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Dynamic Illustration of 45-Degree Trig Values

Dynamic Illustration of 45-Degree Trig Values
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2016-11-27 11:10
CircleAngleFunctionTrigonometryGeometric Construction
In collections Special Right Triangles Special Right Triangles
This example dynamically illustrates the evaluation of trigonometric functions at 45 degrees, showing cosine, sine, and other values through geometric constructions. It involves core concepts like trigonometric definitions, unit circle, and special angle values. GeoGebra commands used include Center, Rotate, Angle, Intersect, Polygon, CircleArc, Point, Segment, Textfield, and Dilate. Suitable for high school mathematics education to understand the geometric relationship between trigonometric functions and special angles.

GeoGebra Commands & Parameters

  1. Center(c) → A
  2. Rotate(B, π / 4, A) → B'_2
  3. Angle(B, A, B'_2) → γ
  4. Intersect(g, i) → D
  5. Polygon(A, D, B'_2) → poly2, b'_2, a_2, d
  6. CircleArc(A, B'_2, B) → p
  7. Point(p) → I
  8. Segment(A, I) → s
  9. Textfield(r) → InputBox1
  10. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], I) → A'_2
  11. Segment(A, A'_2) → t
  12. Rotate(A, If[3.5 < a ≤ 4.5, ((a - 3.5) * π / 3), π / 3], A) → A'_3
  13. CircleArc(A, B, B'_2) → q
  14. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B'_2) → A'
  15. Segment(A, A') → f
  16. Dilate(A, If[1 < a ≤ 2, 2 - a, 0], D) → A'_1
  17. Segment(A, A'_1) → h
  18. Dilate(D, If[2 < a ≤ 3, 3 - a, 0], B'_2) → D'
  19. Segment(D, D') → j
  20. Rotate(A, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → A'_4
  21. Rotate(A'_1, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → C
  22. Segment(A'_4, C) → h'
  23. Angle(B'_2, D, A) → α

AI Drawing Prompt Reference

Create a dynamic illustration showing trigonometric values at 45 degrees, involving a unit circle and geometric constructions. Start by drawing a circle and marking its center, then rotate a point by 45 degrees to form an angle. Find intersection points on the circle, construct a polygon and arc. Draw line segments from the center to points on the circle, adjusting dynamically. Rotate and scale points to create additional segments. Finally, label angles and trigonometric values to demonstrate the geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.