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Coordinates of Special Angle Values on a Unit Circle

Coordinates of Special Angle Values on a Unit Circle
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2022-01-02 21:35
CircleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the coordinate relationships of special angles (0°, 30°, 45°, 60°, 90°, etc.) on a unit circle, involving core concepts like the unit circle, trigonometric functions (sine, cosine), and coordinate geometry. It uses GeoGebra commands such as Circle, Intersect, Rotate, Mirror, Polygon, Segment, Angle, Midpoint, and Vector. Suitable for high school students to understand the geometric meaning of trigonometric functions and for introductory calculus education.

GeoGebra Commands & Parameters

  1. Circle((0, 0), L) → c
  2. Intersect(xAxis, yAxis) → A
  3. Intersect(c, xAxis, 2) → B
  4. Rotate(L, π / 6, A) → C
  5. Rotate(L, π / 4, A) → D
  6. Rotate(L, π / 3, A) → E
  7. Mirror(C, yAxis) → C'
  8. Mirror(D, yAxis) → D'
  9. Mirror(E, yAxis) → E'
  10. Mirror(E', xAxis) → E''
  11. Mirror(D', xAxis) → D''
  12. Mirror(C', xAxis) → C''
  13. Mirror(C, xAxis) → C'_{1}
  14. Mirror(D, xAxis) → D'_{1}
  15. Mirror(E, xAxis) → E'_{1}
  16. Intersect(c, xAxis, 1) → G
  17. Intersect(c, yAxis, 1) → H
  18. If(a ≟ 1, B, a ≟ 6, E', a ≟ 7, D', a ≟ 8, C', a ≟ 9, G, a ≟ 10, C'', a ≟ 2, a ≟ 11, D'', a ≟ 12, E'', a ≟ 13, H, a ≟ 14, E'_{1}, a ≟ 15, D'_{1}, C, a ≟ 16, C'_{1}, a ≟ 3, D, a ≟ 4, E, a ≟ 5, F) → P
  19. Intersect(eq1, xAxis) → I
  20. Polygon(A, I, P) → t1, p, a_1, i
  21. Segment(A, P) → f
  22. Angle(P, I, A) → α
  23. Midpoint(A, P) → J
  24. Segment(A, P) → j
  25. Segment(A, I) → k
  26. Segment(I, P) → l
  27. Angle(I, A, P) → β
  28. Vector(A, I) → u
  29. SurdText(x(I)) → text3
  30. SurdText(y(P)) → text4
  31. Vector(I, P) → v
  32. Angle(B, A, P) → γ
  33. Textfield(q) → InputBox1
  34. Textfield(r) → InputBox2
  35. Textfield(b) → InputBox3
  36. Textfield(d) → InputBox4
  37. Textfield(e) → InputBox5
  38. Textfield(g) → InputBox6
  39. Textfield(h) → InputBox7
  40. Textfield(m) → InputBox8
  41. Textfield(n) → InputBox9
  42. Textfield(o) → InputBox10
  43. Textfield(s) → InputBox11
  44. Intersect(j, xAxis) → K
  45. Point(yAxis) → M
  46. CircleArc((0, 0), (2, 0), (0, 2)) → t
  47. If(c1, "√", "x") → text1
  48. If(c2, "√", "x") → text2
  49. If(c3, "√", "x") → text5
  50. If(c4, "√", "x") → text6
  51. If(c5, "√", "x") → text7
  52. Point(t) → N
  53. Segment((0, 0), N) → f_1
  54. Segment((0, 0), (2, 0)) → g_1
  55. Segment((0, 0), (0, 2)) → h_1

AI Drawing Prompt Reference

Draw a coordinate plane with the x-axis and y-axis, then construct a circle centered at the origin with radius L (unit-circle style). Mark the intersection of the axes to serve as a reference point, and place points on the circle where it meets the x-axis. Using the same center, generate points by rotating a radius position by π/6, π/4, and π/3 to obtain corresponding special-angle points in the first quadrant. Mirror these points across the y-axis to get their symmetric images, then mirror across the x-axis to obtain the remaining symmetric points. Connect the origin to the key endpoints to form a polygon and several segments, and include angle markings at the formed angle locations. Add coordinate-based square-root text labels near the relevant points.
The prompt above was compiled from the original drawing commands and is for reference only.