Geometric Proof of Sine and Cosine of Angle Sum

GeoGebra Commands & Parameters

1. Segment(A, B) → a
2. Point(a) → J
3. Point(a) → D
4. OrthogonalLine(A, a) → b
5. Point(b) → C
6. CircleArc(A, D, C) → d
7. Point(d) → F
8. Line(A, F) → p
9. Point(p) → K
10. CircleArc(A, J, K) → q
11. Segment(A, C) → c
12. Point(q) → E_1
13. Segment(E_1, A) → e
14. Segment(A, F) → f
15. OrthogonalLine(F, e) → g
16. OrthogonalLine(F, a) → h
17. Intersect(g, e) → G
18. OrthogonalLine(G, a) → i
19. Intersect(h, a) → H
20. Segment(F, H) → j
21. Segment(F, G) → k
22. Intersect(i, a) → I
23. Segment(G, I) → l
24. Angle(B, A, E_1) → α
25. Angle(E_1, A, F) → β
26. Angle(F, G, A) → γ
27. Segment(A, G) → m
28. Segment(A, H) → n
29. Segment(A, I) → r
30. Angle(H, F, G) → δ
31. Line(G, a) → t
32. Intersect(t, j) → E
33. Segment(E, G) → a_1
34. Angle(F, H, A) → ε
35. Angle(G, E, F) → ζ
36. Angle(G, I, A) → η
37. Segment(E, F) → b_1
38. Intersect(c, a) → M
39. Segment(G, I) → c_1
40. Segment(E, G) → d_1
41. Angle(I, M, F) → θ
42. Segment(M, I) → e_1
43. Segment(F, H) → f_1
44. Segment(M, H) → g_1
45. Intersect(f_1, e) → L
46. Polygon(M, H, L) → polígono1, l_1, m_1, h_1
47. Polygon(F, L, G) → polígono2, i_1, j_1, k_1
48. Angle(M, L, H) → ι
49. Angle(G, L, F) → κ
50. Segment(F, E) → n_1
51. Segment(E, G) → p_1
52. Segment(F, H) → q_1
53. Segment(G, I) → r_1
54. Segment(F, G) → s_1
55. Segment(M, G) → t_1
56. Segment(G, I) → a_2
57. Segment(M, I) → b_2
58. Segment(F, H) → c_2
59. Segment(M, H) → d_2