Sinusoidal Graphs

Part 1. Period

Enter the following commands to graph y = sin (t).

sinePlot  =  
        Plot[Sin[t], {t, -8, 8}, 
          PlotStyle -> RGBColor[0, 0, 1]];

Solve for locations of peaks on sine curve.

h  =  Solve[Sin[t] == 1, t]

Explore the period of sin (3*t).

c  =  3;
sinePlot  =  
        Plot[Sin[c*t], {t, -8, 8}, 
          PlotStyle -> RGBColor[0, 0, 1]];

h  =  Solve[Sin[c*t] == 1, t]

Part 2. Amplitude

b  =  3;
sinePlot  =  
        Plot[b*Sin[t], {t, -8, 8}, 
          PlotStyle -> RGBColor[0, 0, 1]];

c  =  4;
b  =  3;
sinePlot  =  
        Plot[b*Sin[c*t], {t, -8, 8}, 
          PlotStyle -> RGBColor[0, 0, 1]];

Part 3. Translations

Vertical translations

a  =  0.5;
sinePlot  =  
        Plot[a  +  Sin[t], {t, -8, 8}, 
          PlotStyle -> RGBColor[0, 0, 1]];

h = Solve[Sin[t]==  ,  
        t]

Horizontal translations

sinePlot  =  
        Plot[Sin[t], {t, -8, 8}, PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
t0  =  2;
horizPlot  =  
        Plot[Sin[t - t0], {t, -8, 8}, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
Show[sinePlot, horizPlot, 
        DisplayFunction -> $DisplayFunction];

cosPlot  =  
        Plot[Cos[t], {t, -8, 8}, PlotStyle -> RGBColor[0, 1, 0], 
          DisplayFunction -> Identity];
Show[sinePlot, cosPlot, 
        DisplayFunction -> $DisplayFunction];

a  =  1;
b  =  5;
c  =  0.5;
t0  =  -Pi;
guessPlot  =  
        Plot[a + b*Sin[c*(t - t0)], {t, -8, 8}, 
          PlotStyle -> RGBColor[1, 0, 0]];

Part 4. Fitting Sinusoidal Curves to Data

TamanTemp  =  {55.0, 59.7, 65.1, 72.0, 79.0, 83.8, 83.3, 82.8, 
          79.9, 72.7, 64.0, 56.7};
month  =  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12};
TamanData  =  Thread[{month, TamanTemp}]

<< Graphics`Graphics`
TamanPlot  =  
        ListPlot[TamanData,  PlotStyle -> RGBColor[1, 0, 0], 
          Prolog -> {PointSize[0.015]}];

a  =  70;
b  =  15;
c  =  Pi/6;
t0  =  4;
guessPlot  =  
        Plot[a + b*Sin[c*(t - t0)], {t, 0, 12}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
Show[TamanPlot, guessPlot, 
        DisplayFunction -> $DisplayFunction];

Part 5. Summary