SECTION FIVE:
Experimenting With
Polar Graphs

Mathematica allows you to experiment easily with a wide variety of polar graphs.  Here are a few examples to help you get started.

Graph r = 2Sinθ + Cos2θ.

Clear[r,t,x,y,p, tex]
r[t_] = (Sin[t] + Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
 
p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,
     DisplayFunction->Identity];
     
tex = Graphics[Text["r = Sin[t] + Cos[2t]",
								Scaled[{.85,.97}]]];  
Show[p,tex,
	PlotRange->All,
	DisplayFunction->$DisplayFunction];

Graph r = Sinθ + Sin2θ.

Clear[r,t,x,y,p, tex]
r[t_] = (Sin[t] + Sin[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
 
p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     PlotRange->{{-1.5,1.8},{-.4,2}},
     PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,
     DisplayFunction->Identity];
     
tex = Graphics[Text["r = Sin[t] + Sin[2t]",
								Scaled[{.7,.97}]]];  
Show[p,tex,
	DisplayFunction->$DisplayFunction];

Graph r = Sin¤θ + Cos3θ.

Clear[r,t,x,y,p, tex]
r[t_] = (Sin[t])^2 + Cos[3t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
 
p = ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     PlotRange->All,
     PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,
     DisplayFunction->Identity];
     
tex = Graphics[Text["r = (Sin[t])^2 + Cos[3t]",
								Scaled[{.85,.97}]]];  
Show[p,tex,DisplayFunction->$DisplayFunction];

Try some of your own.  One caution, however:  many people find  once they get started creating these graphs that it is very difficult to stop.  Another note:  to say you are really doing mathematics you should constantly be asking yourself why  the graphs look the way they do.  It might help to think about the same function graphed in rectangular coordinates.