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Explore Sine and Cosine Ratios on the Unit Circle

Explore Sine and Cosine Ratios on the Unit Circle
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2019-03-14 20:36
CircleFunctionTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates how to visualize sine and cosine ratios on the unit circle through angles. It involves core math concepts like the unit circle, trigonometric functions, sine, and cosine ratios. Construction highlights the use of GeoGebra commands and objects such as Polygon, Zip, RemoveUndefined, Sort, Point, IndexOf, Sequence, Element, Translate, LaTeX, Segment, Angle, Length, TableText, and CurveCartesian. Suitable for high school students learning trigonometric definitions and properties.

GeoGebra Commands & Parameters

  1. Polygon({Anchor1Box, Anchor1Box + u, Anchor1Box + u + (3.9 * v), Anchor1Box + (3.9 * v)}) → boxTemplate
  2. Zip((((("\left(" * A) * ",") * B) * "\right)"), A, allCosVals, B, allSinVals) → allPointsDisplay
  3. RemoveUndefined(Sequence[If[Element[showRows, i], Element[allPointsDisplay, i]], i, 1, Length[allPointsDisplay]]) → displayPoints
  4. Sort(Unique[Union[Sequence[i, i, 0, 360, 45], Sequence[i, i, 0, 360, 30]]]) → allAngles
  5. Zip(Rotate[(1, 0), A°, (0, 0)], A, allAngles) → allPoints
  6. Point(allPoints) → P
  7. IndexOf(P, allPoints) → currIndx
  8. RemoveUndefined(Sequence[If[Element[showRows, i], If[currIndx ≟ i, "*", ""]], i, 1, Length[showRows]]) → rowPointer
  9. Sequence(Translate[boxTemplate, Vector[(i * u)]], i, 0, Length[displayPoints]) → boxes
  10. Element(boxes, 2) → box1
  11. Element(boxes, 3) → box2
  12. Element(boxes, 4) → box3
  13. Element(boxes, 5) → box4
  14. Element(boxes, 6) → box5
  15. Element(boxes, 7) → box6
  16. Element(boxes, 8) → box7
  17. Element(boxes, 9) → box8
  18. Translate(boxTemplate, (2.5, 0)) → header2
  19. Element(allPoints, 1) → P1
  20. Element(allPoints, 2) → P2
  21. Element(allPoints, 3) → P3
  22. Element(allPoints, 4) → P4
  23. Element(allPoints, 5) → P5
  24. Element(allPoints, 6) → P6
  25. Element(allPoints, 7) → P7
  26. Element(allPoints, 8) → P8
  27. Element(allPoints, 9) → P9
  28. Element(allPoints, 10) → P10
  29. Element(allPoints, 11) → P11
  30. Element(allPoints, 12) → P12
  31. Element(allPoints, 13) → P13
  32. Element(allPoints, 14) → P14
  33. Element(allPoints, 15) → P15
  34. Element(allPoints, 16) → P16
  35. LaTeX("\text{\textbf{Question: }The point \(P\) is the intersection of the terminal arm of \(\angle θ\) and the unit circle.\\How are the coordinates of \(P\) related to the sine and cosine ratios of θ?}") → MainInstruc
  36. LaTeX("\text{\textbf{Instructions: }Move the point \(P\) to complete the table of values.}") → SpecInstruc
  37. Segment(P, (x(P), 0)) → oppositeSide
  38. Angle(Vector[(1, 0)], Vector[P]) → angleA
  39. LaTeX("\text{Unit Circle}") → lblUnitCircle
  40. Zip(((spacer * A) * "°"), A, allAngles) → allAngleDisplay
  41. RemoveUndefined(Sequence[If[Element[showRows, i], Element[allSinVals, i]], i, 1, Length[allSinVals]]) → displaySinVals
  42. RemoveUndefined(Sequence[If[Element[showRows, i], Element[allCosVals, i]], i, 1, Length[allCosVals]]) → displayCosVals
  43. RemoveUndefined(Sequence[If[Element[showRows, i], Element[allAngleDisplay, i]], i, 1, Length[allAngleDisplay]]) → displayAngles
  44. Length(displayPoints) → test
  45. LaTeX(("P" * Element[allPointsDisplay, currIndx])) → lblPoint
  46. TableText({Join[{{titleTheta}, displayAngles}], Join[{{titlePoints}, displayPoints}], Join[{{titleSin}, displaySinVals}], Join[{{titleCos}, displayCosVals}], Join[{{""}}]}, "|011100vc") → tableA
  47. Zip(Rotate[A, Element[allAngles, currIndx]°, (0, 0)], A, {(0.25, -0.05), (0.3, 0), (0.35, -0.05)}) → chevronPoints
  48. Sequence(Segment[Element[chevronPoints, i], Element[chevronPoints, i + 1]], i, 1, 2) → chevron
  49. CurveCartesian((0.3 * cos(((2 * π) * t))), (0.3 * sin(((2 * π) * t))), t, 0, Element[allAngles, currIndx] / 360) → angleCurve
  50. Polygon({(0, 0), (0.1, 0), (0.1, 0.1), (0, 0.1)}) → rightBox
  51. Translate(rightBox, (If[x(P) < 0, x(P), x(P) - 0.1], If[y(P) < 0, -0.1, 0])) → showRight