Periodic Functions

Radian Measure

Manipulate[ParametricPlot[{Sin[x],Cos[x]},{x,0,2π},PlotRange->{-1.1,1.1},ImageSize->550,Prolog-> {Red,PointSize[Medium],Point[{Cos[t],Sin[t]}],Line[{{0,0},{Cos[t],Sin[t]}}],Circle[{0,0},.18,{0,t}],If[pi ,{
If[t>π/2,{Red,Point[{Cos[π/2],Sin[π/2]}],Thick,Line[{{0,0},{Cos[π/2],Sin[π/2]}}],Text["(π/2)",{0.07,0.9}]}],If[t>π,{Red,Point[{Cos[π],Sin[π]}],Thick,Line[{{0,0},{Cos[π],Sin[π]}}],Text["π",{-1.05,0.05}]}],If[t>3π/2,{Red,Point[{Cos[3π/2],Sin[3π/2]}],Thick,Line[{{0,0},{Cos[3π/2],Sin[3π/2]}}],Text["((3  π)/2)",{0.1,-0.9}]}],If[t==2π,{Red,Point[{Cos[2π],Sin[2π]}],Thick,Line[{{0,0},{Cos[2π],Sin[2π]}}],Text["2π",{1.07,0.05}]}]},{
If[t>1,{Red,Point[{Cos[1],Sin[1]}],Line[{{0,0},{Cos[1],Sin[1]}}],Text["1 rad",{0.25,0.1}]}],If[t>2,{Red,Point[{Cos[2],Sin[2]}],Line[{{0,0},{Cos[2],Sin[2]}}],Text["2 rad",{0,0.25}]}],If[t>3,{Red,Point[{Cos[3],Sin[3]}],Line[{{0,0},{Cos[3],Sin[3]}}],Text["3 rad",{-0.25,0.12}]}],If[t>4,{Red,Point[{Cos[4],Sin[4]}],Line[{{0,0},{Cos[4],Sin[4]}}],Text["4 rad",{-0.25,-0.1}]}],If[t>5,{Red,Point[{Cos[5],Sin[5]}],Line[{{0,0},{Cos[5],Sin[5]}}],Text["5 rad",{-0.05,-0.25}]}],If[t>6,{Red,Point[{Cos[6],Sin[6]}],Line[{{0,0},{Cos[6],Sin[6]}}],Text["6 rad",{0.2,-0.15}]}]}]}],{{t,0,"θ"},0,2π},{{pi,False,"π"},{True,False}}]

Exact Value

Isoceles triangle with the equal side lengths being 1 :

Graphics[{EdgeForm[{Thick,Blue}],White,Polygon[{{0,0},{5,0},{5,5}}],Red,Circle[{0,0},1,{0,π/4}],Text[Style[45°,16],{1.2,0.5}],Text[Style[1,16],{5.2,2.5}],Text[Style[1,16],{2.5,-0.2}],Text[Style["(Sqrt[(1^2) + (1^2)])= (Sqrt[2])",16],{2,3}],Black,Text[Style[Column[{"sin(θ)=(opp/hyp)","sin(45°)=(1/(Sqrt[2]))","","cos(θ)=(adj/hyp)","cos(45°)=(1/(Sqrt[2]))","","tan(θ)=(opp/adj)","tan(45°)=1"}],20],{7,2.8}]},ImageSize->600]

Equilateral triangle with side lengths being 2 :

Graphics[{EdgeForm[{Thick,Blue}],White,Polygon[{{0,0},{2,0},{1,(Sqrt[3])}}],Red,Thick,Circle[{0,0},0.15,{0,π/3}],Circle[{1,(Sqrt[3])},0.18,{3π/2-π/6,3π/2}],Text[Style[30°,16],{0.94,1.48}],Text[Style[60°,16],{0.2,0.1}],Text[Style[2,16],{0.4,0.9}],Text[Style[2,16],{1,-0.1}],Text[Style[1,16],{0.5,0.05}],Text[Style[Column[{"(Sqrt[(2^2) - (1^2)])","= (Sqrt[3])"}],16],{1.2,0.8}],Black,Text[Style[Column[{"sin(θ)=(opp/hyp)","sin(60°)=((Sqrt[3])/2)","","cos(θ)=(adj/hyp)","cos(60°)=(1/2)","","tan(θ)=(opp/adj)","tan(60°)=(Sqrt[3])"}],20],{2.2,1}],Text[Style[Column[{"sin(θ)=(opp/hyp)","sin(30°)=(1/2)","","cos(θ)=(adj/hyp)","cos(30°)=((Sqrt[3])/2)","","tan(θ)=(opp/adj)","tan(30°)=(1/(Sqrt[3]))"}],20],{3,1}],Dashed,Blue,Line[{{1,0},{1,(Sqrt[3])}}]},ImageSize->700]

Exact values on the unit circle :

Manipulate[ParametricPlot[{Sin[x],Cos[x]},{x,0,2π},PlotRange->{-1.5,1.5},ImageSize->550,Prolog-> {Black,PointSize[Medium],Point[{Cos[t],Sin[t]}],Thick,Line[{{0,0},{Cos[t],Sin[t]}}],Circle[{0,0},.2,{0,t}],Dashed,Red,Line[{{Cos[t],0},{Cos[t],Sin[t]}}],Line[{{0,0},{Cos[t],0}}],Text[Row[{"(",Cos[t],", ",Sin[t],")"}],{Cos[t]+0.25,Sin[t]+0.08
}],Black,Text[Style[Row[{t 180/π,"°",", ",t}],{Italic,10}],{0.35,0.1}]}],{{t,0,"θ"},0,2π,π/12,Appearance->"Labeled"}]

Symmetry

Manipulate[
ParametricPlot[{Sin[x],Cos[x]},{x,0,2π},PlotRange->{-1.5,1.5},ImageSize->550,Prolog-> {Black,PointSize[Medium],Point[{Cos[t],Sin[t]}],Red,Line[{{0,0},{Cos[t],Sin[t]}}],Circle[{0,0},.2,{0,t}],Dashed,Line[{{Cos[t],0},{Cos[t],Sin[t]}}],Line[{{0,0},{Cos[t],0}}],Black,
If[t≥π/3,{Point[{Cos[π/3],Sin[π/3]}],Thick,Dashing[0.0001],Line[{{0,0},{Cos[π/3],Sin[π/3]}}],Dashed,Purple,Line[{{Cos[π/3],0},{Cos[π/3],Sin[π/3]}}],Line[{{0,0},{Cos[π/3],0}}],Text[Row[{"(",Cos[π/3],", ",Sin[π/3],")"}],{Cos[π/3]+0.2,Sin[π/3]+0.08}],Text["θ",{0.25,0.1}]}],
If[t≥2π/3,{Point[{Cos[2π/3],Sin[2π/3]}],Thick,Dashing[0.0001],Line[{{0,0},{Cos[2π/3],Sin[2π/3]}}],Dashed,Purple,Line[{{Cos[2π/3],0},{Cos[2π/3],Sin[2π/3]}}],Line[{{0,0},{Cos[2π/3],0}}],Text[Row[{"(",Cos[2π/3],", ",Sin[2π/3],")"}],{Cos[2π/3]-0.2,Sin[2π/3]+0.08}],Line[{{Cos[π/3],Sin[π/3]},{Cos[2π/3],Sin[2π/3]}}],Text["θ",{-0.25,0.1}]}],
If[t≥4π/3,{Point[{Cos[4π/3],Sin[4π/3]}],Thick,Dashing[0.0001],Line[{{0,0},{Cos[4π/3],Sin[4π/3]}}],Dashed,Purple,Line[{{Cos[4π/3],0},{Cos[4π/3],Sin[4π/3]}}],Line[{{0,0},{Cos[4π/3],0}}],Text[Row[{"(",Cos[4π/3],", ",Sin[4π/3],")"}],{Cos[4π/3]-0.2,Sin[4π/3]-0.08}],Text["θ",{-0.25,-0.1}]}],
If[t≥5π/3,{Point[{Cos[5π/3],Sin[5π/3]}],Thick,Dashing[0.0001],Line[{{0,0},{Cos[5π/3],Sin[5π/3]}}],Dashed,Purple,Line[{{Cos[5π/3],0},{Cos[5π/3],Sin[5π/3]}}],Line[{{0,0},{Cos[5π/3],0}}],Text[Row[{"(",Cos[5π/3],", ",Sin[5π/3],")"}],{Cos[5π/3]+0.2,Sin[5π/3]-0.08}],Line[{{Cos[4π/3],Sin[4π/3]},{Cos[5π/3],Sin[5π/3]}}],Text["θ",{0.25,-0.1}]}]
}],{{t,0,"θ"},0,2π,π/12,Appearance->"Labeled"}]

({{Quadrant 1, Quadrant 2, Quadrant 3, Quadrant 4}, {sin θ, sin(π-θ)=sin θ, sin(π+θ)=-sin θ, sin(2π-θ)=-sin θ}, {cos θ, cos(π-θ)=-cos θ, cos(π+θ)=-cos θ, cos(2π-θ)=cos θ}, {tan θ, tan(π-θ)=-tan θ, tan(π+θ)=tan θ, tan(2π-θ)=-tan θ}})

As long as the angle is expressed as π±θ or 2π±θ, the trig function (sin, cos or tan) remains the same as does the magnitude. Only the sign of the value changes.

Trigonometric Graphs

Manipulate[Row[{ParametricPlot[{Sin[x],Cos[x]},{x,0,2π},PlotRange->{-1.5,1.5},ImageSize->400,Prolog-> {Red,PointSize[Medium],Point[{Cos[t],Sin[t]}],Red,Line[{{0,0},{Cos[t],Sin[t]}}],Circle[{0,0},.2,{0,t}],Dashed,Line[{{Cos[t],0},{Cos[t],Sin[t]}}],Line[{{0,0},{Cos[t],0}}]
}],Plot[Sin[x],{x,0,2π},ImageSize->400,PlotRange->1,Ticks->{{0,π/4,π/2,3π/4,π,5π/4,3π/2,7π/4,2π},{-1,-0.5,0,0.5,1}},Prolog->{Red,PointSize[Medium],Point[{t,Sin[t]}],Dashed,Line[{{t,0},{t,Sin[t]}}]}]}],{{t,0,"θ"},0,2π,π/12,Appearance->"Labeled"}]