We begin by observing that we may draw parametric curves as follows:

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

We have animated this graph below

Animate[ParametricPlot[{Cos[4*t]*Cos[t],Cos[4*t]*Sin[t]},{t,0,s},PlotRange->2],{s,0,2*Pi},AnimationRate->0.1]

Animate the following graphs

1)  r =1 + Cos[t]

2)  r = 2 - Sin[t]

3)  r = 1 - 3*Cos[t]

4)  r = Sin[2*t]

5)  r = Cos[3*t]

The parametric graphs are given  below

1)

ParametricPlot[{1+Cos[t])*Cos[t],(1+Cos[t])*Sin[t]},{t,0,2*Pi}]

2)

ParametricPlot[{2-Sin[t])*Cos[t],(2-Sin[t])*Sin[t]},{t,0,2*Pi}]

3)

ParametricPlot[{1-3*Cos[t])*Cos[t],(1-3*Cos[t])*Sin[t]},{t,0,2*Pi}]

4)

ParametricPlot[{Sin[2*t])*Cos[t],(Sin[2*t])*Sin[t]},{t,0,2*Pi}]

5)

ParametricPlot[{Cos[3*t])*Cos[t],(Cos[3*t])*Sin[t]},{t,0,2*Pi}]