{NotebookFileName[],DateString[]}

Sound demo

functions8Dir=FileNameDrop[NotebookFileName[]]

A440=440

middleC=(2^(-9/12))*440.0

((2^(1/12))-1)*middleC

Clear[envelope];
envelope[
t_,
l_:5
]:=
Exp[-t/l]-Exp[-t/0.1]

Plot[envelope[t],{t,0,20}
,PlotRange->{0,All}
]

Play[envelope[t,10]Sin[2π*A440*t],{t,0,20}]

Hz441=Play[envelope[t,10]Sin[2π*(440+1)*t],{t,0,20}]

Export[FileNameJoin[{functions8Dir,"Hz441.wav"}],Hz441]

Hz439=Play[envelope[t,10]Sin[2π*(A440-1)*t],{t,0,20}]

Export[FileNameJoin[{functions8Dir,"Hz439.wav"}],Hz439]

Hz441p439=Play[envelope[t,10](Sin[2π*(A440-1)*t]+Sin[2π*(A440+1)*t]),{t,0,20}]

Export[FileNameJoin[{functions8Dir,"Hz441p439.wav"}],Hz441p439]

Plot[
Sin[2π*(A440-1)*t]+Sin[2π*(A440+1)*t]-2Sin[2π*A440*t]Cos[2π t],
{t,0,5}
]

Plot[Evaluate@{
2Sin[2π*A440*t]Cos[2π t],
2Cos[2π t]
},
{t,0,1}
,PlotLegends->"Expressions"
,PlotPoints->500
,PlotStyle->{Automatic,Thick}
,TicksStyle->{
Directive[Bold,White,FontFamily->"Arial",FontSize->14],
Automatic
}
]

Mouse liver mass

styletext[
t_
]:=
Style[t,Bold,FontFamily->"Arial",FontSize->14]

m1[
t_,
mn_:1.5,
mx_:2.25,
tmn_:20,
tmx_:8
]:=
((mx+mn)/2)+(((mx-mn)/2))Cos[((2π)/24)(t-tmx)]

Block[{mn=1.5,mx=2.25,tmn=20,tmx=8},
Plot[m1[t],{t,0,24}
,PlotRange->{0,mx+0.2}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{0,mn},{24,mn}}],
Text[styletext[ToString[mn]<>" g"],{12,mn},{0,1}],
Line[{{0,mx},{24,mx}}],
Text[styletext[ToString[mx]<>" g"],{12,mx},{0,-1}]
}
]
]

Block[{mn=1.5,mx=2.25,tmn=20,tmx=8},
Plot[m1[t],{t,0,24}
,PlotRange->{0,mx+0.2}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{0,mn},{24,mn}}],
Text[styletext[ToString[mn]<>" g"],{12,mn},{0,1}],
Line[{{0,mx},{24,mx}}],
Text[styletext[ToString[mx]<>" g"],{12,mx},{0,-1}],
PointSize[Large],
Point[{{tmn,mn}}],
Text[styletext[ToString[tmn]<>":00"],{tmn,mn},{0,1}],
Point[{{tmx,mx}}],
Text[styletext[ToString[tmx]<>":00"],{tmx,mx},{0,-1}]
}
]
]

Block[{mn=1.5,mx=2.25,tmn=20,tmx=8},
Plot[m1[t],{t,0,24}
,PlotRange->{0,mx+0.2}
(*,PlotStyle->Opacity[0]*)
,Epilog->{
Dashed,
Line[{{0,mn},{24,mn}}],
Text[styletext[ToString[mn]<>" g"],{12,mn},{0,1}],
Line[{{0,mx},{24,mx}}],
Text[styletext[ToString[mx]<>" g"],{12,mx},{0,-1}],
PointSize[Large],
Point[{{tmn,mn}}],
Text[styletext[ToString[tmn]<>":00"],{tmn,mn},{0,1}],
Point[{{tmx,mx}}],
Text[styletext[ToString[tmx]<>":00"],{tmx,mx},{0,1}]
}
]
]

Solve[{1.5==-A+D,2.25==A+D},{A,D}]

Sketching a sin function

f1[
x_
]:=
3Sin[6π(x-2)]-1.5

Block[{mn=-4.5,mx=1.5,mid,xmn,xmx,x,xmin=0,xmax=3},
mid=((mn+mx)/2);
xmn=x/.Solve[6π(x-2)==-(π/2),x];
xmx=x/.Solve[6π(x-2)==(π/2),x];
Plot[f1[x],{x,xmin,xmax}
,PlotRange->{mn-0.4,mx+0.4}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{xmin,mn},{xmax,mn}}],
Text[styletext[ToString[mn]],{((xmin+xmax)/2),mn},{0,1}],
Line[{{xmin,mx},{xmax,mx}}],
Text[styletext[ToString[mx]],{((xmin+xmax)/2),mx},{0,-1}],
Line[{{xmin,mid},{xmax,mid}}],
Text[styletext[ToString[mid]],{((xmin+xmax)/2),mid},{0,-1}]
}
]
]

Block[{mn=-4.5,mx=1.5,mid,xmn,xmx,x,xmin=0,xmax=3},
mid=((mn+mx)/2);
xmn=x/.Solve[6π(x-2)==-(π/2),x];
xmx=x/.Solve[6π(x-2)==(π/2),x];
Plot[f1[x],{x,xmin,xmax}
,PlotRange->{mn-0.4,mx+0.4}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{xmin,mn},{xmax,mn}}],
Text[styletext[ToString[mn]],{((xmin+xmax)/2),mn},{0,1}],
Line[{{xmin,mx},{xmax,mx}}],
Text[styletext[ToString[mx]],{((xmin+xmax)/2),mx},{0,-1}],
Line[{{xmin,mid},{xmax,mid}}],
Text[styletext[ToString[mid]],{((xmin+xmax)/2),mid},{0,-1}],
PointSize[Large],
Point[{{2,mid}}],
Text[styletext["x = 2"],{2,mid},{0,1}],
}
]
]

Block[{mn=-4.5,mx=1.5,mid,xmn,xmx,x,xmin=0,xmax=3},
mid=((mn+mx)/2);
xmn=x/.Solve[6π(x-2)==-(π/2),x];
xmx=x/.Solve[6π(x-2)==(π/2),x];
Plot[f1[x],{x,xmin,xmax}
,PlotRange->{mn-0.4,mx+0.4}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{xmin,mn},{xmax,mn}}],
Text[styletext[ToString[mn]],{((xmin+xmax)/2),mn},{0,1}],
Line[{{xmin,mx},{xmax,mx}}],
Text[styletext[ToString[mx]],{((xmin+xmax)/2),mx},{0,-1}],
Line[{{xmin,mid},{xmax,mid}}],
Text[styletext[ToString[mid]],{((xmin+xmax)/2),mid},{0,-1}],
PointSize[Large],
Text[styletext["x = 2"],{2,mid},{0,1}],
Table[
Point[{{2+(1/3)n,mid}}],
{n,3(xmin-2),3(xmax-2)}
]
}
]
]

Block[{mn=-4.5,mx=1.5,mid,xmn,xmx,x,xmin=0,xmax=3},
mid=((mn+mx)/2);
xmn=x/.Solve[6π(x-2)==-(π/2),x];
xmx=x/.Solve[6π(x-2)==(π/2),x];
Plot[f1[x],{x,xmin,xmax}
,PlotRange->{mn-0.4,mx+0.4}
,PlotStyle->Opacity[0]
,Epilog->{
Dashed,
Line[{{xmin,mn},{xmax,mn}}],
Text[styletext[ToString[mn]],{((xmin+xmax)/2),mn},{0,1}],
Line[{{xmin,mx},{xmax,mx}}],
Text[styletext[ToString[mx]],{((xmin+xmax)/2),mx},{0,-1}],
Line[{{xmin,mid},{xmax,mid}}],
Text[styletext[ToString[mid]],{((xmin+xmax)/2),mid},{0,-1}],
PointSize[Large],
Text[styletext["x = 2"],{2,mid},{0,1}],
Table[
Point[{{2+(1/3)n,mid}}],
{n,3(xmin-2),3(xmax-2)}
]
,Red,
Table[
Point[{{2+(1/6)+(1/3)n,mid}}],
{n,3(xmin-2),3(xmax-2)}
]
}
]
]

Block[{mn=-4.5,mx=1.5,mid,xmn,xmx,x,xmin=0,xmax=3},
mid=((mn+mx)/2);
xmn=x/.Solve[6π(x-2)==-(π/2),x];
xmx=x/.Solve[6π(x-2)==(π/2),x];
Plot[f1[x],{x,xmin,xmax}
,PlotRange->{mn-0.4,mx+0.4}
(*,PlotStyle->Opacity[0]*)
,Epilog->{
Dashed,
Line[{{xmin,mn},{xmax,mn}}],
Text[styletext[ToString[mn]],{((xmin+xmax)/2),mn},{0,1}],
Line[{{xmin,mx},{xmax,mx}}],
Text[styletext[ToString[mx]],{((xmin+xmax)/2),mx},{0,-1}],
Line[{{xmin,mid},{xmax,mid}}],
Text[styletext[ToString[mid]],{((xmin+xmax)/2),mid},{0,-1}],
PointSize[Large],
Text[styletext["x = 2"],{2,mid},{0,1}],
Table[
Point[{{2+(1/3)n,mid}}],
{n,3(xmin-2),3(xmax-2)}
]
,Red,
Table[
Point[{{2+(1/6)+(1/3)n,mid}}],
{n,3(xmin-2),3(xmax-2)}
]
}
]
]

Identities

((Tan[θ])^2)+1==((Sec[θ])^2)//Simplify

((Cot[θ])^2)+1==((Csc[θ])^2)//Simplify

Sin[x+y]//TrigExpand

Cos[x+y]//TrigExpand

ticks=Table[
{x,x},
{x,-4π,4π,π}
]

Plot[{
((Sin[(θ/2)])^2),((Cos[(θ/2)])^2)
},{θ,-4π,4π}
,Ticks->{ticks,Automatic}
,PlotLegends->"Expressions"
,AspectRatio->1/2
]

Sin[π/12]

Sin[π/12]==(1/2)(Sqrt[2-(Sqrt[3])])//Simplify

Plot[(1/2)Sin[((π x)/3)],{x,-6,6}]

AM waves

ω1=50π

Plot[Sin[ω1 t],{t,0,1}
,PlotLegends->"Expressions"
,PlotLabel->styletext[Sin[ω1 t]]
]

env1[
t_
]:=
Exp[-2t]

Plot[Evaluate@{
env1[t]Sin[50π t],
env1[t]
},{t,0,1}
,PlotLegends->"Expressions"
,PlotLabel->styletext[Exp[-2t]Sin[50π t]]
]

env2[
t_
]:=
Cos[6t]

Plot[Evaluate@{
env2[t]Sin[50π t],
env2[t]
},{t,0,1}
,PlotLegends->"Expressions"
,PlotLabel->styletext[env2[t]Sin[50π t]]
]

ArcTan[(Sqrt[3])]

Play[envelope[t,10]Sin[50π t],{t,0,20}]

Answers to problems

ArcSin[((Sqrt[3])/2)]

Solve[Sin[θ]==((Sqrt[3])/2),θ]

ArcCos[((Sqrt[3])/2)]

ArcTan[(Sqrt[3])]

ArcTan[Tan[((2π)/3)]]

Circle diagrams for inv trig functions

Block[{x,y=((Sqrt[3])/2),arcr=0.2,θ},
x=(Sqrt[1-(y^2)]);
θ=ArcSin[y];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[0.03],Thick,
Line[{{-1.5,y},{1.5,y}}],
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(x,y)",{x,y},{-1.5,-1.5}],
Text[styletext@"arcsin(y)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]

Block[{x,y=-((Sqrt[3])/2),arcr=0.2,θ},
x=(Sqrt[1-(y^2)]);
θ=ArcSin[y];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[0.03],Thick,
Line[{{-1.5,y},{1.5,y}}],
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(x,y)",{x,y},{-1.5,-1.5}],
Text[styletext@"-arcsin(y)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]

Block[{x=((Sqrt[3])/2),y,arcr=0.2,θ},
y=(Sqrt[1-(x^2)]);
θ=ArcCos[x];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[0.03],Thick,
Line[{{x,-1.5},{x,1.5}}],
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(x,y)",{x,y},{-1.5,-1.5}],
Text[styletext@"arccos(x)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]

Block[{x=-((Sqrt[3])/2),y,arcr=0.2,θ},
y=(Sqrt[1-(x^2)]);
θ=ArcCos[x];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[0.03],Thick,
Line[{{x,-1.5},{x,1.5}}],
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(x,y)",{x,y},{-1.5,-1.5}],
Text[styletext@"arccos(x)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]

Block[{x,y=1,arcr=0.2,θ},
x=1;
θ=ArcTan[y];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,PlotRange->{Automatic,{Min[-1,1.2y],Max[1,1.2y]}}
,Epilog->{
PointSize[0.03],Thick,
{Dashed, Line[{{1,0},{1,y}}]},
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(1,y)",{1,y},{1.5,-1.5}],
Text[styletext@"arctan(y)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]

Block[{x,y=-1,arcr=0.2,θ},
x=1;
θ=ArcTan[y];
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,PlotRange->{Automatic,{Min[-1,1.2y],Max[1,1.2y]}}
,Epilog->{
PointSize[0.03],Thick,
{Dashed, Line[{{1,0},{1,y}}]},
Line[{{0,0},{x,y}}],
Circle[{0,0},arcr,{0,θ}],
Point[{{x,y}}],
Text[styletext@"(1,y)",{1,y},{1.5,1.5}],
Text[styletext@"-arctan(y)",arcr{Cos[(θ/2)],Sin[(θ/2)]},{-1.1,0}]
}
]
]