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Geometric Investigation of Trig Identity

Geometric Investigation of Trig Identity
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2021-10-26 09:53
VectorTrigonometryGeometric Construction
This example demonstrates trigonometric identities through dynamic geometry, revealing the intrinsic connection between cosine, sine, and geometric figures. Key concepts include trigonometric functions, geometric transformations, and vector operations. GeoGebra commands like Point, Rotate, Circle, Segment, OrthogonalLine, Intersect, Mirror, Polygon, Semicircle, Line, Dilate, Angle, CircleSector, RotateText, Vector, and Translate are used to construct the example. It is suitable for high school mathematics education, enhancing understanding of trigonometric identities and geometric intuition.

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. Dilate(D'_{1}, If[14 < a ≤ 15, 15 - a, 0], H) → R
  50. Vector(D'_{1}, R) → u
  51. Translate(D'_{1}, u) → S
  52. Translate(H', u) → H''
  53. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], L) → H'_{1}
  54. Vector(H, H'_{1}) → v
  55. Translate(D'_{1}, v) → T
  56. Translate(Q, v) → Q'
  57. Rotate(A, If[17 < a ≤ 18, ((-(a - 17)) * π / 2), ((-π)) / 2], H) → A'_{5}
  58. Segment(H, A'_{5}) → b_1
  59. Dilate(H, If[18 < a ≤ 19, 19 - a, 0], I) → H'_{2}
  60. Vector(H, H'_{2}) → w
  61. Translate(A, w) → A'_{6}
  62. Translate(A'_{4}, w) → U
  63. Segment(A'_{6}, U) → f_2
  64. Dilate(I, If[19 < a ≤ 20, 20 - a, 0], J) → I'
  65. Vector(I, I') → u_1
  66. Translate(H, u_1) → H'_{3}
  67. Translate(A'_{5}, u_1) → V
  68. Segment(H'_{3}, V) → g_2
  69. Angle(F, A, D') → ε
  70. Rotate(D', If[20 < a ≤ 21, ((-(a - 20)) * ε), (-ε)], A) → D''_{1}
  71. Segment(A, D''_{1}) → i_2
  72. Dilate(A, If[21 < a ≤ 22, 22 - a, 0], F) → A'_{7}
  73. Vector(A, A'_{7}) → v_1
  74. Translate(A, v_1) → A'_{8}
  75. Translate(A'_{3}, v_1) → W
  76. Segment(A'_{8}, W) → j_2
  77. Dilate(F, If[22 < a ≤ 23, 23 - a, 0], G) → F'
  78. Vector(F, F') → w_1
  79. Translate(A, w_1) → A'_{9}
  80. Translate(D''_{1}, w_1) → Z
  81. Segment(A'_{9}, Z) → k_2
  82. Polygon(A, E, G, F) → q3, a_2, e_2, g_3, f_3
  83. Dilate(D'_{1}, If[26 < a ≤ 27, 27 - a, 0], N) → A_1
  84. Vector(D'_{1}, A_1) → u_2
  85. Translate(H', u_2) → H''_{1}
  86. Line(H''_{1}, L) → n_2
  87. OrthogonalLine(H, n_2) → p_2
  88. Intersect(n_2, p_2) → B_1
  89. Dilate(L, If[28 < a ≤ 29, 29 - a, 0], B_1) → L'
  90. Vector(L, L') → v_2
  91. Translate(H''_{1}, v_2) → C_1
  92. Dilate(H, If[30 < a ≤ 31, 31 - a, 0], A) → H'_{4}
  93. Vector(H, H'_{4}) → w_2
  94. Angle(B_1, H, I) → ζ
  95. Translate(N, w_2) → N'
  96. Translate(C_1, w_2) → D_1
  97. Translate(L', w_2) → L''
  98. Translate(H, w_2) → H'_{5}
  99. Polygon(N', D_1, L'', H'_{5}) → q5', n', r_2, l'', h'_{5}
  100. Rotate(N', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → N''
  101. Rotate(D_1, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → E_1
  102. Rotate(L'', If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → L'''
  103. Rotate(H'_{5}, If[30 < a ≤ 31, ((a - 30) * ζ), ζ], H'_{5}) → F_1
  104. Line(A, E) → b_2
  105. Circle(A, Distance[I, J]) → c_2
  106. Segment(A, B) → k_4
  107. Segment(A, M) → l_4
  108. Dilate(V_1, If[9 < a ≤ 10, 10 - a, 0], A) → W_1
  109. Rotate(A, If[11 < a ≤ 12, ((a - 11) * (π / 2 - (2 * α))), π / 2 - (2 * α)], A) → A'_{11}
  110. Rotate(A'_{1}, If[11 < a ≤ 12, ((a - 11) * (π / 2 - (2 * α))), π / 2 - (2 * α)], A) → A_2
  111. Segment(A'_{11}, A_2) → n_4
  112. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], Z_1) → A'_{12}
  113. Vector(A, A'_{12}) → u_4
  114. Translate(A'_{11}, u_4) → B_2
  115. Translate(A_2, u_4) → C_2
  116. Segment(B_2, C_2) → m_4
  117. Rotate(B_2, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], C_2) → D_2
  118. Rotate(C_2, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], C_2) → E_2
  119. Segment(D_2, E_2) → p_4
  120. Dilate(C_2, If[14 < a ≤ 15, 15 - a, 0], Z_1) → F_2
  121. Vector(C_2, F_2) → v_4
  122. Translate(D_2, v_4) → G_2
  123. Translate(E_2, v_4) → H_2
  124. Segment(G_2, H_2) → q_4
  125. Dilate(C_2, If[15 < a ≤ 16, 16 - a, 0], D_2) → I_2
  126. Vector(C_2, I_2) → w_4
  127. Translate(B_2, w_4) → J_2
  128. Translate(C_2, w_4) → K_2
  129. Segment(J_2, K_2) → r_4
  130. Polygon(G_2, Z_1, C_2, D_2) → q8, s_4, z_1, t_4, a_4
  131. Rotate(U, If[23 < a ≤ 24, ((a - 23) * π), π], A'_{5}) → U'
  132. Rotate(H, If[23 < a ≤ 24, ((a - 23) * π / 2), π / 2], A'_{5}) → H'_{6}
  133. Rotate(A'_{5}, If[23 < a ≤ 24, ((a - 23) * π / 2), π / 2], A'_{5}) → M_2
  134. Segment(H'_{6}, M_2) → s_1
  135. Dilate(A'_{5}, If[24 < a ≤ 25, 25 - a, 0], H'_{6}) → L_2
  136. Vector(A'_{5}, L_2) → r_1
  137. Translate(H, r_1) → H'_{7}
  138. Translate(A'_{5}, r_1) → N_2
  139. Segment(H'_{7}, N_2) → t_1
  140. RotateText(text3, π / 2) → text5
  141. Circle(A, (2 * f_2)) → h_2
  142. Line(A'_{8}, W) → l_2
  143. Intersect(h_2, l_2, 1) → O_2
  144. Dilate(A'_{8}, If[28 < a ≤ 29, 29 - a, 0], O_2) → P_2
  145. Vector(A'_{8}, P_2) → m_2
  146. Translate(W, m_2) → W'
  147. Polygon(A, A'_{3}, W', P_2) → q2, a_5, a'_{3}, w', q_2
  148. Line(O_2, a'_{3}) → h_3
  149. OrthogonalLine(A'_{3}, h_3) → l_3
  150. Intersect(h_3, l_3) → Q_2
  151. Dilate(A, If[30 < a ≤ 31, 31 - a, 0], Q_2) → A'_{13}
  152. Vector(A, A'_{13}) → s_2
  153. Translate(O_2, s_2) → R_2
  154. Polygon(W', R_2, A'_{13}, A'_{3}) → q4, t_2, n_3, a'_{13}, c_3
  155. Line(W', A'_{3}) → d_3
  156. Dilate(W', If[32 < a ≤ 33, 33 - a, 0], N_2) → W''
  157. Vector(W', W'') → e_3
  158. Translate(W', e_3) → W''_{1}
  159. Translate(R_2, e_3) → S_2
  160. Translate(A'_{13}, e_3) → T_2
  161. Translate(A'_{3}, e_3) → U_2
  162. Polygon(W''_{1}, S_2, T_2, U_2) → q4', w''_{1}, h_4, b_4, c_4
  163. Line(W', f_2) → d_4
  164. Point(d_4) → V_2
  165. Angle(V_2, W', A'_{3}) → θ
  166. Rotate(W''_{1}, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → W_2
  167. Rotate(S_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → Z_2
  168. Rotate(T_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → A_3
  169. Rotate(U_2, If[32 < a ≤ 33, ((-(a - 32)) * θ), (-θ)], W'') → B_3
  170. Polygon(W_2, Z_2, A_3, B_3) → q4'', e_4, z_2, g_5, h_5
  171. Circle(Z_1, (2 * f_2)) → d_2
  172. Line(C_2, p_4) → i_3
  173. Intersect(d_2, i_3) → G_1, M_1
  174. Line(G_2, i_3) → j_3
  175. Dilate(C_2, If[34 < a ≤ 35, 35 - a, 0], M_1) → H_1
  176. Vector(C_2, H_1) → u_3
  177. Translate(D_2, u_3) → I_1
  178. Polygon(I_1, H_1, Z_1, G_2) → q5, k_3, m_3, p_3, q_3
  179. Line(M_1, Z_1) → r_3
  180. Line(I_1, G_2) → s_3
  181. OrthogonalLine(G_2, s_3) → t_3
  182. Intersect(t_3, r_3) → J_1
  183. Dilate(Z_1, If[36 < a ≤ 37, 37 - a, 0], J_1) → K_1
  184. Vector(Z_1, K_1) → v_3
  185. Translate(M_1, v_3) → L_1
  186. Polygon(I_1, L_1, K_1, G_2) → q6, a_3, b_3, f_4, g_4
  187. Dilate(J_1, If[38 < a ≤ 39, 39 - a, 0], Z_2) → O_1
  188. Vector(J_1, O_1) → w_3
  189. Translate(I_1, w_3) → P_1
  190. Translate(L_1, w_3) → Q_1
  191. Translate(K_1, w_3) → R_1
  192. Translate(G_2, w_3) → S_1
  193. Polygon(P_1, Q_1, R_1, S_1) → q6', i_4, j_4, f_5, i_5
  194. Angle(A'_{3}, Z_1, M_1) → η
  195. Rotate(P_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → T_1
  196. Rotate(Q_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → U_1
  197. Rotate(R_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], O_2) → C_3
  198. Line(Z_2, s_1) → j_5
  199. Point(j_5) → D_3
  200. Rotate(P_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → E_3
  201. Rotate(Q_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → F_3
  202. Rotate(R_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → G_3
  203. Rotate(S_1, If[38 < a ≤ 39, ((-(a - 38)) * η), (-η)], R_1) → H_3
  204. Polygon(E_3, F_3, G_3, H_3) → q6'', k_5, l_5, m_5, n_5

AI Drawing Prompt Reference

On a coordinate plane, start at the origin and mark a point on the positive x-axis. Rotate this point by 45 degrees around the origin to get a second point, and draw a circular arc centered at the origin passing through both points. Pick an arbitrary point on this arc, rotate it around the origin by the same angle as the first point, and connect the origin to this rotated point with a segment. From this rotated point, drop a perpendicular to the x-axis and mark its intersection with the axis. Mirror the arc point across the y-axis, drop another perpendicular to the x-axis from the mirrored point, and mark that intersection. Construct squares using the origin, these axis intersections, and the mirrored points. Draw a small circle centered at the origin, place a point on it, rotate it to form a sector, and add several auxiliary segments and angles to illustrate the trigonometric relationships. Finally, rotate a text label by 90 degrees.
The prompt above was compiled from the original drawing commands and is for reference only.