ParametricPlot3D[{10Cos[t],10Sin[t],t}, {t,-4Pi,4Pi}, PlotStyle->{Thick,Blue}]

ParametricPlot3D[{t Cos[t],t Sin[t],t}, {t,0 , 10Pi}, PlotStyle->{Thick,Blue}]

Animate[Plot[Sin[x-a],{x,0,10}],{a,0,5},AnimationRunning->False]

Animate[Plot[Sin[x-a],{x,0,10}],{a,0,2Pi},AnimationRunning->False]

Animate[Plot[Sin[a x]+Sin[b x],{x,0,10},PlotRange->2],{a,1,5},{b,1,5},AnimationRunning->False]

cycloid[]:=Animate[Show[Plot[0,{x,-2,4Pi+2}, PlotStyle->Black], ParametricPlot[{t-Sin[t],1-Cos[t]},{t,-3,k}], Graphics[{Circle[{k,1},1],{Gray,Table[Line[{{k,1},{k+Cos[s],1+Sin[s]}}],{s,Pi/12,2Pi,Pi/12}]}, 
{PointSize[0.015], Red, Point[{k-Sin[k], 1-Cos[k]}]}}], Ticks->None, Axes->False, AspectRatio->Automatic,PlotRange->{{-1,13.5},{0,2.1}},ImageSize->500],{k,-2 ,4Pi+2},AnimationRunning->False]

cycloid[]

polarGrid[r_,dr_]:= Graphics[{Thickness[0.0005], Table[Circle[{0,0},k], {k, dr, r, dr}],Table[Line[{{0,0}, r{Cos[t],Sin[t]}}],{t,Pi/12,2Pi,Pi/12}]}]

polarGrid[1,.1]

polarGrid[r_,dr_]:= Graphics[{Thickness[0.0005], Table[Circle[{0,0},k], {k, dr, r, dr}],Table[Line[{{0,0}, r{Cos[t],Sin[t]}}],{t,Pi/12,2Pi,Pi/12}]}]
polarCurve[r_,{t_,t1_,t2_}, rmax_,opts___]:=Module[{r2=r/.t->t2,pt2={Cos[t2], Sin[t2]}}, Show [polarGrid[rmax,rmax/5], PolarPlot[r,{t,t1,t2},PlotStyle->AbsoluteThickness[3]],Graphics[{{Purple,Thickness[.005],Line[{{0,0},rmax pt2}]},{If[r2>0,Green,Red],Thick,Opacity[.5],Line[{{0,0},r2 pt2}]},{PointSize[.025],Point[r2 pt2]}}], opts,PlotRange->1.02 rmax{{-1,1},{-1,1}}]]

polarCurve[Cos[t],{t,0,Pi/3},1]

polarGrid[r_,dr_]:= Graphics[{Thickness[0.0005], Table[Circle[{0,0},k], {k, dr, r, dr}],Table[Line[{{0,0}, r{Cos[t],Sin[t]}}],{t,Pi/12,2Pi,Pi/12}]}]
polarCurve[r_,{t_,t1_,t2_}, rmax_,opts___]:=Module[{r2=r/.t->t2,pt2={Cos[t2], Sin[t2]}}, Show [polarGrid[rmax,rmax/5], PolarPlot[r,{t,t1,t2},PlotStyle->AbsoluteThickness[3]],Graphics[{{Purple,Thickness[.005],Line[{{0,0},rmax pt2}]},{If[r2>0,Green,Red],Thick,Opacity[.5],Line[{{0,0},r2 pt2}]},{PointSize[.025],Point[r2 pt2]}}], opts,PlotRange->1.02 rmax{{-1,1},{-1,1}}]]
polarMovie[r_,{t_,t1_,t2_},rmax_]:=
Animate[GraphicsRow[{polarCurve[r,{t,t1,k},rmax],Plot[r,{t,t1,k},Ticks->{{Pi,2Pi},None},PlotRange->{{t1,t2},{-rmax,rmax}},ImageSize->250]},20],{k,t1+Pi/12,t2,Pi/12},AnimationRunning->False]

polarMovie[Cos[2 t],{t,0,2 Pi},1]

polarMovie[Sin[4t],{t,0,2 Pi},1]

polarMovie[2(1-Cos[t])/5,{t,0,2 Pi},1]