正弦和余弦的和公式证明：几何作图与三角函数

GeoGebra 命令及参数

1. Intersect(xAxis, yAxis) → O
2. Point(xAxis) → T
3. OrthogonalLine(T, xAxis) → a
4. Rotate(T, α, O) → B'
5. Angle(T, O, B') → γ
6. Line(O, B') → b
7. Intersect(b, a) → S
8. OrthogonalLine(S, b) → c
9. Rotate(S, β, O) → C'
10. Angle(S, O, C') → δ
11. Line(O, C') → d
12. Intersect(d, c) → P
13. OrthogonalLine(P, xAxis) → e
14. OrthogonalLine(S, e) → f
15. Intersect(e, f) → R
16. Segment(O, T) → g
17. Segment(T, S) → h
18. Segment(S, O) → i
19. Segment(S, P) → j
20. Segment(P, O) → k
21. Segment(S, R) → l
22. Intersect(g, xAxis) → F
23. Intersect(e, g) → Q
24. Segment(P, Q) → m
25. Angle(P, Q, O) → ε
26. Angle(S, T, Q) → ζ
27. Angle(P, S, O) → η
28. Angle(S, R, P) → θ
29. Angle(T, O, S) → ι
30. Angle(S, O, P) → κ
31. Polygon(O, T, S) → poly1, s, o, t
32. Polygon(O, S, P) → poly2, p, o_1, s_1
33. Polygon(O, Q, P) → poly3, p_1, o_2, q
34. Polygon(S, R, P) → poly4, p_2, s_2, r
35. Angle(R, P, S) → λ
36. Angle(R, S, O) → μ
37. Angle(P, S, R) → ν
38. RotateText("\cos(\beta)", α) → text3
39. RotateText("\sin(\alpha) \cos(\beta)", -90*°) → text5
40. RotateText("\cos(\alpha) \sin(\beta)", -90*°) → text9
41. RotateText("\sin(\alpha+\beta)", -90*°) → text7
42. RotateText("1", α + β) → text1
43. RotateText("\sin(\beta)", α - 90°) → text2
44. Segment(O, Q) → a_1
45. Point(a_1) → B
46. Segment(B, T) → c_1
47. Segment(P, Q) → d_1
48. Segment(P, R) → e_1
49. Segment(S, T) → b_1
50. Segment(R, S) → f_1