Some example Mathematica commands for plotting polar curves:

PolarPlot[Sin[3t],{t,0,Pi}]

PolarPlot[{1,1+1/10 Sin[10t]},{t,0,2Pi},PlotStyle->{Green,Directive[Dashed,Thick,Orange]}]

#16, p. 687

PolarPlot[2Sec[t],{t,0,2Pi}]

#67, p. 688

PolarPlot[1+2Sin[t/2],{t,0,4Pi}]

Plot[3/4*(1.5-Cos[2t]),{t,0,2Pi}]
PolarPlot[3/4*(1.5-Cos[2t]),{t,0,2Pi}]

#64, p. 688

xmin=-2
xmax=2
ymin=-2
ymax=2
t0=-2Pi
t1=1
r[t_]:=Exp[t]
Manipulate[
PolarPlot[r[t],{t,t0,tstop},PlotRange->{{xmin,xmax},{ymin,ymax}}],
{tstop,t0,t1}
]