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Geometric Proof of the Cosine Subtraction Formula

Geometric Proof of the Cosine Subtraction Formula
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2014-04-04 23:53
CircleAngleFunctionTrigonometryGeometric Construction
This GeoGebra applet demonstrates and rigorously proves the cosine subtraction formula cos(u−v)=cos u·cos v−sin u·sin v through geometric construction. The construction uses GeoGebra commands including Intersect, Circle, Point, Segment, OrthogonalLine, Angle, CircleArc, and Midpoint to build a visual proof on the unit circle. By representing two angles u and v geometrically and analyzing the resulting line segments and areas, the applet reveals why the formula holds. Ideal for high school trigonometry courses, student self-study, and interactive classroom demonstrations to help learners understand the geometric origin of trigonometric identities.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, B) → c
  3. Point(c) → C
  4. Segment(A, C) → a
  5. Point(c) → D
  6. Segment(A, D) → b
  7. Segment(A, B) → d
  8. OrthogonalLine(C, d) → e
  9. Intersect(e, d) → E
  10. Angle(B, A, C) → v
  11. Angle(B, A, D) → u
  12. Angle(C, A, D) → w
  13. OrthogonalLine(D, xAxis) → g
  14. OrthogonalLine(D, a) → h
  15. Intersect(g, xAxis) → G
  16. Intersect(g, d) → I
  17. Segment(A, G) → i
  18. OrthogonalLine(G, a) → j
  19. Intersect(a, j) → K
  20. Segment(A, K) → k
  21. Segment(G, D) → l
  22. OrthogonalLine(G, h) → m
  23. Intersect(h, m) → L
  24. Segment(G, L) → n
  25. Intersect(a, h) → F
  26. Segment(A, F) → p
  27. Angle(G, D, L) → α
  28. Segment(G, K) → q
  29. Segment(L, F) → r
  30. Angle(D, G, A) → β
  31. Angle(D, L, G) → γ
  32. Angle(A, K, G) → δ
  33. CircleArc(A, B, C) → f
  34. Segment(L, D) → s
  35. CircleArc(A, C, D) → t
  36. CircleArc(A, B, D) → c_1
  37. Midpoint(D, G) → H
  38. Midpoint(A, G) → J