SetOptions[EvaluationNotebook[],CellEpilog:>SetOptions[EvaluationCell[],CellLabel->""]]

Basic fiddling around

Plot[Sec[x],{x,((-π)/2),(π/2)}]

(π/4)+(π/6)

Sin[(π/4)+(π/6)]

ArcCos

Cos[ArcSin[θ]]

Basic plots

raGraphic={
Line[{{0,1},{0,0},{1,0}}],
Line[{{0,0.5},{0.5,0.5},{0.5,0}}]
};
raImage=Image[Graphics[raGraphic],ImageSize->20]

Plot[{
Sin[θ],
Cos[θ]
},{θ,0,π/2}
,Frame->{{True,False},{True,False}}
,FrameTicks->{{Automatic,None},{{{0,0},{(π/2),raImage}},None}}
,PlotLegends->"Expressions"
]

ParametricPlot[{Sin[θ],Cos[θ]},{θ,0,(π/2)}
,PlotLegends->{"(cos(θ),sin(θ))"}
]

Block[{is=400,θd=(π/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,(π/2)}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Text[Style["θ",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,0}],
Text[Style["(cos(θ), sin(θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=(π/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,(π/2)}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Line[{{Cos[θd],0},{Cos[θd],Sin[θd]}}],
Line[{{Cos[θd],0},{0,0}}],
Text[Style["x = cos(θ)",Bold,FontSize->14],{((Cos[θd])/2),0},{0,-1}],
Text[Style["y = sin(θ)",Bold,FontSize->14],{Cos[θd],((Sin[θd])/2)},{1.1,0}],
Text[Style["r = 1",Bold,FontSize->14],(1/2){Cos[θd],Sin[θd]},{-1.1,1.1}],
Circle[{0,0},arcr,{0,θd}],
Text[Style["θ",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,0}],
Text[Style["(cos(θ), sin(θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=((4π)/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,π}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Text[Style["θ",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,0}],
Text[Style["\"(cos(θ), sin(θ))\"??",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=((7π)/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Text[Style["θ",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,-1}],
Text[Style["(cos(θ), sin(θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=-(π/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Text[Style["-θ",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,0}],
Text[Style["(cos(-θ), sin(-θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0.5,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=2π+(π/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Circle[{0,0},1.5arcr,{0,θd-2π}],
Text[Style["1 circle",Bold,FontSize->14],1.5arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{1,0}],
Text[Style["+θ",Bold,FontSize->14],1.5arcr{Cos[((θd-2π)/2)],Sin[((θd-2π)/2)]},{-1.5,0}],
Text[Style["(cos(1 circle+θ), sin(1 circle+θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0.8,-1}]
}
,ImageSize->is
]
]

The Unit Circle:

Block[{is=400,θd=2π+(π/6),arcr=0.1},
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,2π}
,ImageSize->is
]
]

Why radians?

Approximation of a circular arc with a Bezier curve, from https://math.stackexchange.com/questions/1671588/b%C3%A9zier-curve-approximation-of-a-circular-arc, last answer. I need this because the Mathematica Arrow function doesn’t work on a circular arc drawn with Circle.

Block[{is=400,θd=(π/6),arcr=0.1,p0,p1,q0,q1,q2,q3,l,t0,t1},
p0={1,0};
p1={Cos[θd],Sin[θd]};
t0={0,1};
t1={-Sin[θd],Cos[θd]};
l=(4/3)Tan[((θd)/4)];
q0=p0;
q1=p0+l t0;
q2=p1-l t1;
q3=p1;
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,(π/2)}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Arrowheads[{-.05,.05}],
Arrow[BezierCurve[{q0,q1,q2,q3}]],
Text[Style["s = arc length",Bold,FontSize->14],{Cos[((θd)/2)],Sin[((θd)/2)]},{1.1,0}],
Text[Style["r = radius",Bold,FontSize->14],(1/2){Cos[θd],Sin[θd]},{-1.1,0}],
Text[Style["θ = (s/r) radians",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.1,-0.5}],
Text[Style["(cos(θ), sin(θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

Block[{is=400,θd=1,arcr=0.1,p0,p1,q0,q1,q2,q3,l,t0,t1},
p0={1,0};
p1={Cos[θd],Sin[θd]};
t0={0,1};
t1={-Sin[θd],Cos[θd]};
l=(4/3)Tan[((θd)/4)];
q0=p0;
q1=p0+l t0;
q2=p1-l t1;
q3=p1;
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,(π/2)}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Arrowheads[{-.05,.05}],
Arrow[BezierCurve[{q0,q1,q2,q3}]],
Text[Style["s = arc length",Bold,FontSize->14],{Cos[((θd)/2)],Sin[((θd)/2)]},{1.1,0}],
Text[Style["r = radius",Bold,FontSize->14],(1/2){Cos[θd],Sin[θd]},{-1.1,0}],
Text[Style["θ = (s/r) = 1 radian",Bold,FontSize->14],arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.1,-0.1}],
Text[Style["(cos(θ), sin(θ))",Bold,FontSize->14],{Cos[θd],Sin[θd]},{0,-1}]
}
,ImageSize->is
]
]

N[1/Degree]

Block[{is=400,θd=(π/12),arcr=0.1,p0,p1,q0,q1,q2,q3,l,t0,t1},
p0={1,0};
p1={Cos[θd],Sin[θd]};
t0={0,1};
t1={-Sin[θd],Cos[θd]};
l=(4/3)Tan[((θd)/4)];
q0=p0;
q1=p0+l t0;
q2=p1-l t1;
q3=p1;
ParametricPlot[{Cos[θ],Sin[θ]},{θ,0,θd}
,PlotRange->{{0,1.4},{0,Sin[θd]}}
,Epilog->{
PointSize[Large],
Point[{{Cos[θd],Sin[θd]}}],
Thick,
Line[{{0,0},{Cos[θd],Sin[θd]}}],
Circle[{0,0},arcr,{0,θd}],
Arrowheads[{-.05,.05}],
{Red,
Arrow[Line[{{Cos[θd],0},{Cos[θd],Sin[θd]}}]],
Text[Style["sin(θ)",Bold,FontSize->14],{Cos[θd],Sin[((θd)/2)]},{1.1,0}]
},
{Blue,
Arrow[BezierCurve[{q0,q1,q2,q3}]],
Text[Style["θ = arc length",Bold,FontSize->14],{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.1,0}]
},

Text[Style["r = 1",Bold,FontSize->14],(1/2){Cos[θd],Sin[θd]},{-1.1,0}],
Text[Style["θ",Bold,FontSize->14],2arcr{Cos[((θd)/2)],Sin[((θd)/2)]},{-1.5,0}],
}
,ImageSize->is
]
]

Plot[{
Sin[x],
x
},{x,0,(π/2)}
,PlotLegends->{"sin(x) (with x in radians)","slope=1"}
,AspectRatio->Automatic
,Frame->{{True,False},{True,False}}
,FrameTicks->{{{0,1,π/2},None},{{0,π/2},None}}
,Epilog->{
PointSize[Large],
Point[{{0,0}}]
}
]

CellLabel

Graphs

Block[{funcs={{Sin,Tan,Sec},{Cos,Cot,Csc}},pfunc},
pfunc=Function[f,
Plot[f[θ],{θ,0,(π/2)}
,Axes->False
,Frame->{{True,False},{True,False}}
,FrameTicks->{{Automatic,None},{{{0,0},{(π/4),(π/4)},{(π/2),(π/2)}},None}}
,PlotLabel->Style[TraditionalForm[f],Bold,FontFamily->"Arial",FontSize->14]
]
];
Map[pfunc,funcs,{2}]
]//Grid

ticks1=Table[
{θ,θ},
{θ,-2π,2π,(π/2)}
]

Plot[Sin[θ],{θ,0,π}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Sin[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Sin[θ],{θ,0,2π}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Sin[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Sin[θ],{θ,-2π,2π}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Sin[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Cos[θ],{θ,-(π/2),(π/2)}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Cos[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Cos[θ],{θ,-((3π)/2),((3π)/2)}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Cos[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Cos[θ],{θ,-2π,2π}
,PlotRange->{{-2π,2π},{-1,1}}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Cos[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Tan[θ],{θ,-2π,2π}
,PlotRange->{{-2π,2π},Automatic}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Tan[θ]],Bold,FontFamily->"Arial",FontSize->14]
,Epilog->{
Dashed,
Table[
Line[{{x,-100},{x,100}}],
{x,-((3π)/2),((3π)/2),π}
]
}
]

Plot[Sin[θ],{θ,0,2π}
,PlotRange->{{0,2π},Automatic}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Sin[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Cos[θ],{θ,0,2π}
,PlotRange->{{0,2π},Automatic}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Cos[θ]],Bold,FontFamily->"Arial",FontSize->14]
]

Plot[Tan[θ],{θ,0,2π}
,PlotRange->{{0,2π},Automatic}
,Ticks->{ticks1,Automatic}
,ImageSize->400
,PlotLabel->Style[TraditionalForm[Tan[θ]],Bold,FontFamily->"Arial",FontSize->14]
,Epilog->{
Dashed,
Table[
Line[{{x,-100},{x,100}}],
{x,-((3π)/2),((3π)/2),π}
]
}
]

Misc

{Sin[((16π)/6)],Cos[((16π)/6)],Tan[((16π)/6)]}

Sin[(π/6)]

{Sin[((3π)/4)],Cos[((3π)/4)],Tan[((3π)/4)]}

{Sin[((8π)/3)],Cos[((8π)/3)],Tan[((8π)/3)]}

{Sin[-((2π)/3)],Cos[-((2π)/3)],Tan[-((2π)/3)]}

Solve[Cos[θ]==Sin[θ],θ]

Sin[θ]==Cos[θ-(π/2)]//Simplify

Sin[(π/12)]