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New Trig Identities from Similar Right Triangles

New Trig Identities from Similar Right Triangles
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2020-03-12 04:03
TriangleTrigonometryGeometric Construction
This example uses similar right triangles to derive new trigonometric identities and includes key questions. It demonstrates the properties of similar triangles and the concept of trigonometric functions. Using GeoGebra commands such as Intersect, Circle, Rotate, etc., it dynamically explores the geometric meaning of trigonometric identities. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Intersect(c, b) → E, F
  4. Rotate(F, (o * π / 180), A) → B
  5. Ray(A, B) → a
  6. OrthogonalLine(B, b) → d
  7. Intersect(b, d) → C
  8. Intersect(b, xAxis) → D
  9. Line(F, d) → e
  10. Angle(b, a) → θ
  11. Segment(A, B) → f
  12. Angle(B, C, A) → α
  13. Segment(A, C) → g
  14. Segment(C, B) → h
  15. Intersect(e, a) → G
  16. Segment(F, G) → i
  17. Angle(G, F, A) → β
  18. Line(G, b) → j
  19. Segment(A, G) → l
  20. Point(c) → H
  21. Segment(A, H) → radius
  22. OrthogonalLine(I, e) → m
  23. Intersect(m, a) → J
  24. Segment(I, J) → n
  25. Angle(J, I, A) → γ
  26. Angle(I, J, A) → δ
  27. OrthogonalLine(J, a) → p
  28. Line(A, p) → q
  29. Point(q) → L
  30. OrthogonalLine(L, q) → s
  31. Intersect(p, s) → M
  32. Segment(L, M) → t
  33. Angle(J, M, L) → ε
  34. Angle(M, L, A) → ζ
  35. Angle(A, J, M) → η
  36. Segment(J, M) → r
  37. Segment(A, L) → f_1

AI Drawing Prompt Reference

Draw a right triangle ABC with A as the right angle. Draw a circle c with A as the center and radius 1. Let c intersect line b at points E and F. Rotate F around A by π/180 to get B. Draw line a through B, and let it intersect b at C. Let the x-axis intersect b at D. Draw line e through F, and let it intersect a at G. Connect AG. Draw line j through G, and let it intersect b at I. Connect AI. Draw line m through I, and let it intersect a at J. Connect IJ. Draw line p through J, and let it intersect a at L. Connect AL. Draw line s through L, and let it intersect p at M. Connect LM.
The prompt above was compiled from the original drawing commands and is for reference only.