Export["siney.gif",
 Table[
  Show[Graphics[
    Translate[
     Rotate[{Circle[], Thick, Blue, Line[{{0, 0}, {0, -1}}], Red,
       PointSize[.02], Point[{0, -1}]}, t], { 2 Pi, 0}],
    PlotRange -> {{0, 4 Pi}, {-2, 2}}, ImageSize -> {Large, Tiny},
    Axes -> {True, False}, AxesOrigin -> {0, -1}],
   Graphics[
    Plot[Sin[x + t - Sin[t] + 3*Pi/2], {x, -4 Pi, 4 Pi}]
    ]
   ], {t, 0, 4*Pi - 0.25, 0.25}]
 , ImageSize -> 1200]

 Export["cycloid.gif" ,
 Table[Show[
   Graphics[
    Translate[
     Rotate[{Circle[], Thick, Blue, Line[{{0, 0}, {0, -1}}], Red,
       PointSize[.02], Point[{0, -1}]}, t], { 2 Pi, 0}],
    PlotRange -> {{0, 4 Pi}, {-2, 2}}, ImageSize -> {Large, Tiny},
    Axes -> {True, False}, AxesOrigin -> {0, -1}],
   Graphics[
    ParametricPlot[{-t + x + Cos[x] + Pi/2, Sin[x]}, {x, -4 Pi,
      8 Pi}]
    ]
   ], {t, 0, 2*Pi - 0.25, 0.25}]
 , ImageSize -> 1200]