Etymology of Trig Functions

Development

ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi}]

Graphics[Line[{{0,0},{Cos[Pi/6],Sin[Pi/6]}}]]

Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi}],Graphics[Line[{{0,0},{Cos[Pi/6],Sin[Pi/6]}}]]]

Manipulate[Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi}],Graphics[{Opacity[1],Red,Line[{{0,0},{Cos[theta],Sin[theta]}}]}]],{{theta,1},0,2 Pi}]

Manipulate[Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi}],Graphics[{Opacity[1],Red,Line[{{0,0},{Cos[theta],Sin[theta]}}],Line[{{Cos[theta],0},{Cos[theta],Sin[theta]}}]}]],{{theta,1},0,2 Pi}]

Manipulate[{Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->Tiny,Ticks->{{-1,1},{-1,1}}],Graphics[{Opacity[1],Red,Line[{{0,0},{Cos[theta],Sin[theta]}}],Line[{{Cos[theta],0},{Cos[theta],Sin[theta]}}]}]],Show[Plot[Sin[t],{t,-2 Pi, 2 Pi},AspectRatio->1/(4 Pi),Ticks->{{-2 Pi, 2 Pi},{-1,1}},ImageSize->Large]]},{{theta,-2 Pi},-2Pi,2 Pi}]

Straight lines

point - slope form

ptsl[m_,p_,x_]:=m x+ p[[2]]-m p[[1]]

ptsl[m_,p_,x_]:=m (x-p[[1]])+ p[[2]]//Simplify

ptsl[3,{1,2},x]

two-point form

tpl[p_,q_,x_]:=(p[[2]]-q[[2]])/(p[[1]]-q[[1]])(x-p[[1]])+p[[2]]//Simplify

tpl[{1,2},{0,-1},x]

tpl[{0,-1},{1,2},x]

tpl[{1,4},{5,2},x]

Secant lines

(1.5)^2-1/5

Plot[x^3-x,{x,-1.5,1.5},AspectRatio->4.1/3]

cu[x_]:=x^3-x;cud[x_]:=cu'[x]

tpl[{a,cu[a]},{b,cu[b]},x]

Manipulate[
Plot[
{cu[x],
tpl[{a,cu[a]},{b,cu[b]},x]
},
{x,-1.5,1.5},PlotRange->{{-1.5,1.5},{-4.2,4.2}},PlotStyle->{Blue,Red},
AspectRatio->4.1/3
],
{{a,-1},-1.5,1.5,Appearance->"Labeled"},{{b,.5},-1.5,1.5,Appearance->"Labeled"},
SaveDefinitions->True
]

ptsl[cud[1],{1,cu[1]},x]

Manipulate[
Plot[
{cu[x],
If[
a==b,
ptsl[cud[a],{a,cu[a]},x],
tpl[{a,cu[a]},{b,cu[b]},x]
]
},
{x,-1.5,1.5},PlotRange->{{-1.5,1.5},{-2.2,2.2}},PlotStyle->{Blue,Red},
AspectRatio->4.4/3
],
{{a,-1},-1.5,1.5},{{b,.5},-1.5,1.5},
SaveDefinitions->True
]

Manipulate[
Show[
Plot[
{cu[x],
If[
a==b,
ptsl[cud[a],{a,cu[a]},x],
tpl[{a,cu[a]},{b,cu[b]},x]
]
},
{x,-1.5,1.5},PlotRange->{{-1.5,1.5},{-2.2,2.2}},PlotStyle->{Blue,Red},
AspectRatio->4.4/3
],
Graphics[
{Black,
PointSize[.02],
Point[{a,cu[a]}],
Point[{b,cu[b]}]
}
]
],
{{a,-1},-1.5,1.5},{{b,.5},-1.5,1.5},
SaveDefinitions->True
]

Tangent lines

Manipulate[
Show[
Plot[
{cu[x],
ptsl[cud[a],{a,cu[a]},x]
},
{x,-1.5,1.5},PlotRange->{{-1.5,1.5},{-2.2,2.2}},PlotStyle->{Blue,Red},
AspectRatio->4.4/3
],
Graphics[
{Black,
PointSize[.02],
Point[{a,cu[a]}],
}
]
],
{{a,-1},-1.5,1.5},
SaveDefinitions->True
]

Not a tangent line

csp[x_]:=Piecewise[{{-(x-1/2)^2+7/4,x<0},{-(x+1/2)^2+7/4,x≥0}}]

csp[0]

csp[1]

csp'[x]

Plot[
csp[x],
{x,-1.5,1.5},
PlotRange->{{-1.5,1.5},{0,2.2}},
PlotStyle->{Red,Blue},
AspectRatio->2.2/3
]

Manipulate[
Show[
Plot[
{csp[x],ptsl[m,{0,1.5},x]
},
{x,-1.5,1.5},
PlotRange->{{-1.5,1.5},{0,3}},
PlotStyle->{Blue,Red},
AspectRatio->1,ImageSize->Small
]
],
{{m,0},-1,1},
SaveDefinitions->True
]

Hugging Line

hug[x_]:=Piecewise[{{-(x-1/2)^2+7/4,x<0},{x+3/2,1>=x≥0},{(x-1/2)^2+9/4,x≥1}}]

hug[x]//TraditionalForm//TeXForm

Plot[hug[x],{x,-2,4}]

Plot[hug[x],{x,-1,2},PlotRange->{{-1,2},{-1,4}},
PlotStyle->{Red,Blue},
AspectRatio->4/3,PlotPoints->200,ImageSize->Small,Ticks->{{-1,0,1,2},{-1,0,1,2,3,4}}]

Manipulate[
Show[
Plot[{hug[x],ptsl[hug'[a],{a,hug[a]},x],hug'[x]},{x,-1,2},PlotRange->{{-1,2},{-1,4}},
PlotStyle->{Red,Blue,Green},
AspectRatio->5/3,PlotPoints->200,ImageSize->200,Ticks->{{-1,0,1,2},{-1,0,1,2,3,4}}
],
Graphics[
{Black,
PointSize[.02],
Point[{a,hug[a]}],
}
]
],
{{a,-1/2},-1,2},
SaveDefinitions->True
]

Sine

Manipulate[Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->Small,PlotRange->{{-1,1},{-1.1,1.1}}],Graphics[{Opacity[1],Blue,Line[{{0,0},{Cos[theta],Sin[theta]}}],Red,Line[{{Cos[theta],0},{Cos[theta],Sin[theta]}}]}]],{{theta,0},0,2 Pi}]

Manipulate[Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->Medium,Ticks->{{-1,1,2,3,4,5,6},{-1,1}},PlotRange->{{-1,2 Pi},{-1.1,1.1}}],Graphics[{Opacity[1],Blue,Line[{{0,0},{Cos[theta],Sin[theta]}}],Red,Line[{{Cos[theta],0},{Cos[theta],Sin[theta]}}],Green,Line[{{theta,0},{theta,Sin[theta]}}]}],Plot[Sin[t],{t,-1, 2 Pi},AspectRatio->1/(4 Pi),ImageSize->Large]],{{theta,0},0,2 Pi}]

Manipulate[Show[ParametricPlot[{ Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->Small,PlotRange->{{-1,1},{-1.1,1.1}}],Graphics[{Opacity[1],Blue,Line[{{0,0},{Cos[theta],Sin[theta]}}],Line[{{0,0},{Cos[theta],-Sin[theta]}}],Red,Line[{{Cos[theta],-Sin[theta]},{Cos[theta],Sin[theta]}}]}]],{{theta,0},0,2 Pi}]

Tangent

whichpiece[theta_]:=If[(Pi/2<theta<3Pi/2),-1,1]

Manipulate[
Show[
ParametricPlot[
{Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->100,PlotRange->{{-1,1},{-3,3}},Ticks->{{-1,1},{-3,-2,-1,1,2,3}},AspectRatio->3
],
Graphics[
{
Opacity[1],
Blue,
Line[{{0,0},{1,Tan[theta]}}],
Red,
Line[{{1,0},{1,Tan[theta]}}]
}
]
],
{{theta,Pi/6},-Pi/2,3  Pi/2},SaveDefinitions->True
]

Manipulate[
Show[
ParametricPlot[
{Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->300,Ticks->{{-1,1,2,3,4,5,6},{-1,1}},PlotRange->{{-Pi/2,2 Pi},{-4,4}},AspectRatio->8/(2 Pi+Pi/2)
],
Graphics[
{
Opacity[1],
Blue,
Line[{{0,0},{whichpiece[theta],whichpiece[theta]*Tan[theta]}}],
Red,
Line[{{whichpiece[theta],0},{whichpiece[theta],whichpiece[theta]*Tan[theta]}}],
Green,
Line[{{theta,0},{theta,Tan[theta]}}]
}
],
Plot[
Tan[t],{t,-Pi/2,2 Pi}
]
],
{{theta,0},-Pi/2,2 Pi}
]

Tan[{Pi/2-.1, Pi/2+.1}]

ttan[theta_]:=Sin[theta]/Cos[theta]

ttan[Pi/2+.2]

Secant

{Sec[-Pi/3],Sec[-Pi/4],Sec[-Pi/6],Sec[0],Sec[Pi/6],Sec[0],Sec[Pi/6],Sec[Pi/4],Sec[Pi/3],Sec[Pi/2],Sec[Pi]}

%//N

Manipulate[
Show[
ParametricPlot[
{Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->100,Ticks->{{-1,1},{-3,-2,-1,1,2,3}},PlotRange->{{-1,1},{-3,3}},AspectRatio->3
],
Graphics[
{
Opacity[1],
Red,
Line[{{0,0},{1,Tan[theta]}}],
Blue,
Line[{{1,0},{1,Tan[theta]}}]
}
]
],
{{theta,Pi/4},-Pi/2,3 Pi/2}]

whichpiece[theta_]:=If[(Pi/2<theta<3Pi/2),-1,1]

whichpiece[Pi]

Manipulate[
Show[
ParametricPlot[
{Cos[t],Sin[t]},{t,0,2 Pi},ImageSize->300,Ticks->{{-1,1,2,3,4,5,6},{-1,1}},PlotRange->{{-Pi/2,2 Pi},{-4,4}},AspectRatio->8/(2 Pi+Pi/2)
],
Graphics[
{
Opacity[1],
Red,
Line[{{0,0},{whichpiece[theta],whichpiece[theta]*Tan[theta]}}],
Blue,
Line[{{whichpiece[theta],0},{whichpiece[theta],whichpiece[theta]*Tan[theta]}}],
Green,
Line[{{theta,0},{theta,Sec[theta]}}]
}
],
Plot[
Sec[t],{t,-Pi/2,2 Pi}
]
],
{{theta,0},-Pi/2,2 Pi},SaveDefinitions->True
]

.8{510,715}