Geometric Investigation of Trig Identities via Rotation and Reflection

GeoGebra Commands & Parameters

1. Point(Ray[(0, 0), (1, 0)]) → B
2. Rotate(B, π / 4, A) → B'
3. Circle(A, B) → c
4. Point(CircleArc[A, B, B']) → D
5. Rotate(D, Angle[B, A, D], A) → D'
6. Segment(A, D') → f
7. OrthogonalLine(D', xAxis) → g
8. Intersect(g, xAxis) → E
9. Intersect(h, yAxis) → F
10. Intersect(g, h) → G
11. Mirror(D, yAxis) → D'_{1}
12. OrthogonalLine(D'_{1}, xAxis) → i
13. Segment(A, D'_{1}) → j
14. Intersect(i, xAxis) → H
15. Polygon(A, H, 4) → poly1, k, l, m, n, I, J
16. Polygon(H, D'_{1}, 4) → poly2, p, q, r, s, K, L
17. Semicircle(M, B) → d
18. Polygon(A, E, G, F) → q1, a_1, e, g_1, f_1
19. Line(D'_{1}, K) → t
20. Circle(H, Distance[H, I]) → c_1
21. Intersect(c_1, t, 2) → N
22. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D) → A'
23. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], D') → A'_{1}
24. Segment(A, A'_{1}) → h_1
25. Segment(A, A') → i_1
26. Circle(A, 0.4) → d_1
27. Intersect(d_1, i_1, 1) → O
28. Angle(B, A, D) → α
29. Angle(B, A, O) → β
30. Angle(D, A, D') → γ
31. Intersect(d_1, xAxis, 2) → P
32. Rotate(P, If[2 < a ≤ 3, ((a - 2) * α), α], A) → P'
33. CircleSector(A, P, P') → e_1
34. Rotate(e_1, If[4 < a ≤ 5, ((a - 4) * α), α], A) → k_1
35. Angle(D'_{1}, A, M) → δ
36. Rotate(k_1, If[6 < a ≤ 7, ((a - 6) * (π - (2 * α))), π - (2 * α)], A) → p_1
37. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], D'_{1}) → A'_{2}
38. Segment(A, A'_{2}) → j_1
39. Dilate(D'_{1}, If[9 < a ≤ 10, 10 - a, 0], H) → Q
40. Dilate(D', If[9 < a ≤ 10, 10 - a, 0], E) → D''
41. Segment(D', D'') → l_1
42. Segment(D'_{1}, Q) → m_1
43. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], E) → A'_{3}
44. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], H) → A'_{4}
45. Segment(A, A'_{3}) → n_1
46. Segment(A, A'_{4}) → q_1
47. RotateText(text1, π / 2) → text2
48. Rotate(H, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], D'_{1}) → H'
49. Segment(D'_{1}, H') → r_1
50. Dilate(D'_{1}, If[14 < a ≤ 15, 15 - a, 0], H) → R
51. Vector(D'_{1}, R) → u
52. Translate(D'_{1}, u) → S
53. Translate(H', u) → H''
54. Segment(S, H'') → s_1
55. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], L) → H'_{1}
56. Vector(H, H'_{1}) → v
57. Translate(D'_{1}, v) → T
58. Translate(Q, v) → Q'
59. Segment(T, Q') → t_1
60. Polygon(H, D'_{1}, H', L) → q2, h_2, d'_{1}, h', l_2
61. Rotate(A, If[17 < a ≤ 18, ((-(a - 17)) * π / 2), ((-π)) / 2], H) → A'_{5}
62. Segment(H, A'_{5}) → b_1
63. Dilate(H, If[18 < a ≤ 19, 19 - a, 0], I) → H'_{2}
64. Vector(H, H'_{2}) → w
65. Translate(A, w) → A'_{6}
66. Translate(A'_{4}, w) → U
67. Segment(A'_{6}, U) → f_2
68. Dilate(I, If[19 < a ≤ 20, 20 - a, 0], J) → I'
69. Vector(I, I') → u_1
70. Translate(H, u_1) → H'_{3}
71. Translate(A'_{5}, u_1) → V
72. Segment(H'_{3}, V) → g_2
73. Angle(F, A, D') → ε
74. Rotate(D', If[20 < a ≤ 21, ((-(a - 20)) * ε), (-ε)], A) → D''_{1}
75. Segment(A, D''_{1}) → i_2
76. Dilate(A, If[21 < a ≤ 22, 22 - a, 0], F) → A'_{7}
77. Vector(A, A'_{7}) → v_1
78. Translate(A, v_1) → A'_{8}
79. Translate(A'_{3}, v_1) → W
80. Segment(A'_{8}, W) → j_2
81. Dilate(F, If[22 < a ≤ 23, 23 - a, 0], G) → F'
82. Vector(F, F') → w_1
83. Translate(A, w_1) → A'_{9}
84. Translate(D''_{1}, w_1) → Z
85. Segment(A'_{9}, Z) → k_2
86. Polygon(A, E, G, F) → q3, a_2, e_2, g_3, f_3
87. Dilate(D'_{1}, If[26 < a ≤ 27, 27 - a, 0], N) → A_1
88. Vector(D'_{1}, A_1) → u_2
89. Translate(H', u_2) → H''_{1}
90. Polygon(H, A_1, H''_{1}, L) → q4, h_3, m_2, h''_{1}, l_3
91. Line(H''_{1}, L) → n_2
92. OrthogonalLine(H, n_2) → p_2
93. Intersect(n_2, p_2) → B_1
94. Dilate(L, If[28 < a ≤ 29, 29 - a, 0], B_1) → L'
95. Vector(L, L') → v_2
96. Translate(H''_{1}, v_2) → C_1
97. Polygon(N, C_1, L', H) → q5, n_3, q_2, l', h_4
98. Dilate(H, If[30 < a ≤ 31, 31 - a, 0], A) → H'_{4}
99. Vector(H, H'_{4}) → w_2
100. Angle(B_1, H, I) → ζ
101. Translate(N, w_2) → N'
102. Translate(C_1, w_2) → D_1
103. Translate(L', w_2) → L''
104. Translate(H, w_2) → H'_{5}
105. Polygon(N', D_1, L'', H'_{5}) → q5', n', r_2, l'', h'_{5}
106. Rotate(N', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → N''
107. Rotate(D_1, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → E_1
108. Rotate(L'', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → L'''
109. Rotate(H'_{5}, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → F_1
110. Polygon(N'', E_1, L''', F_1) → q5'', n'', s_2, l''', t_2
111. Line(A, E) → b_2
112. Circle(A, Distance[I, J]) → c_2
113. Line(D', E) → d_2
114. Intersect(c_2, d_2) → I_1, J_1
115. Dilate(E, If[32 < a ≤ 33, 33 - a, 0], J_1) → E'
116. Vector(E, E') → u_3
117. Translate(G, u_3) → G'
118. Polygon(A, E', G', F) → q6, a_3, e', g', f_4
119. Line(A, J_1) → i_3
120. OrthogonalLine(G', i_3) → j_3
121. Intersect(i_3, j_3) → G_1
122. Dilate(J_1, If[34 < a ≤ 35, 35 - a, 0], G_1) → H_1
123. Vector(J_1, H_1) → v_3
124. Translate(A, v_3) → A'_{10}
125. Polygon(F, G', H_1, A'_{10}) → q7, f_5, k_3, m_3, a'_{10}
126. Dilate(G_1, If[36 < a ≤ 37, 37 - a, 0], J) → K_1
127. Line(G', f_2) → p_3
128. Point(p_3) → N_1
129. Angle(F, G', N_1) → η
130. Dilate(G', If[36 < a ≤ 37, 37 - a, 0], J) → G''
131. Vector(G', G'') → w_3
132. Translate(F, w_3) → F'_{1}
133. Translate(G', w_3) → G''_{1}
134. Translate(H_1, w_3) → L_1
135. Translate(A'_{10}, w_3) → M_1
136. Polygon(F'_{1}, G''_{1}, L_1, M_1) → q7', f'_{1}, g''_{1}, q_3, r_3
137. Rotate(G''_{1}, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → P_1
138. Rotate(L_1, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → Q_1
139. Rotate(M_1, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → R_1
140. Rotate(F'_{1}, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → O_1
141. Rotate(G''_{1}, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → S_1
142. Rotate(L_1, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → T_1
143. Rotate(M_1, If[36 < a ≤ 37, ((a - 36) * η), η], G''_{1}) → U_1
144. Polygon(O_1, S_1, T_1, U_1) → q7'', o_1, s_3, t_3, b_3
145. Line(D'_{1}, H') → c_3
146. Line(H''_{1}, B_1) → d_3
147. Line(H, d_3) → e_3
148. Line(A, F) → g_4
149. Line(F, G') → i_4
150. Line(J_1, G_1) → j_4
151. Segment(A, B) → k_4
152. Segment(A, M) → l_4