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Dynamic Proof of sin(A+B) and cos(A+B)

Dynamic Proof of sin(A+B) and cos(A+B)
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2013-06-06 18:00
AngleFunctionTrigonometryGeometric Construction
In collections Trig
This example dynamically demonstrates the derivations of the sine and cosine angle addition formulas, sin(A+B) and cos(A+B). It involves core mathematical concepts such as trigonometric functions, geometric constructions, and the relationships between angles and side lengths. The construction utilizes GeoGebra commands/objects like Line, Segment, Midpoint, Circle, Point, OrthogonalLine, Intersect, and Angle. It is suitable for high school students to learn and understand the geometric meaning of trigonometric formulas, as well as for teachers to use in classroom demonstrations.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(C, a) → b
  3. Segment(A, C) → c
  4. Midpoint(A, C) → D
  5. Circle(D, C) → d
  6. Point(d) → E
  7. Segment(A, E) → e
  8. Segment(E, C) → f
  9. OrthogonalLine(E, b) → g
  10. Intersect(b, g) → F
  11. Intersect(a, g) → G
  12. Segment(F, E) → h
  13. Segment(E, G) → i
  14. OrthogonalLine(C, b) → j
  15. Intersect(a, j) → H
  16. Segment(C, H) → k
  17. OrthogonalLine(A, a) → l
  18. Point(l) → I
  19. Line(I, a) → m
  20. Intersect(m, g) → K
  21. Segment(I, K) → n
  22. Line(E, b) → p
  23. Intersect(p, k) → J
  24. Segment(J, E) → q
  25. Midpoint(A, E) → L
  26. Midpoint(A, H) → M
  27. Midpoint(J, E) → N
  28. Midpoint(E, C) → O
  29. Midpoint(F, E) → P
  30. Midpoint(E, G) → Q
  31. Midpoint(I, K) → R
  32. OrthogonalLine(B, a) → r
  33. Intersect(r, b) → S
  34. Segment(S, B) → s
  35. Midpoint(S, B) → T
  36. Angle(E, A, B) → α
  37. Angle(E, A, C) → β
  38. Angle(C, E, A) → γ
  39. Angle(C, J, E) → δ
  40. Angle(E, G, A) → ε

AI Drawing Prompt Reference

Draw a dynamic proof of the sine and cosine angle addition formulas. Construct triangle ABC, find midpoint D of AC, draw a circle centered at D with radius DC, and mark the intersection point E. Connect AE and EC. Draw perpendiculars to AB and AC, marking points F, G, H, and J. Draw a perpendicular to AB, marking points I and K, and connect IK. Find midpoints of segments and angles, labeling α, β, γ, δ, ε to show the geometric relationships with trigonometric functions.
The prompt above was compiled from the original drawing commands and is for reference only.