Solutions to Polar Graphs

by Wayne Brown
Blue Ridge High School

The following is a demonstration to show to the students.  It will show why a graph of two polar equations may have intersections that are not solutions to the system of equations.  The example here is:
  r = sin 2t
  r = 1

The graph of this system of equations is shown below.

Clear[w,t];w=0;
ParametricPlot[{{(Sin[2 t]) Cos[t],(Sin[2 t]) Sin[t]},
                 {Cos[t],Sin[t]}},
               {t,0,2 Pi},PlotRange->{{-2,2},{-2,2}},
               PlotStyle->{{RGBColor[1,0,0]},{RGBColor[0,0,1]}},
               AspectRatio->Automatic];

This graph has four intersections, but the following animation will show why the system of equations has only two solutions for angles between 0 and 2 Pi.

Clear[w,t,g1,g2];
g1:=ParametricPlot[{{(Sin[2 t]) Cos[t],(Sin[2 t]) Sin[t]},
                 {Cos[t],Sin[t]}},
               {t,0,w Pi/8},
               PlotStyle->{{RGBColor[1,0,0]},{RGBColor[0,0,1]}},
               PlotRange->{{-2,2},{-2,2}},
               AspectRatio->Automatic,PlotStyle->Thickness[.005],
               DisplayFunction->Identity];
               
 g2:=ListPlot[{{(Sin[2 w Pi/8]) Cos[w Pi/8],
              (Sin[2 w Pi/8]) Sin[w Pi/8]},
               {Cos[w Pi/8],Sin[w Pi/8]}},
                PlotStyle->{{RGBColor[1,0,0]},{RGBColor[0,0,1]}},
              PlotRange->{{-2,2},{-2,2}},
               AspectRatio->Automatic,
               PlotStyle->PointSize[.025],
               DisplayFunction->Identity]; 
               
Do[Show[{g1,g2,Graphics[{Text["Theta=",{.5,1.5},{-1,0}],
                        Text[N[w/8,3],{1.2,1.5},{-1,0}],
                        Text["Pi",{1.85,1.5},{-1,0}]}]},
DisplayFunction->$DisplayFunction],{w,1,16,1}]

This animation shows that the graphs are only at the same place twice.

The End.