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sin (x)  ;   cos (x): Periódus idő: 2π

plt=Plot[Sin[t],{t,0,2Pi},AspectRatio->Automatic];
PlotStickSin[u_]:=Graphics[{PointSize[0.015],Thickness[0.006],Red,Point[{u,Sin[u]}],Line[{{u,0},{u,Sin[u]}}]}];
PlotStickCos[u_]:=Graphics[{PointSize[0.015],Thickness[0.006],Blue,Point[{u,Cos[u]}],Line[{{u,0},{u,Cos[u]}}]}];
PlotStick1[u_]:=Graphics[{PointSize[0.05],Point[{Cos[u],Sin[u]}],Thickness[0.02],{Red,Line[{{Cos[u],0},{Cos[u],Sin[u]}}]},
{Blue,Line[{{0,0},{Cos[u],0}}]},{Line[{{0,0},{Cos[u],Sin[u]}}]}}]
Manipulate[
Row[
{Show[Graphics[{Text[StringJoin["φ=",ToString[Style[u,Bold]]],{0.1,0.1}]}],
ParametricPlot[{Cos[t],Sin[t]},{t,0,u},AspectRatio->Automatic,
PlotStyle->{Thickness[0.02]}],PlotStick1[u],PlotRange->{{-1.1,1.1},{-1.1,1.1}},AxesOrigin->{0,0},ImageSize->{180,180},Axes->True],Spacer[15],
Show[Plot[{Cos[t],Sin[t]},{t,0,u},AspectRatio->Automatic,PlotStyle->{{Blue,Thickness[0.006]},{Red,Thickness[0.006]}},
Ticks->{Range[0,4Pi,Pi/4],Automatic}],PlotStickSin[u],PlotStickCos[u+0.03],PlotRange->{{0,2Pi},{-1.1,1.1}},AxesOrigin->{0,0},ImageSize->{550,190}]}
],
{{u,0.001,"φ"},0.001,2Pi,Appearance->{"Open","Labeled"}},SaveDefinitions->True]

tg(x)=((sin x)/(cos x)),  ctg(x)=((cos x)/(sin x)): periódus idő: π

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Manipulate[
Plot[fv[x],{x,-dx,dx},AspectRatio->aspectratio,PlotRange->Automatic,ImageSize->{400,Automatic},PlotStyle->Thick,Ticks->{Range[-2Pi,2Pi,Pi/2],Automatic}],{fv,{Sin,Cos, Tan,Cot}},{{dx,2Pi},Pi/3,4Pi },{aspectratio,{Automatic, 1/GoldenRatio}}]

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cos(x)=sin((π/2)+x),

Plot[{Sin[x],Sin[(π/2)+x]},{x,-2 π,2 π},AspectRatio->Automatic,
Ticks->{{-2Pi,-Pi,0,Pi,2Pi},Automatic},PlotRange->{-1.1,1.1},PlotStyle->{Red, Blue}]

ctg(x)=tg((π/2)-x)

Plot[{Tan[x],Tan[(π/2)-x]},{x,- π, π},AspectRatio->Automatic,
Ticks->{{-2Pi,-Pi,0,Pi,2Pi},Automatic},PlotRange->{-3,3},PlotStyle->{Red, Blue}]

ctg(x)=-tg(x-(π/2))

Plot[{Tan[x],-Tan[x-(π/2)]},{x,- π, π},AspectRatio->Automatic,
Ticks->{{-2Pi,-Pi,0,Pi,2Pi},Automatic},PlotRange->{-3,3},PlotStyle->{Red, Blue}]

Fontos összehasonlítás: (x^3) és tg(x)

Manipulate[
Plot[{x^3,Tan[x]},{x,p[[1]]-d,p[[1]]+d},AspectRatio->Automatic,PlotRange->{{p[[1]]-d,p[[1]]+d},{p[[2]]-2d,p[[2]]+2d}},PlotStyle->{{Thick,Black}, {Thick,Red}}],
{{p,{0,0}},{0,0},{4,4},Slider2D},{{d,3},0.2,5},ControlPlacement->Left]

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Sin (c x)

Manipulate[Plot[Sin[c x], {x, 0, 2 Pi}, PlotStyle -> {Thickness[0.02]},
  PlotLabel -> (TraditionalForm[Sin[c x]]), Ticks -> {Range[0, 2 Pi, Pi/4], Automatic}], {c, 1, 5}]

Ha a T periódus idő adott akkor a képlet

sin (((2π)/T)t)

Manipulate[Plot[Sin[((2π)/T)t], {t, 0, 10}, PlotStyle -> {Thickness[0.02]},
  PlotLabel -> (TraditionalForm[Sin[((2π)/T)t]])], {T, 1, 10}]

A kezdeti fázisszög (Subscript[φ, 0]) bevezetésével az általános sin-cos rezgés alakja

sin ((Subscript[φ, 0])+((2π)/T)t)

Manipulate[Plot[Sin[φ0+((2π)/T)t], {t, 0, 10}, PlotStyle -> {Thickness[0.02]},
  PlotLabel -> (TraditionalForm[Sin[(Subscript[φ, 0])+((2π)/T)t]])], {T, 1, 10},{{φ0,0,"(Subscript[φ, 0])"},0,2T}]

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Szuperpozíció: A Sin (c x) + B Sin (d (x+(Subscript[x, 0]))), interferencia, lebegés

Manipulate[Plot[A Sin[c  x]+B Sin[d  (x+x0)],{x,0,2Pi},PlotLabel->TraditionalForm[A Sin[c  x]+B Sin[d  (x+x0)]],PlotRange->{-3,3},
PlotPoints->30],
{{c,50},1,200,1,Appearance->"Labeled",ImageSize->Small},{{A,1},0,2,Appearance->"Labeled",ImageSize->Small},{{d,52},1,200,1,Appearance->"Labeled",ImageSize->Small},{{B,1},0,1,Appearance->"Labeled",ImageSize->Small},{{x0,0,"(Subscript[x, 0])"},-Pi,Pi,Appearance->"Labeled",ImageSize->Small},ControlPlacement->Left]

Hangok

Manipulate[
Play[A Sin[c 2Pi x]+B Sin[d 2Pi  x],{x,0,4},SampleRate->10000],{{c,440},20,10000,0.5},{{d,880},20,10000,0.5},{{A,1},0,2,0.1},{{B,1},0,2,0.1}]

Hullámok, mint trigonometrikus függvények összegei