Trig Functions
Sine, Cosine, Tangent

Overview

The subject matter of TRIGONOMETRY is based upon six trigonometric functions.  In this section we will introduce the first three--the sine, cosine and tangent functions (abbreviated sin, cos and tan).

Definitions

1.  Let q be any angle.
2.  Place q in standard position (vertex at the origin)
3.  Choose any point P = (x,y), other than (0,0), on q.
4.  Draw a perpendicular from P to the x-axis forming a right triangle.
5.  Let r be the distance from P to the origin.
6.  By the Pythagorean theorem:  r€ = x€ + y€
7.  sin q = y/r
    cos q = x/r
    tan q = y/x

Selection of specific point P not critical

It does not matter which point P on the terminal side of q is chosen.  The value of sin q, cos q and tan q are independent of the point P.  An similar triangles argument will easily show that sin q depends only upon q and not on which point P was chosen.
  Note:  For this reason we will choose our point P on the unit circle making r = 1.  The coordinates of P will thus be (cos q, sin q).

The Sine function

The y-coordinate of the point on the terminal side of an angle q intersecting with the unit circle sin q.

(* SIN curve display *)

Clear[pCircle,pTrig,pAxes,s1,s2,x,y];
num=4;
x[t_]= Cos[t];
y[t_]= Sin[t];
pAxes=Graphics[{Thickness[0.002],
                RGBColor[0,0,0],
                Line[{{-1.5,-1.1},{-1.5,1.1}}]}];        
pCircle=ParametricPlot[{x[t]-1.5,y[t]},{t,0,2 Pi},
        DisplayFunction->Identity,
        PlotStyle->{RGBColor[0,1,0],Thickness[0.005]},
        Axes->True,
        Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi},None}];
pTrig:=Plot[y[t],{t,0,2Pi},
       DisplayFunction->Identity,
       Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi}},
       PlotStyle->{RGBColor[1,0,0],Thickness[0.01]},
       PlotRange->{{-1,2Pi},{-1,1}}];        
Do[s1=Graphics[{Thickness[0.005],
                RGBColor[0,0,1],
                Line[{{-1.5,0},{x[theta]-1.5,y[theta]}}],
                RGBColor[0,.5,.4],
                Line[{{x[theta]-1.5,y[theta]},{theta,y[theta]}}],
                Line[{{x[theta]-1.5,y[theta]},{x[theta]-1.5,0}}],
                Line[{{theta,y[theta]},{theta,0}}],
                PointSize[0.015],
                RGBColor[1,0,0],
                Point[{x[theta]-1.5,y[theta]}],
                Point[{theta,y[theta]}]
              }];
   Show[pCircle,pTrig,s1,pAxes,
        AspectRatio->Automatic,
        DisplayFunction->$DisplayFunction],
  {theta,0,2Pi,Pi/6}
  ]
  
(* SIN curve display *)

The Cosine function

The x-coordinate of the point on the terminal side of an angle q intersecting with the unit circle cos q.

(* COS curve display *)

Clear[pCircle,pTrig,l1,t,
      p1,p2,p3,p4,p5,p6,s1,x,y,theta,num,n,org];
num=6;
org={-1.5,0};
x[t_]=Cos[t];
y[t_]=Sin[t];
pAxes=Graphics[{Thickness[0.002],
                RGBColor[0,0,0],
                Line[{{-1.5,-1.1},{-1.5,1.1}}]}];
pCircle=ParametricPlot[{x[t]-1.5,y[t]},{t,0,2 Pi},
        DisplayFunction->Identity,
        PlotStyle->{RGBColor[0,1,0],Thickness[0.005]},
        Axes->True,
        Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi},None}];
pTrig=Plot[x[t],{t,0,2Pi},
       DisplayFunction->Identity,
       Ticks->{{Pi/2,Pi,3Pi/2,2Pi},None},
       PlotStyle->{RGBColor[1,0,0],Thickness[0.01]},
       PlotRange->{{-1,2Pi},{-1,1}}];
Do[p1={x[theta]-1.5,y[theta]};
   p2={-1.5,y[theta]};
   p3={theta,0};
   p4={theta,x[theta]};
   s1=Graphics[{Thickness[0.005],
                RGBColor[0,0,1],
                Line[{org,p1}],
                RGBColor[0,.9,.9],
                Thickness[0.001],
                Line[{p1,p2}],
                Line[{p3,p4}],
                PointSize[0.01],
                RGBColor[1,0,1],
                Point[p1],
                Point[p4]}];
   Do[p5=p1+(p3-p1) n;
      p6=p2+(p4-p2) n;
      l1=Graphics[{Thickness[0.001],
                   RGBColor[0,.5,.5],
                   Line[{p5,p6}]}];
      Show[pCircle,pTrig,s1,l1,pAxes,
           AspectRatio->Automatic,
           DisplayFunction->$DisplayFunction];,
     {n,1/num,1-1/num,1/num}
     ],
   {theta,0,2Pi-Pi/6,Pi/6}
  ]
  
  (* COS curve display *)

The Tangent function

The y-coordinate of the point where the terminal side of angle q intersects with the vertical line tangent passing through the point (1,0) is the tan q.
  A proof of the above statement involves similar triangles.

(* TAN curve display *)

Clear[pCircle,pTrig,pAxes,s1,s2,x,y,LongLine];
shift=2;
org={shift,0};
num=4;
x[t_]= Cos[t];
y[t_]= Sin[t];
pAxes=Graphics[{Thickness[0.002],
                RGBColor[0,0,0],
                Line[{{shift,-1.1},{shift,1.1}}]}]; 
LongLine=Graphics[{Thickness[0.002],
                   RGBColor[.2,.2,.2],
                   Line[{{shift+1,-6},{shift+1,6}}]}];                       
pCircle=ParametricPlot[{x[t]+shift,y[t]},{t,-Pi/2, Pi/2},
        DisplayFunction->Identity,
        PlotStyle->{RGBColor[0,1,0],
        Thickness[0.005]}];
pTrig:=Plot[Tan[t],{t,-Pi/2+0.05,Pi/2-0.05},
       DisplayFunction->Identity,
       Ticks->{{-Pi/2,0,Pi/2}},
       PlotStyle->{RGBColor[1,0,0],
       Thickness[0.01]},
       PlotRange->{{-Pi/2-0.01,4.1},{-6,6}}];        
Do[p1={x[theta]+shift,y[theta]};
   p2={shift+1,Tan[theta]};
   p3={theta,Tan[theta]};
   p4={theta,0};
   s1=Graphics[{Thickness[0.005],
                RGBColor[0,0,1],
                Line[{org,p1}],
                RGBColor[0,.5,.4],
                Line[{p1,p2}],
                Line[{p2,p3}],
                Line[{p3,p4}],
                PointSize[0.025],
                RGBColor[1,0,0],
                Point[p1],
                Point[p2],
                Point[p3]
              }];
   Show[pCircle,LongLine,pAxes,s1,pTrig,
        AspectRatio->Automatic,
        DisplayFunction->$DisplayFunction,
        PlotRange->{{-Pi/2-0.01,4.1},{-6,6}}],
  {theta,-5Pi/12,5Pi/12,Pi/12}
  ]
  
(* TAN curve display *)