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Unit-circle comparison of phase shifts: cos(π/2+x), sin(π/2+x), cos(π+x), sin(π+

Unit-circle comparison of phase shifts: cos(π/2+x), sin(π/2+x), cos(π+x), sin(π+
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2021-01-15 04:11
CircleAngleFunctionTrigonometryGeometric Construction
This example demonstrates, on the unit circle, how shifting the angle by π/2+x and π+x transforms the cosine and sine values, and how those transformed quantities correspond visually to the baseline functions sin x and cos x. The key math ideas include the unit-circle mapping from an angle to coordinates, phase-shift identities (for π/2 and π), and geometric symmetry/sign changes. Construction approach: use Intersect, Circle, Point, Line, and Angle to define the unit circle, the angle rays, and the measurement setup; apply OrthogonalLine and Midpoint (with Segment) to obtain projections/connected points that represent function values; employ Rotate/RotateText, Translate, and Mirror to realize the phase-shift configurations; use Vector, CircleArc, CircleSector, and Ray to highlight arcs, sectors, and comparable geometric elements; optionally use Polygon for structured grouping. Multiple constructed points/angle markers and arcs (as in the model’s object list) are used to make the comparisons clear. Suitable for advanced middle school, pre-university, and early college teaching as a dynamic-geometry tool for trigonometric identity learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 1) → c
  3. Intersect(c, xAxis) → C_1, D
  4. Segment(C_1, D) → r
  5. Point(r) → I_1
  6. Rotate(D, j, O) → M
  7. If(j < 0, M) → M1
  8. Intersect(c, yAxis) → K, L
  9. Line(K, L) → s
  10. OrthogonalLine(M, yAxis) → a
  11. Intersect(a, yAxis) → S
  12. Angle(D, O, M) → β
  13. OrthogonalLine(M, xAxis) → b
  14. Intersect(b, xAxis) → C
  15. Segment(O, S) → d
  16. Segment(O, C) → e
  17. Segment(M, S) → g
  18. Segment(M, C) → h
  19. Midpoint(M, S) → G
  20. Midpoint(M, C) → H
  21. Vector(M, G) → u
  22. Vector(M, H) → v
  23. Rotate(D, 0.52, O) → I
  24. Angle(I, O, D) → γ
  25. Rotate(D, 1.0471975511965976, O) → J
  26. Angle(D, O, J) → δ
  27. Mirror(J, yAxis) → M_1
  28. Mirror(I, yAxis) → N
  29. Mirror(M_1, xAxis) → O_2
  30. Mirror(J, xAxis) → P
  31. Mirror(I, xAxis) → Q
  32. Mirror(Q, yAxis) → R
  33. Rotate(D, 0.7853981633974483, O) → T
  34. Angle(D, O, T) → ε
  35. Mirror(T, yAxis) → U
  36. Mirror(T, xAxis) → V
  37. Mirror(U, xAxis) → W
  38. If(Distance[M, K] < 0.1, F) → s_4
  39. Segment(R, I) → i
  40. Segment(T, W) → k
  41. Segment(J, O_2) → l
  42. Segment(Q, N) → m
  43. Segment(P, M_1) → p
  44. Segment(L, K) → q
  45. Line(C_1, D) → t
  46. Segment(O, M) → b_1
  47. Point(t) → Z
  48. Point(t) → E_1
  49. Point(s) → C_2
  50. Point(s) → D_2
  51. Point(s) → E_2
  52. Point(s) → F_2
  53. Segment(U, V) → c_1
  54. Point(c) → B
  55. CircleArc(O, M1, D) → k_1
  56. If(j > 0, M) → M2
  57. CircleArc(O, D, M2) → k_2
  58. Angle(I_1, O, M2) → α
  59. Angle(M1, O, I_1) → ζ
  60. Segment(J, O) → f
  61. Segment(T, O) → a1
  62. Segment(I, O) → b1
  63. Segment(U, O) → a_1
  64. Segment(O, W) → d_1
  65. Segment(O, V) → e_1
  66. Intersect(c, d_1, 1) → A
  67. Intersect(c, d_1, 1) → E
  68. CircleArc(F_1, G_1, H_1) → f_1
  69. Point(f_1) → J_1
  70. Point(f_1) → K_1
  71. Vector(K_1, J_1) → w
  72. FractionText(dir) → texte3
  73. Segment(M_1, P) → g_1
  74. Segment(R, I) → h_1
  75. Segment(J, O_2) → i_1
  76. Segment(D, B_1) → j_1
  77. Point(j_1) → L_1
  78. Polygon(B_1, D_1, L_1) → t1, l_1, m_1, n_1
  79. FractionText(mesp) → texte4
  80. FractionText(mesp) → texte5
  81. Point(c) → R_1
  82. Segment(N, Q) → m_2
  83. Rotate(R_1, (-(60°)), O) → W_1
  84. Segment(O, W_1) → n_2
  85. OrthogonalLine(R_1, n_2) → p_2
  86. Intersect(p_2, n_2) → Z_1
  87. Circle(O, 1.25) → c_2
  88. Circle(O, 1) → d_2
  89. Sequence(Rotate[M + (0.35, 0), i°, O], i, 0, 360, 1) → l1
  90. Circle(O, 1.25) → s_2
  91. Sequence(Text[i + "°", Element[l1, i] - (0.065, 0.015)], i, 0, 360, 10) → l2
  92. Circle(O, 1.45) → e_2
  93. Circle(O, 1.45) → t_2
  94. Sequence(Rotate[M + (0.25, 0), i°, O], i, 0, 360, 5) → l3
  95. CircleSector(O, K, C_1) → c_3
  96. CircleSector(O, C_1, L) → d_3
  97. CircleSector(O, L, D) → e_3
  98. Circle(O, 0.15) → f_3
  99. Rotate(B_3, 60°, A_3) → C_3
  100. Ray(A_3, C_3) → n_3
  101. Segment(A_3, B_3) → p_3
  102. OrthogonalLine(B_3, p_3) → q_3
  103. Intersect(q_3, n_3) → K_3
  104. Midpoint(J, O) → M_3
  105. Midpoint(O, U) → N_3
  106. Midpoint(O, W) → O_3
  107. Midpoint(O, V) → P_3
  108. Segment(D, D + (0.45, 0)) → a_2
  109. OrthogonalLine(R_1, r) → p_1
  110. Intersect(p_1, r) → P_1
  111. Polygon(R_1, O, P_1) → t2, q_1, r_1, o
  112. Angle(Q_1, A_3, S_1) → η
  113. Polygon(S_1, Q_1, A_3) → t3, a_3, s_1, t_1
  114. Angle(R_1, P_1, O) → θ
  115. Angle(P_1, O, R_1) → ι
  116. Rotate(R_1, o_1, O) → T_1
  117. Rotate(P_1, o_1, O) → U_1
  118. Polygon(O, T_1, U_1) → t4, u_1, o_2, f_2
  119. Angle(T_1, U_1, O) → κ
  120. Translate(T_1, Vector[O, If[o_1 ≟ π / 2, V_1, K_2]]) → I_2
  121. Translate(U_1, Vector[O, If[o_1 ≟ π / 2, V_1, K_2]]) → J_2
  122. Translate(O, Vector[O, If[o_1 ≟ π / 2, V_1, K_2]]) → O'
  123. Polygon(O', I_2, J_2) → t4', j_2, o', i_2
  124. Angle(U_1, O, T_1) → λ
  125. Angle(P_1, O, U_1) → μ
  126. Angle(S_1, Q_1, A_3) → ν
  127. Angle(I_2, J_2, If[o_1 ≟ π / 2, V_1, K_2]) → ξ
  128. Point(i_2) → N_2
  129. Point(s_1) → P_2
  130. Point(a_3) → Q_2
  131. Point(o') → R_2
  132. RotateText(texte26, 90°) → texte27
  133. RotateText(texte29, 90°) → texte30
  134. Midpoint(s_1) → S_2

AI Drawing Prompt Reference

On the unit circle, illustrate how shifting the angle by π/2+x and π+x transforms cosine and sine values, and how these transformed quantities visually correspond to the baseline functions sin x and cos x. Start by drawing the unit circle and coordinate axes, then mark key angles and rays. Use orthogonal projections and connected points to represent function values. Apply rotations, translations, and reflections to demonstrate phase shifts, and highlight geometric elements with vectors, arcs, and sectors. Finally, connect all elements with line segments and points to clearly show the comparisons.
The prompt above was compiled from the original drawing commands and is for reference only.