More Polar Graphs

Graphs of Form r = a Sinθ   or   r = a Cosθ

A.	Graph r = Sinθ

Clear[r,t,x,y]
r[t_] = Sin[t]; 
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic
     ];

B.	Graph r = Cosθ

Clear[r,t,x,y]
r[t_] =Cos[t]; 
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic
     ];

C.	A bunch all together.

Clear[r1,r2,r3,r4,t,x1,x2,x3,x4,y1,y2,y3,y4]
r1[t_] = Cos[t];
x1[t_] = r1[t] Cos[t];
y1[t_] = r1[t] Sin[t];

r2[t_] = Sin[t]; 
x2[t_] = r2[t] Cos[t];
y2[t_] = r2[t] Sin[t];

r3[t_] = - Cos[t]; 
x3[t_] = r3[t] Cos[t];
y3[t_] = r3[t] Sin[t];

r4[t_] = - Sin[t] ;
x4[t_] = r4[t] Cos[t];
y4[t_] = r4[t] Sin[t];


 
ParametricPlot[
							  																																											{{x1[t],y1[t]},{x2[t],y2[t]},{x3[t],y3[t]},{x4[t],y4[t]}},
							  																																											{t,0,2Pi}, 
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic
     ];

Graphs of Form r = aSin(bθ)+ c Cos(dθ)

r = Sin(θ)+  Cos(θ), etc.

Graph r = Sin(θ) + Cos(θ), r = Sin(θ) - Cos(θ), r = -Sin(θ) + Cos(θ),
 r = -Sin(θ) - Cos(θ) all together

Clear[r1,t,x1,y1,p1]
r1[t_] = (Sin[t] + Cos[t]);
x1[t_] = r1[t] Cos[t];
y1[t_] = r1[t] Sin[t];
 
p1=ParametricPlot[{x1[t],y1[t]},{t,0, 2Pi},
					PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     AspectRatio->Automatic,DisplayFunction->Identity];
Clear[r2,t,x2,y2,p2]
r2[t_] = (Sin[t] - Cos[t]);
x2[t_] = r2[t] Cos[t];
y2[t_] = r2[t] Sin[t];

 
p2=ParametricPlot[{x2[t],y2[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,DisplayFunction->Identity
     ];
Clear[r3,t,x3,y3,p3]
r3[t_] = (- Sin[t] - Cos[t]);
x3[t_] = r3[t] Cos[t];
y3[t_] = r3[t] Sin[t];

 
p3=ParametricPlot[{x3[t],y3[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,DisplayFunction->Identity

     ];
Clear[r4,t,x4,y4,p4]
r4[t_] = (- Sin[t] + Cos[t]);
x4[t_] = r4[t] Cos[t];
y4[t_] = r4[t] Sin[t];

 
p4=ParametricPlot[{x4[t],y4[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,DisplayFunction->Identity
     ];
Show[p1,p2,p3,p4,DisplayFunction->$DisplayFunction];

r = Sin(2θ) -  Cos(2θ), etc.

A.	Graph  r = Sin(2θ) - Cos(2θ)

Clear[r,t,x,y]
r[t_] = (Sin[2t] - Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

B.	Graph	r = Sin(2θ) + Cos(2θ)

Clear[r,t,x,y]
r[t_] = (Sin[2t] + Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

r = 2Sin(θ) +  Cos(2θ), etc.

A. 	Graph r = 2Sin(θ) + Cos(2θ).

Clear[r,t,x,y]
r[t_] = (2Sin[t] + Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic
     ];

B. 	Graph r = Sin(θ) + Cos(2θ).

Clear[r,t,x,y]
r[t_] = (Sin[t] + Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->50
     ];

C.	Graph r = Sin(θ) + Sin(2θ)

Clear[r,t,x,y]
r[t_] = (Sin[t] + Sin[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->50
     ];

D.	Graph r = Sin(2θ) + Cos(θ)

Clear[r,t,x,y]
r[t_] = (Sin[2t] + Cos[t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

E.	Just for fun, show r = Sin(θ) + Cos(2θ) and r = -Sin(θ) + Cos(2θ)
		together.

Clear[r1,t,x1,y1,q1]
r1[t_] = (Sin[t] + Cos[2t]);
x1[t_] = r1[t] Cos[t];
y1[t_] = r1[t] Sin[t];

 
q1=ParametricPlot[{x1[t],y1[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic, PlotPoints->50
				,DisplayFunction->Identity
     ];
Clear[r2,t,x2,y2,q2]
r2[t_] = (- Sin[t] + Cos[2t]);
x2[t_] = r2[t] Cos[t];
y2[t_] = r2[t] Sin[t];

 
q2=ParametricPlot[{x2[t],y2[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->50,
     DisplayFunction->Identity
 ];
Show[q1,q2,DisplayFunction->$DisplayFunction];

Miscellaneous Silly Stuff

A.	r = Sin(.5θ) - Cos(2θ)

Clear[r,t,x,y]
r[t_] = (Sin[.5t] - Cos[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
					PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     AspectRatio->Automatic
     ];

B.	r = Sin¤(θ) + Sin(3θ)

Clear[r,t,x,y]
r[t_] = (((Sin[t])^2) + Sin[3t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

C.	r = Sin¤(θ) + 3Sin(.5θ)

Clear[r,t,x,y]
r[t_] = (((Sin[t])^2) + 3Sin[.5t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

D.	r = Sin¤(θ) - Cos(3θ)

Clear[r,t,x,y]
r[t_] = (((Sin[t])^2) - Cos[3t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];

E.	r = Sin¤(θ) + Sin(2θ)

Clear[r,t,x,y]
r[t_] = (((Sin[t])^2) + Sin[2t]);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

 
ParametricPlot[{x[t],y[t]},{t,0, 2Pi},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
					PlotStyle->{RGBColor[1,0,0]},
     AspectRatio->Automatic,PlotPoints->100
     ];