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Sine Function via Colorful Circular Sectors

Sine Function via Colorful Circular Sectors
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2020-06-03 04:40
CircleAngleFunctionTrigonometryGraphing
① This example demonstrates how to construct the sine function graph by mapping colorful circular sectors on the unit circle to corresponding points on the sine curve. ② Core mathematical concepts include trigonometric functions (sine), the unit circle, angle-radian correspondence, and function graphing. ③ Construction highlights: using GeoGebra commands such as Tangent, Ray, CircleArc, OrthogonalLine, Point, Angle, Intersect, Segment, Line, Textfield, Function, Sequence, and Element, the applet creates multiple auxiliary points (e.g., M_0 to M_7), arcs, triangles, and a dynamic angle β, with distinct colors for each sector, and links them to generate the sine curve. ④ Suitable for high school or early college students learning the relationship between trigonometric functions and the unit circle, and for teachers to demonstrate periodicity, amplitude, and phase shifts in an interactive way.

GeoGebra Commands & Parameters

  1. Tangent(M, c_3) → b_1, d
  2. Ray(M, A) → e
  3. CircleArc(O, I, M) → k
  4. OrthogonalLine(B, yAxis) → i
  5. Point(i) → C
  6. Ray(B, C) → j
  7. Angle(I, O, M) → β
  8. Intersect(yAxis, xAxis) → D
  9. Segment(D, P) → g
  10. Segment(O, I) → h
  11. CircleArc(O, M, I) → p
  12. Point(c_3) → E
  13. Segment(I, E) → l
  14. Line(D, J) → m
  15. Line(O, m) → n
  16. Intersect(c_3, n) → F, G
  17. Line(O, j) → q
  18. Intersect(c_3, q) → H, K
  19. Segment(H, K) → r
  20. Segment(S, M) → s
  21. Segment(M, L) → t_1
  22. Segment(E, F) → a1
  23. Segment(O, M) → b1
  24. Segment(P, L) → c1
  25. Segment(O, S) → d1
  26. Angle(B, I, M) → α
  27. Line(E, J) → e1
  28. Line(F, A) → g1
  29. Textfield(pas) → ChampTexte1
  30. Function(f, 0, aa) → f1
  31. Sequence((i, f(i)), i, 0, aa, (pas * π / 360)) → liste1
  32. Sequence(I + (cos((2 * i)), f(i)), i, 0, aa, (pas * π / 360)) → liste2
  33. Sequence(CircleSector[I, Element[liste2, i], Element[liste2, i + 1]], i, 1, Length[liste2]) → liste3
  34. Element(liste3, A1) → B1
  35. Element(liste3, A2) → B2
  36. Element(liste3, A3) → B3
  37. Element(liste3, A4) → B4
  38. Element(liste3, A5) → B5
  39. Element(liste3, A6) → B6
  40. Element(liste3, A7) → B7
  41. Element(liste3, A8) → B8
  42. Element(liste3, A9) → B9
  43. Element(liste3, A10) → B10
  44. Element(liste3, A11) → B11
  45. Element(liste3, A12) → B12
  46. Element(liste3, A13) → B13
  47. Element(liste3, A14) → B14
  48. Element(liste3, A15) → B15
  49. Element(liste3, A16) → B16
  50. Element(liste3, A17) → B17
  51. Element(liste3, A18) → B18
  52. Element(liste3, A19) → B19
  53. Element(liste3, A20) → B20
  54. Element(liste3, A21) → B21
  55. Element(liste3, A22) → B22
  56. Element(liste3, A23) → B23
  57. Element(liste3, A24) → B24
  58. Element(liste3, A25) → B25
  59. Element(liste3, A26) → B26
  60. Element(liste3, A27) → B27
  61. Element(liste3, A28) → B28
  62. Element(liste3, A29) → B29
  63. Element(liste3, A30) → B30
  64. Element(liste3, A31) → B31
  65. Element(liste3, A32) → B32
  66. Element(liste3, A33) → B33
  67. Element(liste3, A34) → B34
  68. Element(liste3, A35) → B35
  69. Element(liste3, A36) → B36
  70. Element(liste3, A37) → B37
  71. Element(liste3, A38) → B38
  72. Element(liste3, A39) → B39
  73. Element(liste3, A40) → B40
  74. Element(liste3, A41) → B41
  75. Element(liste3, A42) → B42
  76. Element(liste3, A43) → B43
  77. Element(liste3, A44) → B44
  78. Element(liste3, A45) → B45
  79. Element(liste3, A46) → B46
  80. Element(liste3, A47) → B47
  81. Element(liste3, A48) → B48
  82. Element(liste3, A49) → B49
  83. Element(liste3, A50) → B50
  84. Element(liste3, A51) → B51
  85. Element(liste3, A52) → B52
  86. Element(liste3, A53) → B53
  87. Element(liste3, A54) → B54
  88. Element(liste3, A55) → B55
  89. Element(liste3, A56) → B56
  90. Element(liste3, A57) → B57
  91. Element(liste3, A58) → B58
  92. Element(liste3, A59) → B59
  93. Element(liste3, A60) → B60
  94. Sequence((i, 0), i, 0, aa, (pas * π / 360)) → l1
  95. Sequence(Polygon[Element[l1, i], Element[liste1, i], Element[liste1, i + 1], Element[l1, i + 1]], i, 1, Length[l1]) → l2
  96. Element(l2, A1) → C1
  97. Element(l2, A2) → C2
  98. Element(l2, A3) → C3
  99. Element(l2, A4) → C4
  100. Element(l2, A5) → C5
  101. Element(l2, A6) → C6
  102. Element(l2, A7) → C7
  103. Element(l2, A8) → C8
  104. Element(l2, A9) → C9
  105. Element(l2, A10) → C10
  106. Element(l2, A11) → C11
  107. Element(l2, A12) → C12
  108. Element(l2, A13) → C13
  109. Element(l2, A14) → C14
  110. Element(l2, A15) → C15
  111. Element(l2, A16) → C16
  112. Element(l2, A17) → C17
  113. Element(l2, A18) → C18
  114. Element(l2, A19) → C19
  115. Element(l2, A20) → C20
  116. Element(l2, A21) → C21
  117. Element(l2, A22) → C22
  118. Element(l2, A23) → C23
  119. Element(l2, A24) → C24
  120. Element(l2, A25) → C25
  121. Element(l2, A26) → C26
  122. Element(l2, A27) → C27
  123. Element(l2, A28) → C28
  124. Element(l2, A29) → C29
  125. Element(l2, A30) → C30
  126. Element(l2, A31) → C31
  127. Element(l2, A32) → C32
  128. Element(l2, A33) → C33
  129. Element(l2, A34) → C34
  130. Element(l2, A35) → C35
  131. Element(l2, A36) → C36
  132. Element(l2, A37) → C37
  133. Element(l2, A38) → C38
  134. Element(l2, A39) → C39
  135. Element(l2, A40) → C40
  136. Element(l2, A41) → C41
  137. Element(l2, A42) → C42
  138. Element(l2, A43) → C43
  139. Element(l2, A44) → C44
  140. Element(l2, A45) → C45
  141. Element(l2, A46) → C46
  142. Element(l2, A47) → C47
  143. Element(l2, A48) → C48
  144. Element(l2, A49) → C49
  145. Element(l2, A50) → C50
  146. Element(l2, A51) → C51
  147. Element(l2, A52) → C52
  148. Element(l2, A53) → C53
  149. Element(l2, A54) → C54
  150. Element(l2, A55) → C55
  151. Element(l2, A56) → C56
  152. Element(l2, A57) → C57
  153. Element(l2, A58) → C58
  154. Element(l2, A59) → C59
  155. Element(l2, A60) → C60
  156. Element(liste3, A61) → B61
  157. Element(l2, A61) → C61
  158. Element(liste3, A62) → B62
  159. Element(l2, A62) → C62
  160. Element(liste3, A63) → B63
  161. Element(l2, A63) → C63
  162. Element(liste3, A64) → B64
  163. Element(l2, A64) → C64
  164. Element(liste3, A65) → B65
  165. Element(l2, A65) → C65
  166. Element(liste3, A66) → B66
  167. Element(l2, A66) → C66
  168. Element(liste3, A67) → B67
  169. Element(l2, A67) → C67
  170. Element(liste3, A68) → B68
  171. Element(l2, A68) → C68
  172. Element(liste3, A69) → B69
  173. Element(l2, A69) → C69
  174. Element(liste3, A70) → B70
  175. Element(l2, A70) → C70
  176. Element(liste3, A71) → B71
  177. Element(l2, A71) → C71
  178. Element(liste3, A72) → B72
  179. Element(l2, A72) → C72
  180. Element(liste3, A73) → B73
  181. Element(l2, A73) → C73
  182. Element(liste3, A74) → B74
  183. Element(l2, A74) → C74
  184. Element(liste3, A75) → B75
  185. Element(l2, A75) → C75
  186. Element(liste3, A76) → B76
  187. Element(l2, A76) → C76
  188. Element(liste3, A77) → B77
  189. Element(l2, A77) → C77
  190. Element(liste3, A78) → B78
  191. Element(l2, A78) → C78
  192. Element(liste3, A79) → B79
  193. Element(l2, A79) → C79
  194. Element(liste3, A80) → B80
  195. Element(l2, A80) → C80
  196. Element(liste3, A81) → B81
  197. Element(l2, A81) → C81
  198. Element(liste3, A82) → B82
  199. Element(l2, A82) → C82
  200. Element(liste3, A83) → B83
  201. Element(l2, A83) → C83
  202. Element(liste3, A84) → B84
  203. Element(l2, A84) → C84
  204. Element(liste3, A85) → B85
  205. Element(l2, A85) → C85
  206. Element(liste3, A86) → B86
  207. Element(l2, A86) → C86
  208. Element(liste3, A87) → B87
  209. Element(l2, A87) → C87
  210. Element(liste3, A88) → B88
  211. Element(l2, A88) → C88
  212. Element(liste3, A89) → B89
  213. Element(l2, A89) → C89
  214. Element(liste3, A90) → B90
  215. Element(l2, A90) → C90
  216. Element(liste3, A91) → B91
  217. Element(l2, A91) → C91
  218. Element(liste3, A92) → B92
  219. Element(l2, A92) → C92
  220. Element(liste3, A93) → B93
  221. Element(l2, A93) → C93
  222. Element(liste3, A94) → B94
  223. Element(l2, A94) → C94
  224. Element(liste3, A95) → B95
  225. Element(l2, A95) → C95
  226. Element(liste3, A96) → B96
  227. Element(l2, A96) → C96
  228. Element(liste3, A97) → B97
  229. Element(l2, A97) → C97
  230. Element(liste3, A98) → B98
  231. Element(l2, A98) → C98
  232. Element(liste3, A99) → B99
  233. Element(l2, A99) → C99
  234. Element(liste3, A100) → B100
  235. Element(l2, A100) → C100

AI Drawing Prompt Reference

Draw a unit circle centered at the origin with a horizontal radius to the right. From the center, draw a ray that intersects the circle at a moving point. As this point moves around the circle, generate colorful circular sectors on the circle, each filled with a distinct color. To the right of the circle, set up a Cartesian coordinate system with the horizontal axis representing angle in radians and the vertical axis representing sine values. Map the angle of the moving point to the horizontal axis and plot the corresponding sine value as a point. Connect these points to form a smooth sine curve starting from the origin, oscillating with amplitude 1 and period 2π. Connect the colorful sectors on the circle to the points on the curve with thin lines to visually show the correspondence between angle and sine value.
The prompt above was compiled from the original drawing commands and is for reference only.