SECTION ONE:
The Basics

Introduction

If a polar coordinate system and a rectangular coordinate system are 
		superimposed in the standard manner any point P can be indicated 
		either by rectangular coordinates, P(x,y), or by polar coordinates, P(r,θ), 
		where the relationship between (x,y) and (r,θ) is given by:

x = rCosθ

y = rSinθ.

A polar equation is an equation in the variable r and θ, where (r,θ) represents coordinates of a point in a polar coordinate plane, for example
																																																																																	r = Cos2θ,												
																																																																																	r = 1 - 2Sinθ,or
																																																																																	r = 1 + Cosθ. 
																						
		A graph of a polar equation in a polar coordinate plane is the set of all points 
		in the plane having at least one pair of polar coordinates that satisfy the 
		equation.

Graphing Polar Equations

Mathematica uses the equations 	x = rCosθ  and y = rSinθ  to graph a polar 
		equation. For example, the equation 
r = 1 - 2Sinθ  is graphed using the system 			
																																																																																x = (1 - 2Sinθ )Cosθ 																																							
																																																																																y = (1 - 2Sinθ )Sinθ .

Let's look at how Mathematica does this:

Clear[r,t,x,y,g]
r[t_] = 1 - 2Sin[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Clear[lim]
lim = ParametricPlot[{x[t],y[t]},{t,0,2Pi},
    	PlotStyle->{{RGBColor[1,0,0],Thickness[0.0053]}},
    	PlotRange->{{-2,2},{-3.2,1.3}},
    	Axes->True,
     	AspectRatio->Automatic,
		DisplayFunction->Identity];

g = Graphics[Text["r = 1 - 2Sint",{1,.5}]];

Show[g,lim,PlotRange->{{-2,2},{-3.2,1.3}},
    	Axes->True,
     	AspectRatio->Automatic];

Now let's look at the same graph with a polar grid background.  This is just
		 to give an idea of what the graph would look like on
polar graph paper.  It won't be used on subsequent graphs.

Clear[r,t,x,y,lim2]
r[t_] = 1 - 2Sin[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

lim2 = ParametricPlot[{x[t],y[t]},{t,0,2Pi},
    	PlotStyle->{{RGBColor[1,0,0],Thickness[0.01]}},
    	Axes->False,
     	AspectRatio->Automatic,
     	DisplayFunction->Identity];

plrgrd = Graphics[{
			Circle[{0,0},.5],Circle[{0,0},1],
			Circle[{0,0},1.5],Circle[{0,0},2],
			Line[{{0,0},{4,0}}],
			Text["r = 1 - 2Sint",{3,2}],
			Text["Polar Axis",{3,.2}],
			Text["0.5",{0.6,.1}],Text["1",{1.1,.1}],
				Text["1.5",{1.6,.1}],Text["2",{2.1,.1}]},
			AspectRatio->Automatic];

Show[plrgrd,lim2];