ParametricPlot[{Cos[t],Sin[t]},{t,0,Pi},
PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{Sin[t],Cos[t]},{t,0,Pi},
PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{t Cos[t],t Sin[t]},{t,0,10},
PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{3+2 Cos[t], 2 + 2 Sin[t]},{t,0,2Pi},PlotRange->{-1,5},
PlotStyle->{Black,Thickness[.015]}]

circle=ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi}];
arrows=Graphics[{Thickness[.01],Arrow[{{1,0},{1,1}}],Arrow[{{Sqrt[2]/2,Sqrt[2]/2},{0,Sqrt[2]}}]}];
points=Graphics[{PointSize[.04],Point[{1,0}],Point[{Sqrt[2]/2,Sqrt[2]/2}]}];
Show[circle,arrows,points,PlotRange->{{-1,1},{-1,2}}]
lines=ParametricPlot[{{1,t},{Sqrt[2]/2 (1-t),Sqrt[2]/2(1+t)}},{t,-1,2},PlotStyle->Directive[Red,Thick]];
Show[circle,lines,PlotRange->{{-1,1},{-1,2}}]

circle=ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi}];
arrows=Graphics[{Thickness[.01],Arrow[{{1,0},{0,0}}],Arrow[{{Sqrt[2]/2,Sqrt[2]/2},{0,0}}]}];
points=Graphics[{PointSize[.04],Point[{1,0}],Point[{Sqrt[2]/2,Sqrt[2]/2}]}];
Show[circle,arrows,points,PlotRange->{{-1,1},{-1,1}}]
parabolas=ParametricPlot[
{
{1-t^2/2,t},
{Sqrt[2]/2 (1 -t -t^2/2), Sqrt[2]/2 (1+t-t^2/2)}
},
{t,-2,2},PlotStyle->{Directive[Red,Thick],Directive[Orange,Thick]}];
Show[circle,parabolas]

x^3+x

ParametricPlot[{t + Sin[3t],t+Sin[5t]},{t,0,4 Pi},
PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{Cos[t],Sin[3t]},{t,0,2 Pi},
PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{2Cos[10t] + Cos[23t],2Sin[10t]+Sin[23t]},{t,0,2 Pi},
PlotStyle->{Black,Thickness[.01]}]

ParametricPlot[{
Cos[t] ( Sin[t] Sqrt[Abs[Cos[t]]]/(Sin[t] + 7/5) -2 Sin[t] +2),
Sin[t]( Sin[t] Sqrt[Abs[Cos[t]]]/(Sin[t] + 7/5) -2 Sin[t] +2)
},
{t,0,2 Pi},
PlotStyle->{Black,Thickness[.015]}]

parametricLine=ParametricPlot[{2+4t^3+4t,6+3t^3+3t},{t,0,1},
PlotStyle->{Black,Thick}];
x=NSolve[t^3+t==1,t][[All,1,2]][[3]];
arrows=Graphics[{Thickness[.01],Red,
Arrow[{{2,6},{2+4/3,6+3/3}}],
Arrow[{{6,9},{6+(4+12x^2)/3,9+(3+9x^2)/3}}
],Arrow[{{Sqrt[2]/2,Sqrt[2]/2},{0,0}}]}];
points=Graphics[{PointSize[.03],Point[{2,6}],Point[{6,9}]}];
Show[parametricLine,arrows,points]

ParametricPlot[{2 (t - Sin[t]),2 (1 - Cos[t])},{t,-2Pi,4 Pi},
PlotStyle->{Black,Thickness[.015]}]

PolarPlot[1,{t,0,2Pi},
Epilog->{
Black,PointSize[.04],Point[{-1,0}],
Text[Style[{1,Pi},FontSize->18],
Offset[{40,-20},{-1,0}]],
Black,PointSize[.04],Point[{0,1}],
Text[Style[{1,Pi/2},FontSize->18],
Offset[{40,-35},{0,1}]],
Black,PointSize[.04],Point[{-Sqrt[2]/4,Sqrt[2]/4}],
Text[Style[{1/2,3Pi/4},FontSize->18],
Offset[{-40,30},{-Sqrt[2]/4,Sqrt[2]/4}]]
}
]

PolarPlot[1,{t,0,2Pi},PlotRange->{{-2,2},{-2,2}},
Epilog->{
Black,PointSize[.04],Point[{-1,1}],
Text[Style[{Sqrt[2],3Pi/4},FontSize->18],
Offset[{40,40},{-1,1}]],
Black,PointSize[.04],Point[{Sqrt[3],1}],
Text[Style[{2,Pi/6},FontSize->18],
Offset[{-40,-35},{Sqrt[3],1}]]
}
]

PolarPlot[3,{t,0,2Pi},PlotStyle->{Black,Thickness[.015]}]

ParametricPlot[{t,t},{t,-5,5},PlotStyle->{Black,Thickness[.015]}]

RegionPlot[x>=0&&y>=0&&x^2+y^2<=1,{x,-1,1},{y,-1,1}]

RegionPlot[x<=0&&y<=0&&1<=x^2+y^2<=4,{x,-2,2},{y,-2,2}]

PolarPlot[2Cos[t],{t,0,2 Pi},PlotStyle->{Black,Thickness[.015]}]

PolarPlot[1+Cos[t],{t,0,2 Pi},PlotStyle->{Black,Thickness[.015]}]

PolarPlot[Cos[2t],{t,0,2 Pi},PlotStyle->{Black,Thickness[.015]}]

ContourPlot[{(x-1)^2  + y^2 ==1, y - Sqrt[3]/2 == -1/Sqrt[3] (x-3/2)},{x,-.5,2.5},{y,-1.5,1.5},
ContourStyle->Directive[Black,Thickness[.015]]]

RegionPlot[{x^2+y^2<=1,x^2+y^2<=1&&2Pi/10<=ArcTan[y/x]<=3Pi/10&&y>=0,
x^2+y^2<=1&&3Pi/10<=ArcTan[y/x]<=4Pi/10&&x>=0,
x^2+y^2<=1&&4Pi/10<=ArcTan[y/x]<=5Pi/10&&y>=0
},{x,-1,1},{y,-1,1}]

PolarPlot[{3 Sin[t],1 + Sin[t]},{t,0,2 Pi},PlotStyle->Thickness[.01]]

Integrate[1/2 (3 Sin[t])^2,{t,Pi/6,5Pi/6}]
Integrate[1/2 (1 + Sin[t])^2,{t,Pi/6,5Pi/6}]

Show[
ParametricPlot[{3Cos[t]+1,3Sin[t]-1},{t,0,4Pi},
PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0,.1,0,0,.1}],Arrow[x]},
ListPlot[{{1,-1},{4,-1}},PlotStyle->{Black,PointSize[.04]}]
]

Show[ParametricPlot[{{Cos[t]^3,Sin[t]^3}},{t,0,2 Pi}],
ContourPlot[y + 3 Sqrt[3]/8==Sqrt[3] (x-1/8)
,{x,-1,1},{y,-1,1}]]

Sin[-Pi/3]^3