], F) → B'''
  25. Segment(B, B'_2) → r
  26. CircleArc(F, B'_2, B''') → s
  27. Dilate(B, If[6 < m ≤ 7, 7 - m, 0], C''') → B'_3
  28. Vector(B, B'_3) → w
  29. Translate(B'_2, w) → G
  30. Vector(A, C) → t
  31. Vector(A, C') → u_1
  32. Vector(A', C'') → v_1
  33. Vector(D, C''') → w_1
  34. Vector(B, E) → a_1
  35. Vector(B'_3, G) → b_1
  36. Segment(B', C) → f_1
  37. OrthogonalLine(B, xAxis) → i_1
  38. Ray(A, B'_4) → g_1
  39. Intersect(i_1, g_1) → H
  40. Segment(A, B'_4) → h_1
  41. Vector(J, B'_4) → c_1
  42. Vector(B, H) → d_1
  43. Dilate(J, If[0 < m ≤ 1, 1 - m, 0], D) → J'
  44. Vector(J, J') → e_1
  45. Translate(B'_4, e_1) → I
  46. Vector(J', I) → k_1
  47. Rotate(I, If[1 < m ≤ 2, ((-(m - 1)) * π / 2), ((-π)) / 2], D) → I'
  48. Vector(D, I') → l_1
  49. CircleArc(A, B'_4, B) → p_1
  50. CircleArc(A, B, B'_4) → q_1
  51. Dilate(B, If[3 < m ≤ 4, 4 - m, 0], L) → B'_5
  52. Vector(B, B'_5) → m_1
  53. Translate(c, m_1) → c'_1
  54. Center(c'_1) → N
  55. Segment(B, B'_5) → n_1
  56. Translate(B'_4, m_1) → P
  57. Rotate(P, If[3 < m ≤ 4, ((-(m - 3)) * β), (-β)], N) → P'
  58. CircleArc(N, B'_5, P') → r_1
  59. Mirror(r_1, xAxis) → s_1
  60. Mirror(B, xAxis) → B'_6
  61. Mirror(B'_5, xAxis) → Q
  62. Segment(B'_6, Q) → t_1
  63. Mirror(I, xAxis) → I'_1
  64. Mirror(B'_4, xAxis) → R
  65. Mirror(I', D) → I''
  66. Mirror(q_1, xAxis) → c_2
  67. Mirror(A, xAxis) → A'_1
  68. Mirror(B'_4, xAxis) → S
  69. Segment(A'_1, S) → f_2
  70. Mirror(c_1, xAxis) → u_2
  71. Mirror(k_1, xAxis) → v_2
  72. Mirror(l_1, D) → w_2
  73. Dilate(B, If[5 < m ≤ 6, 6 - m, 0], I') → B'_7
  74. Vector(B, B'_7) → b_2
  75. Vector(B, P') → d_2
  76. Translate(P', b_2) → P''
  77. Translate(B, b_2) → B'_8
  78. Vector(B'_8, P'') → e_2
  79. Mirror(P', xAxis) → P''_1
  80. Dilate(B, If[5 < m ≤ 6, 6 - m, 0], I'') → B'_9
  81. Vector(B, B'_9) → g_2
  82. Vector(B, P''_1) → h_2
  83. Translate(P''_1, g_2) → M
  84. Vector(B'_9, M) → i_2
  85. Mirror(B'_8, D) → O
  86. Mirror(I', D) → I''_1
  87. Ray(A, H) → j_2
  88. Mirror(d_1, xAxis) → k_2
  89. Mirror(H, xAxis) → H'
  90. Mirror(A, xAxis) → A'_2
  91. Mirror(H, xAxis) → H'_1
  92. Ray(A'_2, H'_1) → l_2
  93. Dilate(B, If[0 < m ≤ 1, 1 - m, 0], D) → B'_{10}
  94. Vector(B, B'_{10}) → m_2
  95. Translate(H, m_2) → H'_2
  96. Translate(d_1, m_2) → n_2
  97. Translate(H', m_2) → H''
  98. Vector(B'_{10}, H'_2) → p_2
  99. Vector(B'_{10}, H'') → q_2
  100. Segment(A, H) → r_2
  101. Mirror(B, A) → B'_{11}
  102. Segment(A, B) → s_2
  103. Segment(A, H') → t_2
  104. Rotate(H'_2, If[1 < m ≤ 2, ((-(m - 1)) * π / 2), ((-π)) / 2], D) → T
  105. Rotate(H'', If[1 < m ≤ 2, ((-(m - 1)) * π / 2), ((-π)) / 2], D) → H'''
  106. Vector(D, T) → u_3
  107. Vector(D, H''') → v_3
  108. Intersect(c, r_2, 1) → U
  109. Intersect(c, t_2, 1) → V
  110. CircleArc(A, B, U) → c_3
  111. CircleArc(A, V, B) → d_3
  112. Dilate(B, If[3 < m ≤ 4, 4 - m, 0], W) → B'_{12}
  113. Dilate(B, If[3 < m ≤ 4, 4 - m, 0], Z) → B'_{13}
  114. Vector(B, B'_{12}) → w_3
  115. Vector(B, B'_{13}) → a_3
  116. Translate(A, w_3) → A'_3
  117. Translate(A, a_3) → A'_4
  118. Translate(V, a_3) → V'
  119. Translate(U, w_3) → U'
  120. Rotate(U', If[3 < m ≤ 4, ((-(m - 3)) * β), (-β)], A'_3) → U''
  121. Rotate(V', If[3 < m ≤ 4, ((m - 3) * β), β], A'_4) → V''
  122. Segment(B, B'_{12}) → f_3
  123. Segment(B, B'_{13}) → g_3
  124. CircleArc(A'_3, B'_{12}, U'') → e_3
  125. CircleArc(A'_4, V'', B'_{13}) → h_3
  126. Vector(B, V'') → b_3
  127. Vector(B, U'') → i_3
  128. Dilate(B, If[5 < m ≤ 6, 6 - m, 0], T) → B'_{14}
  129. Vector(B, B'_{14}) → j_3
  130. Translate(U'', j_3) → U'''
  131. Vector(B'_{14}, U''') → k_3
  132. Dilate(B, If[5 < m ≤ 6, 6 - m, 0], H''') → B'_{15}
  133. Vector(B, B'_{15}) → l_3
  134. Translate(V'', l_3) → V'''
  135. Vector(B'_{15}, V''') → m_3

AI 绘图提示语参考

画一个动态展示反三角函数图像和性质的图形，包含反三角函数图像、定义域和值域，以及它们的几何意义。首先画一个圆和一条射线，然后通过旋转和垂线构造反三角函数图像，并标出关键点和线段，最后展示几何变换和连续性。
以上提示语为根据原始绘图命令整理而来，仅供参考。