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Unit Circle Diagram of Trigonometric Ratios

Unit Circle Diagram of Trigonometric Ratios
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2019-11-10 12:44
TriangleCircleTrigonometryCoordinate GeometryGeometric Construction
① This example demonstrates how the unit circle visually represents the trigonometric ratios (sine, cosine, tangent) through geometric segments and intersection points. ② Core mathematical concepts include the unit circle definition, trigonometric functions, right-triangle side ratios, the Pythagorean theorem, and coordinate symmetry. ③ Construction details: Using GeoGebra commands Point, Intersect, and Segment, along with multiple objects (eq1, eq2, f, g, etc.), the diagram builds the unit circle, axes, radius segments, and perpendicular lines, marking the lengths corresponding to each trigonometric function. ④ Suitable for high school or introductory college students learning trigonometry, and useful for teachers demonstrating the geometric meaning of sine, cosine, and tangent, as well as for exploring identities and symmetry on the unit circle.

GeoGebra Commands & Parameters

  1. Point(eq1) → B
  2. Intersect(xAxis, yAxis) → A
  3. Point(eq1) → C
  4. Segment(A, C) → h
  5. Point(eq1) → D
  6. Segment(A, D) → i
  7. Point(eq1) → E
  8. Segment(A, E) → j
  9. Point(eq1) → F
  10. Segment(A, F) → k
  11. Intersect(eq3, g) → G
  12. Segment(E, G) → l
  13. Point(eq2) → H
  14. Segment(F, H) → m
  15. Intersect(eq1, h, 1) → I
  16. Point(eq2) → J
  17. Segment(I, J) → n
  18. Point(f) → K
  19. Segment(D, K) → p
  20. Point(g) → L
  21. Segment(H, L) → q
  22. Point(f) → M
  23. Segment(J, M) → r
  24. Point(eq3) → N
  25. Segment(K, N) → s
  26. Point(xAxis) → O
  27. Segment(L, O) → t
  28. Point(xAxis) → P
  29. Segment(M, P) → a
  30. Point(xAxis) → Q
  31. Segment(K, Q) → b
  32. Intersect(eq1, eq3, 1) → R
  33. Segment(G, R) → c
  34. Intersect(eq1, eq2, 1) → S
  35. Segment(S, J) → d
  36. Segment(R, N) → e
  37. Segment(S, H) → f_1
  38. Segment(Q, A) → g_1
  39. Segment(A, O) → h_1
  40. Segment(Q, R) → i_1
  41. Segment(I, T) → j_1
  42. Point(g_1) → U
  43. Segment(D, U) → k_1
  44. Point(g_1) → V
  45. Segment(E, V) → l_1
  46. Point(h_1) → W
  47. Segment(F, W) → m_1
  48. Point(yAxis) → Z
  49. Segment(E, Z) → n_1
  50. Point(yAxis) → A_1
  51. Segment(F, A_1) → p_1
  52. Point(yAxis) → B_1
  53. Segment(I, B_1) → q_1
  54. Point(yAxis) → C_1
  55. Segment(D, C_1) → r_1

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with coordinate axes. Place several points on the circle and draw radius segments from the center to these points. From each point on the circle, drop perpendicular lines to the x-axis, marking the foot points. Also, draw a vertical tangent line outside the circle, and from the center draw segments to points on this tangent, marking intersections. Use distinct segments to represent sine, cosine, and tangent values, and mark intersections with the axes. The overall diagram shows the geometric relationships of trigonometric ratios on the unit circle, including radii, perpendiculars, and tangents.
The prompt above was compiled from the original drawing commands and is for reference only.