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Intuitive Discovery of the Derivative of Sine Function

Intuitive Discovery of the Derivative of Sine Function
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2015-10-07 23:23
FunctionTrigonometryDerivativeGeometric ConstructionGraphing
This example demonstrates how students can intuitively discover that the derivative of the sine function y=sin(x) is the cosine function y=cos(x). It involves core math concepts like derivatives, sine and cosine functions, and geometric constructions. The construction uses GeoGebra commands and objects such as Random, RandomElement, Point, Circle, Intersect, and Vector. It is suitable for high school mathematics teaching to help students understand the geometric meaning of derivatives and properties of trigonometric functions.

GeoGebra Commands & Parameters

  1. Random(1, 2) → b
  2. Random(-3, 3) → c
  3. Random(-3, 3) → d
  4. RandomElement(list1) → a
  5. Point(vert_A) → A
  6. Circle((x(A), g(x(A))), 1) → circle_A
  7. Intersect(circle_A, line_A, 1) → A_2
  8. Intersect(circle_A, line_A, 2) → A_3
  9. Intersect(g, vert_A, (0, 0)) → A_1
  10. Vector(A_1, A_3) → u_A
  11. Vector(A_1, A_2) → v_A
  12. Point(vert_B) → B
  13. Circle((x(B), g(x(B))), 1) → circle_B
  14. Intersect(g, vert_B, (1.5707963267948966, 1)) → B_1
  15. Intersect(circle_B, line_B, 1) → B_2
  16. Intersect(circle_B, line_B, 2) → B_3
  17. Vector(B_1, B_2) → u_B
  18. Vector(B_1, B_3) → v_B
  19. Point(vert_C) → C
  20. Intersect(g, vert_C, (3.141592653589793, 1.2246467991473532E-16)) → C_1
  21. Circle((x(C), g(x(C))), 1) → circle_C
  22. Intersect(circle_C, line_C, 1) → C_2
  23. Intersect(circle_C, line_C, 2) → C_3
  24. Vector(C_1, C_3) → u_C
  25. Vector(C_1, C_2) → v_C
  26. Point(vert_D) → D
  27. Circle((x(D), g(x(D))), 1) → circle_D
  28. Intersect(g, vert_D, (4.71238898038469, -1)) → D_1
  29. Intersect(circle_D, line_D, 1) → D_2
  30. Intersect(circle_D, line_D, 2) → D_3
  31. Vector(D_1, D_2) → u_D
  32. Vector(D_1, D_3) → v_D
  33. Point(vert_E) → E
  34. Circle((x(E), g(x(E))), 1) → circle_E
  35. Intersect(g, vert_E, (6.283185307179586, -2.4492935982947064E-16)) → E_1
  36. Intersect(circle_E, line_E, 1) → E_2
  37. Intersect(circle_E, line_E, 2) → E_3
  38. Vector(E_1, E_2) → u_E
  39. Vector(E_1, E_3) → v_E
  40. Point(vert_F) → F
  41. Circle((x(F), g(x(F))), 1) → circle_F
  42. Intersect(g, vert_F, (-1.5707963267948966, -1)) → F_1
  43. Intersect(circle_F, line_F, 1) → F_3
  44. Intersect(circle_F, line_F, 2) → F_2
  45. Vector(F_1, F_3) → u_F
  46. Vector(F_1, F_2) → v_F
  47. Point(vert_G) → G
  48. Circle((x(G), g(x(G))), 1) → circle_G
  49. Intersect(g, line_G, (-3.141592653589793, -1.2246467991473532E-16)) → G_1
  50. Intersect(circle_G, line_G, 1) → G_2
  51. Intersect(circle_G, line_G, 2) → G_3
  52. Vector(G_1, G_2) → u_G
  53. Vector(G_1, G_3) → v_G
  54. Point(vert_H) → H
  55. Circle((x(H), g(x(H))), 1) → circle_H
  56. Intersect(g, line_H, (-4.71238898038469, 1)) → H_1
  57. Intersect(circle_H, line_H, 1) → H_2
  58. Intersect(circle_H, line_H, 2) → H_3
  59. Vector(H_1, H_2) → u_H
  60. Vector(H_1, H_3) → v_H
  61. Point(vert_I) → I
  62. Circle((x(I), g(x(I))), 1) → circle_I
  63. Intersect(circle_I, line_I, 1) → I_2
  64. Intersect(circle_I, line_I, 2) → I_3
  65. Intersect(g, line_I, (-6.283185307195264, -1.5677880535186782E-11)) → I_1
  66. Vector(I_1, I_2) → u_I
  67. Vector(I_1, I_3) → v_I

AI Drawing Prompt Reference

Draw the graph of the sine function y=sin(x), select multiple random points on the curve, draw circles with radius 1 centered at each point, intersect the circles with tangent lines to the sine curve at two points, and connect the tangent points to the intersection points to form vectors, demonstrating the geometric meaning that the derivative of the sine function is the cosine function with multiple points' vector constructions.
The prompt above was compiled from the original drawing commands and is for reference only.