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Radians, Number Line and the Unit Circle by Irina Boyadzhiev

Radians, Number Line and the Unit Circle by Irina Boyadzhiev
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2015-01-10 01:27
TrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the one-to-one correspondence between points on the number line and points on the unit circle. It involves core mathematical concepts such as radians, trigonometric functions, the unit circle, and the number line. The construction utilizes GeoGebra commands/objects like Vector, Text, Length, Intersect, Point, Circle, If, Rotate, Tangent, Ray, UnitVector, Sequence, CircleArc, Take, Segment, Textfield, and CurveCartesian. It is suitable for high school mathematics education, helping students understand the connection between trigonometric functions and the unit circle, ideal for learners of analytical geometry and trigonometry.

GeoGebra Commands & Parameters

  1. Vector((0, 0), (1, 0)) → u_1
  2. Text(A50) → B50
  3. Text(A49) → B49
  4. Text(A48) → B48
  5. Text(A47) → B47
  6. Text(A46) → B46
  7. Text(A45) → B45
  8. Text(A44) → B44
  9. Text(A43) → B43
  10. Text(A42) → B42
  11. Text(A41) → B41
  12. Text(A40) → B40
  13. Text(A39) → B39
  14. Text(A38) → B38
  15. Text(A37) → B37
  16. Text(A36) → B36
  17. Text(A35) → B35
  18. Text(A34) → B34
  19. Text(A33) → B33
  20. Text(A32) → B32
  21. Text(A31) → B31
  22. Text(A30) → B30
  23. Text(A29) → B29
  24. Text(A28) → B28
  25. Text(A27) → B27
  26. Text(A26) → B26
  27. Text(A25) → B25
  28. Text(A24) → B24
  29. Text(A23) → B23
  30. Text(A22) → B22
  31. Text(A21) → B21
  32. Text(A20) → B20
  33. Text(A19) → B19
  34. Text(A18) → B18
  35. Text(A17) → B17
  36. Text(A16) → B16
  37. Text(A15) → B15
  38. Text(A14) → B14
  39. Text(A13) → B13
  40. Text(A12) → B12
  41. Text(A11) → B11
  42. Text(A10) → B10
  43. Text(A9) → B9
  44. Text(A8) → B8
  45. Text(A7) → B7
  46. Text(A6) → B6
  47. Text(A5) → B5
  48. Text(A4) → B4
  49. Text(A3) → B3
  50. Text(A2) → B2
  51. Text(A1) → B1
  52. Length(numb) → l
  53. Intersect(xAxis, yAxis) → B
  54. Point(xAxis) → O
  55. Circle(O, B) → c
  56. If(WrapPositive ≥ pi / 2, WrapPositive - pi / 2, 0) → f
  57. Rotate(M, f, O) → M'
  58. Tangent(M', c) → b,
  59. Point(b) → D
  60. Ray(M', D) → e
  61. UnitVector(e) → u
  62. Point(xAxis) → C
  63. If(WrapPositive ≤ pi / 2, WrapPositive, 1.5707963267948966) → α
  64. Rotate(B, α, B) → B'
  65. Rotate(C, α, B) → C'
  66. Ray(B', C') → a'
  67. Rotate(B', f, B) → B''
  68. Sequence(Point[M', Vector[If[(k * st) - f ≥ 0, Vector[(u * ((k * st) - f))]]]], k, 0, l) → list1
  69. CircleArc(O, M, M') → d
  70. If(f ≟ 0, RemoveUndefined[list1], Append[(0, 0), RemoveUndefined[list1]]) → ptsR
  71. Sequence((cos(k) - 1, sin(k)), k, 0, f, st) → ptsC
  72. Circle(O, B) → g
  73. UnitVector(a') → v
  74. Sequence(Point[M, Vector[((k * st) * v)]], k, 0, l) → list2
  75. Sequence(Text[Element[numb, k + W + 1], Element[ptsR, k + 1], true, true], k, 0, l - 1) → numR
  76. Take(numR, 2) → NewPtsR
  77. Point(c) → A
  78. Segment(O, A) → ii
  79. Rotate(A, WrapPositive) → A'
  80. If(WrapPositive > pi / 2, Segment[Rotate[O, WrapPositive - pi / 2, O], Rotate[A, WrapPositive - pi / 2, O]]) → j
  81. Textfield(st) → inputBox1
  82. Rotate(A, 6.2831, O) → A'_1
  83. Sequence(Text[Element[numb, k], Element[list2, k], true, true], k, 1, l) → nmbX
  84. Take(Sequence[Text[Element[numb, k], Element[ptsC, k], true, true], k, 1, l], 2) → numC
  85. Text(C1) → D1
  86. Text(C2) → D2
  87. Text(C3) → D3
  88. Text(C4) → D4
  89. Text(C5) → D5
  90. Text(C6) → D6
  91. Text(C7) → D7
  92. Text(C8) → D8
  93. Text(C9) → D9
  94. Text(C10) → D10
  95. Text(C11) → D11
  96. Text(C12) → D12
  97. Text(C13) → D13
  98. Text(C14) → D14
  99. Text(C15) → D15
  100. Text(C16) → D16
  101. Text(C17) → D17
  102. Text(C18) → D18
  103. Text(C19) → D19
  104. Text(C20) → D20
  105. Text(C21) → D21
  106. Text(C22) → D22
  107. Text(C23) → D23
  108. Text(C24) → D24
  109. Text(C25) → D25
  110. Text(C26) → D26
  111. Text(C27) → D27
  112. Text(C28) → D28
  113. Text(C29) → D29
  114. Text(C30) → D30
  115. Text(C31) → D31
  116. Text(C32) → D32
  117. Text(C33) → D33
  118. Text(C34) → D34
  119. Text(C35) → D35
  120. Text(C36) → D36
  121. Text(C37) → D37
  122. Text(C38) → D38
  123. Text(C39) → D39
  124. Text(C40) → D40
  125. Text(C41) → D41
  126. Text(C42) → D42
  127. Text(C43) → D43
  128. Text(C44) → D44
  129. Text(C45) → D45
  130. Text(C46) → D46
  131. Text(C47) → D47
  132. Text(C48) → D48
  133. Text(C49) → D49
  134. Text(C50) → D50
  135. If(WrapNegative ≤ pi / 2, WrapNegative, 1.5707963267948966) → ε
  136. Ray(A, O) → k
  137. If(WrapNegative > pi / 2 + 0.5, WrapNegative - pi / 2 - 0.5, 0) → m
  138. Rotate(A, ε, M) → A'_2
  139. Rotate(O, ε, M) → O'
  140. Ray(A'_2, O') → k'
  141. UnitVector(k') → w
  142. Sequence(Point[M, Vector[((i * st) * w)]], i, 0, l) → ptsXN
  143. Rotate(M, (-m), O) → MM
  144. Tangent(MM, c) → n,
  145. Point(n) → F
  146. Ray(MM, F) → p
  147. UnitVector(p) → q
  148. Sequence(Point[MM, Vector[If[(k * st) - m ≥ 0, Vector[(q * ((k * st) - m))]]]], k, 0, l) → ptsRN
  149. If(m ≟ 0, RemoveUndefined[ptsRN], Append[(0, 0), RemoveUndefined[ptsRN]]) → ptsRN_1
  150. Take(Sequence[Text[Element[numbN, k + WN + 1], Element[ptsRN_1, k + 1], true, true], k, 0, l - 1], 2) → numRN
  151. CircleArc(O, MM, M) → r
  152. Sequence((cos(k) - 1, (-sin(k))), k, 0, m, st) → ptsCN
  153. Take(Sequence[Text[Element[numbN, k], Element[ptsCN, k], true, true], k, 1, l], 2) → numCN
  154. Sequence(Text[Element[numbN, k], Element[ptsXN, k], true, true], k, 1, l) → numXN
  155. If(WrapPositive > pi / 2, Segment[Rotate[O, WrapPositive - pi / 2, O], Rotate[A, WrapPositive - pi / 2, O]]) → s
  156. Circle(O, B) → c_1
  157. Segment(O, A) → a_1
  158. If(a ≟ false ∧ h ≟ false, true, false) → start
  159. Sequence(Point[M, Vector[((i * st) * u_1)]], i, 0, l) → ptsXP
  160. Ray(M, C) → b_1
  161. Sequence(Text[Element[numb, k], Element[ptsXP, k], true, true], k, 1, l) → list3
  162. Ray(A, O) → d_1
  163. Vector((0, 0), (-1, 0)) → v_1
  164. Sequence(Point[M, Vector[((i * st) * v_1)]], i, 0, l) → ptsXNS
  165. Sequence(Text[Element[numbN, k], Element[ptsXNS, k], true, true], k, 1, l) → list4
  166. CurveCartesian(((0.2 / 16 * (t + (6 * pi))) * cos(t + (6 * pi))) - 1, ((0.2 / 16 * (t + (6 * pi))) * sin(t + (6 * pi))), t, 0, WrapPositive - pi / 2) → o
  167. Tangent(End, o) → e_1
  168. UnitVector(e_1) → w_1
  169. Vector(End, Point[End, Vector[(0.12 * w_1)]]) → f_1
  170. CurveCartesian(((0.2 / 16 * (t + (6 * pi))) * cos(t + (6 * pi))) - 1, ((-0.2 / 16 * (t + (6 * pi))) * sin(t + (6 * pi))), t, 0, WrapNegative - pi / 2 - 0.5) → j_1
  171. Tangent(K, j_1) → i
  172. UnitVector(i) → g_1
  173. Vector(K, Point[K, Vector[(0.12 * g_1)]]) → h_1
  174. If(WrapNegative > pi / 2 + 0.5, Segment[Rotate[O, (-WrapNegative) + pi / 2 + 0.5, O], Rotate[A, (-WrapNegative) + pi / 2 + 0.5, O]]) → t

AI Drawing Prompt Reference

Draw a horizontal number line and mark multiple equally spaced points from -2π to 2π on it, labeling each with its radian value. Draw a unit circle centered at the origin and a ray starting from the origin that aligns with the number line. Mark points on the ray at unit distances from the origin and connect each point to its corresponding point on the unit circle. Label the points on the unit circle corresponding to special angles like 0°, 30°, 45°, 60°, 90° with their coordinates.
The prompt above was compiled from the original drawing commands and is for reference only.