Euler's Formula

Euler's Formula
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2022-05-08 18:52
This example demonstrates Euler's formula, involving the relationship between complex numbers and exponential functions. It uses points, circles, vectors, and other GeoGebra commands to show the application of Euler's formula in coordinate geometry, suitable for high school mathematics education.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Circle(A, 1) → c
  3. Vector(A, B) → u
  4. Vector(B, C) → v
  5. Circle(C, t02) → d
  6. OrthogonalLine(C, yAxis) → f
  7. Intersect(d, f, 2) → D
  8. Vector(C, D) → w
  9. Circle(D, t03) → e
  10. OrthogonalLine(D, w) → g
  11. Intersect(e, g) → E, F
  12. Vector(D, F) → a
  13. Circle(F, t04) → h
  14. OrthogonalLine(F, yAxis) → i
  15. Intersect(h, i, 1) → G
  16. Vector(F, G) → b
  17. OrthogonalLine(G, b) → j
  18. Circle(G, t05) → k
  19. Intersect(k, j) → H, I
  20. Vector(G, I) → l
  21. Circle(I, t06) → p
  22. OrthogonalLine(I, l) → m
  23. Intersect(p, m, 2) → J
  24. Vector(I, J) → n
  25. Circle(J, t07) → q
  26. OrthogonalLine(J, n) → r
  27. Intersect(q, r, 2) → K
  28. Vector(J, K) → s
  29. Circle(K, t08) → t
  30. OrthogonalLine(K, s) → f_1
  31. Intersect(t, f_1, 2) → L
  32. Vector(K, L) → u_1
  33. Circle(L, t09) → c_1
  34. OrthogonalLine(L, u_1) → g_1
  35. Intersect(c_1, g_1, 2) → M
  36. Vector(L, M) → v_1
  37. Circle(M, t10) → d_1
  38. OrthogonalLine(M, v_1) → h_1
  39. Intersect(d_1, h_1, 2) → N
  40. Vector(M, N) → w_1
  41. Circle(N, t11) → e_1
  42. OrthogonalLine(N, w_1) → i_1
  43. Intersect(e_1, i_1, 2) → O
  44. Vector(N, O) → a_1
  45. Circle(O, t12) → k_1
  46. OrthogonalLine(O, a_1) → j_1
  47. Intersect(k_1, j_1, 2) → P
  48. Vector(O, P) → b_1
  49. Circle(P, t13) → p_1
  50. OrthogonalLine(P, b_1) → l_1
  51. Intersect(p_1, l_1, 2) → Q
  52. Vector(P, Q) → m_1
  53. Circle(Q, t14) → q_1
  54. OrthogonalLine(Q, m_1) → n_1
  55. Intersect(q_1, n_1, 2) → R
  56. Vector(Q, R) → r_1
  57. Circle(R, t15) → s_1
  58. OrthogonalLine(R, r_1) → t_1
  59. Intersect(s_1, t_1, 2) → S
  60. Vector(R, S) → u_2
  61. Circle(S, t16) → c_2
  62. OrthogonalLine(S, u_2) → f_2
  63. Intersect(c_2, f_2, 2) → T
  64. Vector(S, T) → v_2
  65. Circle(T, t17) → d_2
  66. OrthogonalLine(T, v_2) → g_2
  67. Intersect(d_2, g_2, 2) → U
  68. Vector(T, U) → w_2
  69. Circle(U, t18) → e_2
  70. OrthogonalLine(U, w_2) → h_2
  71. Intersect(e_2, h_2, 2) → V
  72. Vector(U, V) → a_2

AI Drawing Prompt Reference

Draw a graph demonstrating Euler's formula, starting with a point on the y-axis, creating a unit circle centered at this point, drawing vectors from the circle to the y-axis, and repeating the process with circles centered at the vector endpoints, showing the relationship between complex numbers and exponential functions, suitable for high school mathematics education.
The prompt above was compiled from the original drawing commands and is for reference only.