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Geometric Investigation of Trig Identity: Rotation & Area

Geometric Investigation of Trig Identity: Rotation & Area
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2021-09-02 08:57
CircleAngleTrigonometryTransformationGeometric Construction
This GeoGebra applet demonstrates geometric proofs of trigonometric identities through dynamic manipulation. It explores the relationship between sine/cosine functions and area/perimeter using geometric transformations. Construction highlights: Point, Segment, Polygon for basic shapes; Rotate, Vector for vector rotation; Circle, Semicircle, CircleSector for circular elements; Angle, OrthogonalLine for angle measurement; Mirror, Translate for transformations; Dilate for scaling. Covers trigonometric functions, area formulas, and vector operations. Ideal for high school math teachers demonstrating trig identities or students exploring geometric interpretations of trigonometric concepts.

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, 0.6) → 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. RotateText(text1, π / 2) → text2
  47. Rotate(H, If[13 < a ≤ 14, ((-(a - 13)) * π / 2), ((-π)) / 2], D'_{1}) → H'
  48. Segment(D'_{1}, H') → r_1
  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. Segment(S, H'') → s_1
  54. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], L) → H'_{1}
  55. Vector(H, H'_{1}) → v
  56. Translate(D'_{1}, v) → T
  57. Translate(Q, v) → Q'
  58. Segment(T, Q') → t_1
  59. Polygon(H, D'_{1}, H', L) → q2, h_2, d'_{1}, h', l_2
  60. Rotate(A, If[17 < a ≤ 18, ((-(a - 17)) * π / 2), ((-π)) / 2], H) → A'_{5}
  61. Dilate(H, If[18 < a ≤ 19, 19 - a, 0], I) → H'_{2}
  62. Vector(H, H'_{2}) → w
  63. Translate(A, w) → A'_{6}
  64. Translate(A'_{4}, w) → U
  65. Dilate(I, If[19 < a ≤ 20, 20 - a, 0], J) → I'
  66. Vector(I, I') → u_1
  67. Translate(H, u_1) → H'_{3}
  68. Translate(A'_{5}, u_1) → V
  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. Line(D'_{1}, H') → c_3
  107. Segment(A, B) → k_4
  108. Segment(A, M) → l_4
  109. Dilate(A, If[17 < a ≤ 18, 18 - a, 0], M) → A'_{11}
  110. Vector(A, A'_{11}) → u_4
  111. Translate(A, u_4) → A'_{12}
  112. Translate(A'_{1}, u_4) → V_1
  113. Segment(A'_{12}, V_1) → m_4
  114. Rotate(A'_{12}, If[17 < a ≤ 18, ((a - 17) * ((3 * π / 2) - γ)), (3 * π / 2) - γ], A'_{11}) → W_1
  115. Rotate(A'_{12}, If[17 < a ≤ 18, ((a - 17) * ((3 * π / 2) - (2 * γ))), (3 * π / 2) - (2 * γ)], A'_{11}) → Z_1
  116. Rotate(V_1, If[17 < a ≤ 18, ((a - 17) * ((3 * π / 2) - (2 * γ))), (3 * π / 2) - (2 * γ)], A'_{11}) → A_2
  117. Segment(Z_1, A_2) → n_4
  118. Rotate(Z_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], A_2) → N_1
  119. Rotate(A_2, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], A_2) → O_1
  120. Segment(N_1, O_1) → b_1
  121. Rotate(H, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], Q') → H'_{6}
  122. Rotate(D'_{1}, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], Q') → P_1
  123. Rotate(H', If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], Q') → H''_{2}
  124. Rotate(L, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], Q') → L'_{1}
  125. Polygon(H'_{6}, P_1, H''_{2}, L'_{1}) → q2', h'_{6}, f_2, h''_{2}, l'_{1}
  126. Rotate(A'_{4}, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], Q') → Q_1
  127. Segment(Q', Q_1) → g_2
  128. Segment(Q_1, P_1) → p_3
  129. Segment(P_1, H''_{2}) → s_3
  130. Segment(H''_{2}, Q') → t_3
  131. Polygon(H''_{2}, A'_{4}, D'_{1}, P_1) → q8, b_3, a'_{4}, p_4, q_4
  132. Circle(A'_{4}, 1) → c_4
  133. Intersect(c_4, c_3, 1) → R_1
  134. Dilate(D'_{1}, If[26 < a ≤ 27, 27 - a, 0], R_1) → S_1
  135. Vector(D'_{1}, S_1) → v_4
  136. Translate(P_1, v_4) → T_1
  137. Polygon(A'_{4}, S_1, T_1, H''_{2}) → q9, r_4, s_4, t_4, a_4
  138. Line(T_1, H''_{2}) → m_2
  139. Line(R_1, A'_{4}) → h_3
  140. OrthogonalLine(H''_{2}, h_3) → q_2
  141. Intersect(h_3, q_2) → U_1
  142. Dilate(A'_{4}, If[28 < a ≤ 29, 29 - a, 0], U_1) → B_2
  143. Vector(A'_{4}, B_2) → d_3
  144. Translate(A'_{4}, d_3) → C_2
  145. Translate(R_1, d_3) → D_2
  146. Polygon(D_2, T_1, H''_{2}, B_2) → q4, l_3, n_3, e_3, h_4
  147. Dilate(U_1, If[30 < a ≤ 31, 31 - a, 0], O_1) → E_2
  148. Vector(U_1, E_2) → s_2
  149. Translate(D_2, s_2) → F_2
  150. Translate(T_1, s_2) → G_2
  151. Translate(H''_{2}, s_2) → H_2
  152. Translate(B_2, s_2) → I_2
  153. Polygon(F_2, G_2, H_2, I_2) → q4', t_2, b_4, d_4, e_4
  154. Line(U_1, b_1) → g_5
  155. Point(g_5) → J_2
  156. Angle(J_2, U_1, R_1) → η
  157. Rotate(F_2, If[30 < a ≤ 31, ((-(a - 30)) * η), (-η)], E_2) → K_2
  158. Rotate(G_2, If[30 < a ≤ 31, ((-(a - 30)) * η), (-η)], E_2) → L_2
  159. Rotate(H_2, If[30 < a ≤ 31, ((-(a - 30)) * η), (-η)], E_2) → M_2
  160. Rotate(I_2, If[30 < a ≤ 31, ((-(a - 30)) * η), (-η)], E_2) → N_2
  161. Polygon(K_2, L_2, M_2, N_2) → q4'', h_5, i_5, j_5, k_5
  162. Dilate(K_2, If[32 < a ≤ 33, 33 - a, 0], L_2) → O_2
  163. Vector(K_2, O_2) → a_3
  164. Translate(A, a_3) → A'_{13}
  165. Translate(E, a_3) → E'_{1}
  166. Translate(G, a_3) → G'_{1}
  167. Translate(F, a_3) → F'_{2}
  168. Polygon(A'_{13}, E'_{1}, G'_{1}, F'_{2}) → q3', a'_{13}, e'_{1}, g'_{1}, f'_{2}
  169. Rotate(A'_{13}, If[33 < a ≤ 34, ((a - 33) * π / 2), π / 2], O_2) → P_2
  170. Rotate(E'_{1}, If[33 < a ≤ 34, ((a - 33) * π / 2), π / 2], O_2) → Q_2
  171. Rotate(G'_{1}, If[33 < a ≤ 34, ((a - 33) * π / 2), π / 2], O_2) → R_2
  172. Rotate(F'_{2}, If[33 < a ≤ 34, ((a - 33) * π / 2), π / 2], O_2) → S_2
  173. Polygon(P_2, Q_2, R_2, S_2) → q3'', f_4, l_5, m_5, n_5

AI Drawing Prompt Reference

Draw a unit circle centered at the origin. From the positive x-axis, rotate a radius by 45 degrees counterclockwise. Pick a point on the arc and rotate it about the origin by the same angle to get a symmetric point. Drop perpendiculars from this point to the x-axis and y-axis, forming rectangles and triangles. Construct a circular sector inside the circle and rotate it stepwise to show angle changes. Outside the circle, add a semicircle and several polygons, using scaling and mirroring to illustrate relationships between sine, cosine, and area. The final figure includes multiple rotated sectors, rectangles, triangles, and auxiliary lines for a geometric proof of trigonometric identities.
The prompt above was compiled from the original drawing commands and is for reference only.