What we plotted today, one period

PolarPlot[Sin[4*t],{t,-Pi/4,Pi/4}]

Now fill the plane

PolarPlot[Sin[4*t],{t,0,2*Pi}]

Animate the plot above, with angle theta stating from 0 to 2*Pi

r[t_]:=Sin[4*t];ep[s_]:=PolarPlot[r[t],{t,0,s},AxesLabel->{"x","y"},PlotRange->{{-1,1},{-1,1}}];
Animate[ep[s],{s,0.01,2* Pi},AnimationRunning->False]