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Sine, Cosine, and Tangent Curves
by Mona Knight
;[s]
2:0,1;35,0;73,-1;
2:1,26,19,Calculus,0,12,0,0,0;1,45,32,Calculus,0,24,65535,0,0;
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Sine, Cosine, and Tangent Curves
:[font = text; inactive; preserveAspect]
We are going to be graphing the Sine, Cosine, and Tangent curves with changes in the period, amplitude, and phase shift.
First let's graph the normal Sin, Cos, and Tan functions.
:[font = input; preserveAspect]
Plot[Sin[x], {x,-2 Pi,2 Pi},
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-1,-.5,.5,1}}];
:[font = text; inactive; preserveAspect]
What is the period of the above graph?
What is the amplitude of the above graph?
:[font = input; preserveAspect]
Plot[Cos[x],{x,-2 Pi,2 Pi},
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi}, {-1,-.5,.5,1}}];
:[font = text; inactive; preserveAspect]
What is the period of the above graph?
What is the amplitude of the above graph?
:[font = input; preserveAspect]
Plot[Tan[x],{x,-2 Pi,2 Pi},
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi}, Automatic}];
:[font = text; inactive; preserveAspect; endGroup]
What is the period of the above graph?
What is the amplitude of the above graph?
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Changes in Amplitude
:[font = text; inactive; preserveAspect]
Now let's change a few things and see what happens to these curves.
Our first graph was for the function y=Sin x ,
Let's now graph y=3 Sin x
:[font = input; preserveAspect]
Plot[{3 Sin[x], Sin[x]},
{x,-2 Pi,2 Pi},
PlotLabel->"Plots of Sin x (dashed) and 3 Sin x (solid)",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-3,-2,-1,1,2,3}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
How did this curve differ from the first Sin curve that we graphed?
:[font = text; inactive; pageBreak; preserveAspect]
Now let's graph y= .5 Cos x
:[font = input; preserveAspect]
Plot[{.5 Cos[x], Cos[x]},
{x,-2 Pi,2 Pi},
PlotLabel->"Plots of Cos x (dashed) and .5 Cos x (solid)",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
What was different with this graph from the first Cos graph that we did?
Is there anything that we can possibly conclude that happens to the curves of the Sin and Cos curves when we put a number in front of the function in the equation?
Well let's try this with the Tan curve and see what happens.
:[font = input; preserveAspect]
Plot[{4 Tan[x],Tan[x]},
{x,-Pi,Pi},
PlotLabel->"Plots of Tan x (dashed) and 4 Tan x (solid)",
Ticks->{{-Pi,-Pi/2,Pi/2,Pi}, Automatic},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect; endGroup]
Did the amplitude of this curve change? Why or why not?
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Changes in Period
:[font = text; inactive; preserveAspect]
Let's now experiment with changes in the periods of these curves.
What do you think would happen if you graphed y=Sin 3 x
:[font = input; preserveAspect]
Plot[{Sin[3 x],Sin[x]},
{x,0,2 Pi},
PlotLabel->"Plots of Sin x (dashed) and Sin 3x (solid)",
Ticks->{{Pi/2,Pi,3 Pi/2,2 Pi},{-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
What happened to the period of this curve?
Let's graph y=Cos .4 x
:[font = input; preserveAspect]
Plot[{Cos[.4 x],Cos[x]},
{x,-3 Pi,3 Pi},
PlotLabel->"Plots of Cos x (dashed) and Cos .4x (solid)",
Ticks->{{-3 Pi, -2 Pi, -Pi, Pi, 2 Pi, 3 Pi},{-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
What happened to this curve, is the period larger or smaller than the original Cos curve?
Now let's change the equation of the Tangent function, let's graph y=Tan 2 x
:[font = input; preserveAspect]
Plot[{Tan[2 x],Tan[x]},
{x,-Pi,Pi},
PlotLabel->"Plots of Tan x (dashed) and Tan 2x (solid)",
Ticks->{{-Pi,-Pi/2 ,Pi/2, Pi}, Automatic},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect; endGroup]
Again what happened to the period of this function, does it complete it's cycle before or after the normal Tangent curve?
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Phase Shifts
:[font = text; inactive; preserveAspect]
Let's try something different.
What do you think would be changed from the graph of y=Sin x if we changed the equation to y=Sin (x + Pi/9)?
Well let's try it and see!
:[font = input; preserveAspect]
Plot[{Sin[x + Pi/9],Sin[x]},
{x,-2 Pi, 2 Pi},
PlotLabel->"Plots of Sin x (dashed) and Sin (x + Pi/9) (solid)",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi}, {-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
What happened to the new curve?
Why has it moved 20 degrees to the left, did it have anything to do with the original equation?
Now let's play with the Cos curve.
Let's graph y=Cos (x - Pi/3) and see what happens.
:[font = input; preserveAspect]
Plot[{Cos[x-Pi/3],Cos[x]},
{x,-2 Pi, 2 Pi},
PlotLabel->"Plots of Cos x (dashed) and Cos(x-Pi/3) (solid)",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect]
Did the new graph move to the right or to the left of the original curve? Why?
Well, we have experimented with the Sin and Cos curves, now let's try and see what happens when we add or subtract to the Tan curve. Let's graph y=Tan(x+Pi/6)
:[font = input; preserveAspect]
Plot[{Tan[x+Pi/6],Tan[x]},
{x,-Pi,Pi},
PlotLabel->"Plots of Tan x (dashed) and Tan (x+Pi/6) (solid)",
Ticks->{{-Pi,-Pi/2,Pi/2,Pi},Automatic},
PlotStyle->{Dashing[{}],Dashing[{1/20}]}];
:[font = text; inactive; preserveAspect; endGroup]
What conclusions can you draw from all of the previous examples?
What changes the period of these functions?
What changes the amplitude of these functions?
What makes a phase shift in these functions?
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Combinations of Curves
:[font = text; inactive; preserveAspect]
Let's try combining some of these functions and see what happens!
:[font = input; preserveAspect]
Clear[x]
p1=Plot[{Sin[x],Cos[x]},{x,-2 Pi, 2 Pi},
PlotLabel->"Sin x (solid) and Cos x (dashed)",
Ticks->{{-2 Pi,-Pi,Pi, 2 Pi},{-1,-.5,.5,1}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]},
DisplayFunction->Identity];
p3=Plot[Sin[x]+Cos[x],{x,-2 Pi, 2 Pi},
PlotLabel->"Sin x + Cos x",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-1,-.5,.5,1}},
DisplayFunction->Identity];
Show[GraphicsArray[{{p1},{p3}}],
DisplayFunction->$DisplayFunction];
:[font = text; inactive; preserveAspect]
Can you see how all the values of Sin and Cos add together to give you the final graph? If you are confused about this let's run tables of these values and see if it helps you see this process.
:[font = input; preserveAspect]
TableForm[Table[{x 180/Pi,
N[Sin[x]],
N[Cos[x]],
N[Sin[x]]+ N[Cos[x]]},
{x,-2 Pi, 2 Pi,Pi/4}]]
:[font = text; inactive; preserveAspect]
Now let's graph another combined function.
y = Sin x - 2 Cos x
:[font = input; preserveAspect]
Clear[x]
x1=Plot[{Sin[x],2 Cos[x]},{x,-2 Pi, 2 Pi},
PlotLabel->"Sin x (solid) and 2 Cos x (dashed)",
Ticks->{{-2 Pi, -Pi, Pi, 2 Pi},{-2,-1,1,2}},
PlotStyle->{Dashing[{}],Dashing[{1/20}]},
DisplayFunction->Identity];
x2=Plot[Sin[x]-2 Cos[x],{x,-2 Pi, 2 Pi},
PlotLabel->"Sin x - 2 Cos x",
Ticks->{{-2 Pi,-Pi, Pi, 2 Pi}, {-2,-1,1,2}},
DisplayFunction->Identity];
Show[GraphicsArray[{{x1},{x2}}],
DisplayFunction->$DisplayFunction];
:[font = input; preserveAspect]
Again let's look at the numeric output and see if it helps us
determine our points on our final curve.
;[s]
1:0,1;103,-1;
2:0,7,10,Courier,1,12,0,0,0;1,18,16,CalcMath,0,12,0,0,0;
:[font = input; preserveAspect]
TableForm[Table[{x 180/Pi,
N[Sin[x]],
N[2 Cos[x]],
N[Sin[x]]- N[2 Cos[x]]},
{x,-2 Pi, 2 Pi,Pi/4}]]
:[font = text; inactive; preserveAspect; endGroup]
For more practice on the above problems contact Mrs. Knight for selected problems from the textbook.
^*)