Manipulate[
PolarPlot[{1/2 + Cos[t]}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/6,Pi/2, 5Pi/6,Pi,3Pi/2, {k, Red}},{1}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/6}],-1],{1}},TicksStyle->Black, PlotRange->1.7],{k,Pi/50, 2Pi}]

Manipulate[
PolarPlot[{Cos[4t]}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/6,Pi/2, 5Pi/6,Pi,3Pi/2, {k, Red}},{1}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/6}],-1],{1}},TicksStyle->Black, PlotRange->1.7],{k,Pi/50, 2Pi}]

Manipulate[
PolarPlot[{1/2 + Sin[5t]}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/6,Pi/2, 5Pi/6,Pi,3Pi/2, {k, Red}},{1}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/6}],-1],{1}},TicksStyle->Black, PlotRange->2.5],{k,Pi/100, 2Pi}]

Manipulate[
PolarPlot[{t}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/6,Pi/2, 5Pi/6,Pi,3Pi/2, {k, Red}},{1, 2, 3, 4, 5}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/6}],-1],{1, 2, 3, 4, 5}},TicksStyle->Black, PlotRange->5],{k,Pi/12, 2Pi}]

Manipulate[
PolarPlot[{50Cos[2t]}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/6,Pi/2, 5Pi/6,Pi,3Pi/2, {k, Red}},{1, 10, 20 ,30, 40, 50}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/6}],-1],{1, 10, 20 ,30, 40, 50}},TicksStyle->Black, PlotRange->70],{k,Pi/12, 2Pi}]

Manipulate[
PolarPlot[{(Sqrt[50Cos[2t]])}, {t, 0,k},PlotStyle->{Blue},PolarAxes->True,PolarGridLines->{{0,Pi/4,Pi/2, 5Pi/4,3Pi/4,7Pi/4,Pi,3Pi/2, {k, Red}},{1, 5, 7, 10}},
GridLinesStyle->Directive[Dotted,Gray],
PolarTicks->{Drop[Table[i,{i,0,2Pi,Pi/4}],-1],{1,5,7, 10}},TicksStyle->Black, PlotRange->12],{k,Pi/12, 2Pi}]