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Cell["Sinusoidal Functions", "Title",
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Cell["Read Me!  (how to use this program)", "Section",
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  ", a highly sophisticated mathematical software program capable of doing \
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  "You will need to know only a few ",
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to the right of each topic is a short blue bracket sign.  Some brackets have \
a small arrow or triangle at the bottom.  This arrow indicates that there is \
hidden text which can be viewed by clicking with the mouse on the bracket \
containing the arrow.  Try this on one of the arrowed brackets below.  When \
you click on the bracket, new text should appear.  When you are finished, you \
can close that section of the notebook by double-clicking on the same (now \
longer) bracket.  If you are unsure which one it is, scroll up to the \
beginning of the section; it is the one that ",
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anywhere inside the yellow box.  Now depress the \"shift\" and \"return\" \
keys ",
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will have a different notebook from the one you downloaded, and any \
misinformation you might have entered into the notebook will persist.  If for \
any reason, you need to start with a fresh notebook in its original form, you \
can always trash the one you've been working with and download a new one off \
the Web."
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        PlotStyle -> {Hue[n/10], Thickness[0.008]}, 
        AspectRatio -> Automatic], {n, \(-5\), 5, 2}]\)], "Input",
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  StyleBox["What do you notice about the amplitudes of the graphs when ",
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  StyleBox["A ",
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  "is a positive number? a negative number?  How can we generalize these \
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    FontSize->14],
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    FontSize->14,
    FontWeight->"Bold"],
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  StyleBox[" sin(x) ",
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  "that holds true for both positive and negative numbers? \n\tNow see if the \
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    FontWeight->"Bold"],
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  StyleBox["by different numbers",
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  StyleBox[" ",
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  StyleBox["in the following box and press shift-return to graph ",
    FontSize->14],
  StyleBox["f(x)",
    FontSize->14,
    FontWeight->"Bold"],
  StyleBox[" to check your answer. ",
    FontSize->14]
}], "Text",
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          RowBox[{
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            StyleBox[\(Cos[x]\),
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          StyleBox[",",
            FontColor->GrayLevel[0]], 
          StyleBox[\({x, \(-2\)\ Pi, \ 2\ Pi}\),
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          \(PlotStyle -> {Thickness[ .008], Hue[ .7]}\)}], "]"}], ";"}]], 
  "Input",
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Cell["Summary:", "Subsection"],

Cell[TextData[{
  StyleBox["For the functions  f(x) = ",
    FontSize->14,
    FontWeight->"Bold"],
  StyleBox["A",
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    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox[" sin (x) or g(x) = ",
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    FontWeight->"Bold"],
  StyleBox["A",
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    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox[" cos (x) the ",
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    FontWeight->"Bold"],
  StyleBox["amplitude",
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  StyleBox[" = ",
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    FontWeight->"Bold"],
  StyleBox["| A |.\n",
    FontSize->14,
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox["We have the properties:",
    FontSize->14,
    FontWeight->"Bold"],
  StyleBox["\n                     ",
    FontSize->14,
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox[
  "-|A| \[LessEqual] A sin(x) \[LessEqual] |A|  and  -|A| \[LessEqual] A \
cos(x) \[LessEqual] |A|.",
    FontWeight->"Bold"]
}], "Text",
  CellFrame->True,
  FontSize->14,
  Background->GrayLevel[0.849989]],

Cell["\<\
To get an even better intuitive understanding of the amplitude, \
animate the following collection of graphs by first shift-returning the \
yellow box and then double-click on the first graph (you can change the speed \
of animation using the buttons at the lower left hand corner of the \
screen).\
\>", "Text",
  FontSize->14],

Cell[BoxData[
    \(Do[Print["\<The graph of f(x) = \>", n, "\< cos(x)\>"]; 
      Plot[n\ Cos[x], {x, \(-4\)\ Pi, 4\ Pi}, PlotRange -> {\(-5\), 5}, 
        PlotStyle -> {Hue[n/10], Thickness[0.008]}, 
        AspectRatio -> Automatic], {n, \(-4\), 4, 1}]\)], "Input",
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  AnimationCycleOffset->1,
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  FontColor->GrayLevel[0.666667],
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Cell[TextData[{
  "Exercise 1:",
  StyleBox["  What is the range of f(x) = ",
    FontWeight->"Plain"],
  Cell[BoxData[
      FormBox[
        StyleBox[\(-\+3\%7\),
          FontWeight->"Plain"], TraditionalForm]]],
  StyleBox["sin(x)? of g(x) =",
    FontWeight->"Plain"],
  Cell[BoxData[
      FormBox[
        RowBox[{"  ", 
          RowBox[{"-", 
            StyleBox[\(-\+2\%5\),
              FontWeight->"Plain"]}]}], TraditionalForm]]],
  StyleBox["cos(x) ?",
    FontWeight->"Plain"]
}], "Text",
  FontSize->14,
  FontWeight->"Bold"],

Cell["Answer:", "Text"],

Cell[BoxData[
    \(Print[
      "\<The range of f is [-7/3, 7/3] and the range of g is [-5/2, 5/2].\>"]
      \)], "Input",
  FontColor->RGBColor[1, 1, 0],
  Background->RGBColor[1, 1, 0]]
}, Open  ]]
}, Closed]],

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Cell["Period", "Section",
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  Background->RGBColor[0, 0, 1]],

Cell[TextData[{
  "\tThe ",
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    FontSlant->"Italic"],
  " of ",
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  " is defined to be the smallest number ",
  StyleBox["p",
    FontWeight->"Bold"],
  ", for which ",
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    FontWeight->"Bold"],
  " = ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  ", that is the smallest positive number ",
  StyleBox["p",
    FontWeight->"Bold"],
  " such that the function will repeat after ",
  StyleBox["p",
    FontWeight->"Bold"],
  ".  We can detect the period of a function by looking at its graph and \
determining how far we have to travel on x-axis before the graph starts to \
repeat itself.  Look at the graph of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(2",
    FontWeight->"Bold"],
  StyleBox["\[Pi]",
    FontFamily->"Symbol"],
  StyleBox["x)",
    FontWeight->"Bold"],
  " below.  What is its period?  Hint: It's easiest to start at zero and keep \
moving right until the picture starts to repeat itself.  "
}], "Text",
  FontSize->14],

Cell[BoxData[
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      RowBox[{"Plot", "[", 
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          RowBox[{
            StyleBox["Sin",
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            StyleBox["[",
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            StyleBox[\(2 Pi\ x\),
              FontColor->RGBColor[0, 0, 1]], "]"}], ",", 
          \({x, \(-3\), 3}\)}], "]"}], ";"}]], "Input",
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "The following program will plot the graphs of ",
  StyleBox["f(x)",
    FontWeight->"Bold"],
  " = ",
  StyleBox["sin(\[Omega]x) ",
    FontWeight->"Bold"],
  "for various values of ",
  StyleBox["\[Omega]",
    FontWeight->"Bold"],
  ".  See if you can figure out what the period is in each case.  "
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(Do[Print["\<The graph of f(x) = sin(\>", n, "\<x).\>"]; 
      \(Plot[Sin[n\ x], {x, \(-2\)\ Pi, 2  Pi}, PlotRange -> {\(-2\), 2}, 
        Ticks -> {{\(-4\) Pi\ /\((2\ n)\), \(-2\)\ \ Pi, \(-\ Pi\), 0, Pi, 
              2\ Pi, 4\ Pi/\((2\ n)\)}, Automatic}, 
        PlotStyle -> {Thickness[0.008], Hue[2\ n/10]}]; \), {n, 1, 5}]\)], 
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Cell[TextData[{
  StyleBox["Exercise 2: ",
    FontWeight->"Bold"],
  "Try to",
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    FontWeight->"Bold"],
  "find a relation between ",
  StyleBox["\[Omega] ",
    FontWeight->"Bold"],
  "and the period of the function ",
  StyleBox["f(x) = sin (\[Omega]x)",
    FontWeight->"Bold"],
  "."
}], "Text",
  FontSize->14],

Cell[TextData[{
  "   Let's see what happens to the periods when ",
  StyleBox["0 \[LessEqual] \[Omega] \[LessEqual] 1",
    FontWeight->"Bold"],
  ".  Shift-return the commands below:  "
}], "Text",
  FontSize->14],

Cell[BoxData[
    \(Do[Print["\<The graph of f(x) = sin((1/\>", n, 
        "\<)x).  Notice the period is \>", 2\ n, "\<\[Pi].\>"]; 
      \(Plot[Sin[\((1/n)\)\ x], {x, \(-4\)\ Pi, 4\ Pi}, 
        PlotStyle -> Hue[0.7], PlotRange -> {\(-5\), 5}, 
        AspectRatio -> Automatic]; \), {n, 1, 5}]\)], "Input",
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  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "We observe that the periods become longer and longer or more and more \
\"stretched out\" as the constant ",
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    FontWeight->"Bold"],
  "decreases; that is, we see fewer and fewer periods in the same window.  We \
would observe the same thing with the function ",
  StyleBox["cos (\[Omega]x)",
    FontWeight->"Bold"],
  "."
}], "Text",
  FontSize->14],

Cell[CellGroupData[{

Cell["Summary:", "Subsection"],

Cell[TextData[{
  StyleBox["The period of the functions ",
    FontWeight->"Bold"],
  StyleBox["sin (\[Omega]x)",
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    FontColor->RGBColor[0, 0, 1]],
  StyleBox[" and ",
    FontWeight->"Bold"],
  StyleBox["cos (\[Omega]x)",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  StyleBox[" is ",
    FontWeight->"Bold"],
  StyleBox["p = 2\[Pi]/\[Omega].",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]]
}], "Text",
  CellFrame->True,
  FontSize->16,
  Background->GrayLevel[0.849989]],

Cell[TextData[{
  StyleBox["Exercise 3: ",
    FontWeight->"Bold"],
  "What is the period of ",
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    FontWeight->"Plain"],
  Cell[BoxData[
      FormBox[
        StyleBox[\(-\+3\%2\),
          FontWeight->"Plain"], TraditionalForm]]],
  StyleBox["x)? of ",
    FontWeight->"Plain"],
  "sin(6x)",
  StyleBox[" ?",
    FontWeight->"Plain"]
}], "Text",
  FontSize->14],

Cell["Answer:", "Text"],

Cell[BoxData[
    \(Plot[{Cos[2  x/3], \ Sin[8  x]}, \ {x, \ \(-3\) Pi, \ 3  Pi}, \ 
      PlotStyle -> {Hue[0.7], \ Hue[0.1]}, 
      Ticks -> {{\(-3\) Pi, \(-2\) Pi, \(-3\) Pi/2, \(-Pi\), \ \(-Pi\)/3, \ 
            Pi/3, Pi, 3  Pi/2, 2  Pi, 3  Pi}, {\(-1\), \ 1}}]; \n
    Print["\<In the first case \[Omega] = 2/3, so the period is 2\[Pi]/(2/3) \
= 3\[Pi]. In the second case the period is 2\[Pi]/6 = \[Pi]/3.\>"]\)], "Input",\

  FontColor->RGBColor[1, 1, 0],
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  StyleBox["Exercise 4: ",
    FontWeight->"Bold"],
  "What are the amplitude and period of f(x) = -5 sin(4\[Pi]x) ?"
}], "Text",
  FontSize->14],

Cell["Answer:", "Text"],

Cell[BoxData[
    \(Print[
      \*"\"\<The amplitude is |-5| = 5 and the period is \!\(\(2  \
\[Pi]\)\/\(4  \[Pi]\)\) = \!\(1\/2\).  Here's the graph:\>\""]; 
    Plot[\(-5\) Sin[4  Pi*x], \ {x, \ \(-2\), \ 2}, 
      Ticks -> {{\(-2\), \ \(-3\)/2, \(-1\), \ \(-1\)/2, \ 1/2, \ 3/2, \ 2}, {
            \(-5\), \ 5}}, \ PlotStyle -> Hue[0.8]]; \)], "Input",
  FontColor->RGBColor[1, 1, 0],
  Background->RGBColor[1, 1, 0]]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Phase Shifts", "Section",
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  Background->RGBColor[0, 0, 1]],

Cell["Recall the formula:", "Text",
  FontSize->14],

Cell[BoxData[
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          StyleBox[" ",
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          RowBox[{
            StyleBox["(",
              FontWeight->"Bold"], 
            RowBox[{
              StyleBox["x",
                FontWeight->"Bold"], 
              StyleBox[" ",
                FontWeight->"Bold"], 
              StyleBox["-",
                FontWeight->"Bold"], 
              StyleBox[" ",
                FontWeight->"Bold"], 
              FormBox[\(\[Pi]\/2\),
                "TraditionalForm"]}], 
            StyleBox[")",
              FontWeight->"Bold"]}]}], 
        StyleBox[" ",
          FontWeight->"Bold"], 
        StyleBox["=",
          FontWeight->"Bold"], 
        StyleBox[" ",
          FontWeight->"Bold"], 
        StyleBox[\(sin\ \((x)\)\),
          FontWeight->"Bold"]}], TextForm]], "Text",
  CellFrame->True,
  TextAlignment->Center,
  Background->GrayLevel[0.849989]],

Cell[TextData[{
  "What does that mean graphically?  If you recall the graphing techniques, \
to draw the graph of ",
  StyleBox["y",
    FontWeight->"Bold"],
  " = ",
  StyleBox["cos",
    FontWeight->"Bold"],
  Cell[BoxData[
      StyleBox[
        RowBox[{"(", 
          RowBox[{"x", " ", "-", " ", 
            FormBox[\(\[Pi]\/2\),
              "TraditionalForm"]}], ")"}],
        FontWeight->"Bold"]]],
  ", we first graph ",
  StyleBox["y = cos(x) ",
    FontWeight->"Bold"],
  "then shift it",
  Cell[BoxData[
      StyleBox[
        RowBox[{" ", 
          FormBox[\(\[Pi]\/2\),
            "TraditionalForm"]}],
        FontWeight->"Bold"]]],
  "units to the right.  Hence the previous formula shows that the graph of ",
  StyleBox["sine ",
    FontWeight->"Bold"],
  "is the same as the graph of ",
  StyleBox["cosine ",
    FontWeight->"Bold"],
  "shifted ",
  Cell[BoxData[
      FormBox[
        FormBox[\(\[Pi]\/2\),
          "TraditionalForm"], TraditionalForm]]],
  "units to the right.\nTo check it replace ",
  StyleBox["BlueStuff",
    FontColor->RGBColor[0, 0, 1]],
  " by ",
  StyleBox["Sin[x]",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  " and hit shift-return, and then replace it by ",
  StyleBox["Cos[x - Pi/2]",
    FontWeight->"Bold",
    FontColor->RGBColor[0, 0, 1]],
  ":"
}], "Text",
  FontSize->14],

Cell[BoxData[
    RowBox[{\(Clear[f]\), ";", "\n", 
      RowBox[{\(f[x_]\), "=", 
        StyleBox["BlueStuff",
          FontColor->RGBColor[0, 0, 1]]}], ";", " ", "\n", 
      \(Print["\<The graph of \>", f[x], \ "\< :\>"]\), ";", 
      \(Plot[f[x], \ {x, \ \(-2\) Pi, \ 2  Pi}, \ 
        PlotStyle -> {Thickness[ .008], Hue[ .01]}]\), ";"}]], "Input",
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  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Now we want to discuss the graphs of a general ",
  StyleBox["sinusoidal ",
    FontWeight->"Bold"],
  "function, that is the graph of ",
  StyleBox[":",
    FontWeight->"Bold"]
}], "Text",
  FontSize->14],

Cell[TextData[
"y = A sin (\[Omega]x - \[CapitalPhi])  or  y = A cos (\[Omega]x - \
\[CapitalPhi]), where \[Omega] > 0."], "Text",
  CellFrame->True,
  TextAlignment->Center,
  FontSize->14,
  Background->GrayLevel[0.849989]],

Cell[TextData[{
  "The graph of such a function will have a",
  StyleBox["mplitude",
    FontWeight->"Bold"],
  " ",
  StyleBox["|A|",
    FontWeight->"Bold"],
  ", and one period will be traced out as ",
  StyleBox["\[Omega]x - \[CapitalPhi] ",
    FontWeight->"Bold"],
  "varies from ",
  StyleBox["0 ",
    FontWeight->"Bold"],
  "to ",
  StyleBox["2\[Pi].  ",
    FontWeight->"Bold"],
  "This period will begin when ",
  StyleBox["\[Omega]x - \[CapitalPhi] ",
    FontWeight->"Bold"],
  "= ",
  StyleBox["0 ",
    FontWeight->"Bold"],
  " or ",
  StyleBox["x =",
    FontWeight->"Bold"],
  Cell[BoxData[
      FormBox[
        StyleBox[\(\[CapitalPhi]\/\[Omega]\),
          FontSize->16], TraditionalForm]]],
  ", hence the graph of ",
  StyleBox["A sin (\[Omega]x - \[CapitalPhi]) ",
    FontWeight->"Bold"],
  "is nothing more than the graph of ",
  StyleBox["A sin(\[Omega]x) ",
    FontWeight->"Bold"],
  "shifted ",
  Cell[BoxData[
      FormBox[
        RowBox[{"of", 
          StyleBox[\(\[CapitalPhi]\/\[Omega]\),
            FontSize->16]}], TraditionalForm]]],
  " units (to the right if ",
  StyleBox["\[CapitalPhi] > 0, ",
    FontWeight->"Bold"],
  "to the left if ",
  StyleBox["\[CapitalPhi] < 0",
    FontWeight->"Bold"],
  ").  The number ",
  Cell[BoxData[
      FormBox[
        StyleBox[\(\[CapitalPhi]\/\[Omega]\),
          FontSize->16], TraditionalForm]]],
  " is called the ",
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    FontWeight->"Bold"]
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Cell[CellGroupData[{

Cell["Summary:", "Subsection"],

Cell[TextData[
"For the graphs of y = A sin (\[Omega]x - \[CapitalPhi])  or  y = A cos (\
\[Omega]x - \[CapitalPhi]), where \[Omega] > 0\namplitude = |A|,  period = p \
= 2\[Pi]/\[Omega],  phase shift = \[CapitalPhi]/\[Omega]."], "Text",
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  FontSize->14,
  Background->GrayLevel[0.849989]],

Cell[TextData[{
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    FontWeight->"Bold"],
  ": What are the amplitude, period, and phase shift of the function y = -4 \
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    FontWeight->"Bold"],
  " = ",
  StyleBox["-4, \[Omega] = 3 and \[CapitalPhi] = -5.  ",
    FontWeight->"Bold"],
  "Hence the amplitude is ",
  StyleBox["4",
    FontWeight->"Bold"],
  ", the period is ",
  StyleBox["2\[Pi]/3",
    FontWeight->"Bold"],
  ", and the phase shift is ",
  StyleBox["-5/3",
    FontWeight->"Bold"],
  ". \n\nNote that we want ",
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    FontWeight->"Bold"],
  ".  What if we have the function ",
  StyleBox["y = sin (-3x + 4)",
    FontWeight->"Bold"],
  "?  We can always write it in a form so that the coefficient in front of \
the x is positive: \n",
  StyleBox["y = sin (-3x + 4) = sin (-(3x + 4)) = - sin (3x + 4). \n",
    FontWeight->"Bold"],
  "Hence in this case ",
  StyleBox["A",
    FontWeight->"Bold"],
  " = -",
  StyleBox["1, \[Omega] ",
    FontWeight->"Bold"],
  "= ",
  StyleBox["3, and \[CapitalPhi]",
    FontWeight->"Bold"],
  " = ",
  StyleBox["-4.\n\n",
    FontWeight->"Bold"],
  "Here's the graph of  ",
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