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Graphing Sine and Cosine Functions

Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph. (Figure) lists some of the values for the sine function on a unit circle.

Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See (Figure).

Notice how the sine values are positive between 0 and[latex]\,\pi ,\,[/latex]which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between[latex]\,\pi \,[/latex]and[latex]\,2\pi ,\,[/latex]which correspond to the values of the sine function in quadrants III and IV on the unit circle. See (Figure).

Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph. (Figure) lists some of the values for the cosine function on a unit circle.

As with the sine function, we can plots points to create a graph of the cosine function as in (Figure).

Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval[latex]\,\left[-1,1\right].[/latex]

In both graphs, the shape of the graph repeats after[latex]\,2\pi ,\,[/latex]which means the functions are periodic with a period of[latex]\,2\pi .\,[/latex]A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function:[latex]\,f\left(x+P\right)=f\left(x\right)\,[/latex]for all values of[latex]\,x\,[/latex]in the domain of[latex]\,f.\,[/latex]When this occurs, we call the smallest such horizontal shift with[latex]\,P>0\,[/latex]the period of the function. (Figure) shows several periods of the sine and cosine functions.

Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in (Figure), the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because[latex]\,\mathrm{sin}\left(-x\right)=-\mathrm{sin}\,x.\,[/latex]
Now we can clearly see this property from the graph.

(Figure) shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that [latex]\mathrm{cos}\left(-x\right)=\mathrm{cos}\text{ }x.[/latex]

Characteristics of Sine and Cosine Functions

The sine and cosine functions have several distinct characteristics:

• They are periodic functions with a period of[latex]\,2\pi .[/latex]
• The domain of each function is[latex]\,\left(-\infty ,\infty \right)\,[/latex]and the range is[latex]\,\left[-1,1\right].[/latex]
• The graph of[latex]\,y=\mathrm{sin}\text{ }x\,[/latex]is symmetric about the origin, because it is an odd function.
• The graph of[latex]\,y=\mathrm{cos}\text{ }x\,[/latex]is symmetric about the[latex]\,y\text{-}[/latex]axis, because it is an even function.

Investigating Sinusoidal Functions

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are

[latex]\begin{array}{l}y=A\mathrm{sin}\left(Bx-C\right)+D\hfill \\ \text{ and}\hfill \\ y=A\mathrm{cos}\left(Bx-C\right)+D\hfill \end{array}[/latex]

Determining the Period of Sinusoidal Functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.

In the general formula,[latex]\,B\,[/latex]is related to the period by[latex]\,P=\frac{2\pi }{|B|}.\,[/latex]If[latex]\,|B|>1,\,[/latex]then the period is less than[latex]\,2\pi \,[/latex]and the function undergoes a horizontal compression, whereas if[latex]\,|B|<1,\,[/latex]then the period is greater than[latex]\,2\pi \,[/latex]and the function undergoes a horizontal stretch. For example,[latex]\,f\left(x\right)=\mathrm{sin}\left(x\right),\,[/latex][latex]B=1,\,[/latex]so the period is[latex]\,2\pi ,\text{}[/latex]which we knew. If[latex]\,f\left(x\right)=\mathrm{sin}\left(2x\right),\,[/latex]then[latex]\,B=2,\,[/latex]so the period is[latex]\,\pi \,[/latex]and the graph is compressed. If[latex]\,f\left(x\right)=\mathrm{sin}\left(\frac{x}{2}\right),\,[/latex]then[latex]\,B=\frac{1}{2},\,[/latex]so the period is[latex]\,4\pi \,[/latex]and the graph is stretched. Notice in (Figure) how the period is indirectly related to[latex]\,|B|.[/latex]

Period of Sinusoidal Functions

If we let[latex]\,C=0\,[/latex]and[latex]\,D=0\,[/latex]in the general form equations of the sine and cosine functions, we obtain the forms

[latex]y=A\mathrm{sin}\left(Bx\right)[/latex]

[latex]y=A\mathrm{cos}\left(Bx\right)[/latex]

The period is[latex]\,\frac{2\pi }{|B|}.[/latex]

Identifying the Period of a Sine or Cosine Function

Determine the period of the function[latex]\,f\left(x\right)=\mathrm{sin}\left(\frac{\pi }{6}x\right).[/latex]

Show Solution

Let’s begin by comparing the equation to the general form[latex]\,y=A\mathrm{sin}\left(Bx\right).[/latex]

In the given equation,[latex]\,B=\frac{\pi }{6},\,[/latex]so the period will be

[latex]\begin{array}{l}\begin{array}{l}\\ P=\frac{2\pi }{|B|}\end{array}\hfill \\ \text{ }=\frac{2\pi }{\frac{\pi }{6}}\hfill \\ \text{ }=2\pi \cdot \frac{6}{\pi }\hfill \\ \text{ }=12\hfill \end{array}[/latex]

Try It

Determine the period of the function[latex]\,g\left(x\right)=\mathrm{cos}\left(\frac{x}{3}\right).[/latex]

Show Solution

[latex]\,6\pi \,[/latex]

Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable[latex]\,B\,[/latex]relates to the period. Now let’s turn to the variable[latex]\,A\,[/latex]so we can analyze how it is related to the amplitude, or greatest distance from rest.[latex]\,A\,[/latex]represents the vertical stretch factor, and its absolute value[latex]\,|A|\,[/latex]is the amplitude. The local maxima will be a distance[latex]\,|A|\,[/latex]above the horizontal midline of the graph, which is the line[latex]\,y=D;\,[/latex]because[latex]\,D=0\,[/latex]in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If[latex]\,|A|>1,\,[/latex]the function is stretched. For example, the amplitude of[latex]\,f\left(x\right)=4\,\mathrm{sin}\,x\,[/latex]is twice the amplitude of[latex]\,f\left(x\right)=2\,\mathrm{sin}\,x.\,[/latex]If[latex]\,|A|<1,\,[/latex]the function is compressed. (Figure) compares several sine functions with different amplitudes.

Amplitude of Sinusoidal Functions

If we let[latex]\,C=0\,[/latex]and[latex]\,D=0\,[/latex]in the general form equations of the sine and cosine functions, we obtain the forms

[latex]y=A\mathrm{sin}\left(Bx\right)\text{ and }y=A\mathrm{cos}\left(Bx\right)[/latex]

The amplitude is[latex]\,A,\,[/latex]and the vertical height from the midline is[latex]\,|A|.\,[/latex]In addition, notice in the example that

[latex]|A|\text{ = amplitude = }\frac{1}{2}|\text{maximum }-\text{ minimum}|[/latex]

Identifying the Amplitude of a Sine or Cosine Function

What is the amplitude of the sinusoidal function[latex]\,f\left(x\right)=-4\mathrm{sin}\left(x\right)?\,[/latex]Is the function stretched or compressed vertically?

Show Solution

Let’s begin by comparing the function to the simplified form[latex]\,y=A\mathrm{sin}\left(Bx\right).[/latex]

In the given function,[latex]\,A=-4,\,[/latex]so the amplitude is[latex]\,|A|=|-4|=4.\,[/latex]The function is stretched.

Analysis

The negative value of[latex]\,A\,[/latex]results in a reflection across the x-axis of the sine function, as shown in (Figure).

Try It

What is the amplitude of the sinusoidal function[latex]\,f\left(x\right)=\frac{1}{2}\mathrm{sin}\left(x\right)?\,[/latex]Is the function stretched or compressed vertically?

[latex]\frac{1}{2}\,[/latex]compressed

Analyzing Graphs of Variations of y = sin x and y = cos x

Now that we understand how[latex]\,A\,[/latex]and[latex]\,B\,[/latex]relate to the general form equation for the sine and cosine functions, we will explore the variables[latex]\,C\,[/latex]and[latex]\,D.\,[/latex]Recall the general form:

The value[latex]\,\frac{C}{B}\,[/latex]for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If[latex]\,C>0,\,[/latex]the graph shifts to the right. If[latex]\,C<0,\,[/latex]the graph shifts to the left. The greater the value of[latex]\,|C|,\,[/latex]the more the graph is shifted. (Figure) shows that the graph of[latex]\,f\left(x\right)=\mathrm{sin}\left(x-\pi \right)\,[/latex]shifts to the right by[latex]\,\pi \,[/latex]units, which is more than we see in the graph of[latex]\,f\left(x\right)=\mathrm{sin}\left(x-\frac{\pi }{4}\right),\,[/latex]which shifts to the right by[latex]\,\frac{\pi }{4}\,[/latex]units.

While[latex]\,C\,[/latex]relates to the horizontal shift,[latex]\,D\,[/latex]indicates the vertical shift from the midline in the general formula for a sinusoidal function. See (Figure). The function[latex]\,y=\mathrm{cos}\left(x\right)+D\,[/latex]has its midline at[latex]\,y=D.[/latex]

Any value of[latex]\,D\,[/latex]other than zero shifts the graph up or down. (Figure) compares[latex]\,f\left(x\right)=\mathrm{sin}\,x\,[/latex]with[latex]\,f\left(x\right)=\mathrm{sin}\,x+2,\,[/latex]which is shifted 2 units up on a graph.

Identifying the Variations of a Sinusoidal Function from an Equation

Determine the midline, amplitude, period, and phase shift of the function[latex]\,y=3\mathrm{sin}\left(2x\right)+1.[/latex]

Show Solution

Let’s begin by comparing the equation to the general form[latex]\,y=A\mathrm{sin}\left(Bx-C\right)+D.[/latex]

[latex]A=3,\,[/latex]so the amplitude is[latex]\,|A|=3.[/latex]

Next,[latex]\,B=2,\,[/latex]so the period is[latex]\,P=\frac{2\pi }{|B|}=\frac{2\pi }{2}=\pi .[/latex]

There is no added constant inside the parentheses, so[latex]\,C=0\,[/latex]and the phase shift is[latex]\,\frac{C}{B}=\frac{0}{2}=0.[/latex]

Finally,[latex]\,D=1,\,[/latex]so the midline is[latex]\,y=1.[/latex]

Analysis

Inspecting the graph, we can determine that the period is[latex]\,\pi ,\,[/latex]the midline is[latex]\,y=1,\,[/latex]and the amplitude is 3. See (Figure).

Identifying the Equation for a Sinusoidal Function from a Graph

Determine the formula for the cosine function in (Figure).

Show Solution

To determine the equation, we need to identify each value in the general form of a sinusoidal function.

[latex]\begin{array}{l}y=A\mathrm{sin}\left(Bx-C\right)+D\hfill \\ y=A\mathrm{cos}\left(Bx-C\right)+D\hfill \end{array}[/latex]

The graph could represent either a sine or a cosine function that is shifted and/or reflected. When[latex]\,x=0,\,[/latex]the graph has an extreme point,[latex]\,\left(0,0\right).\,[/latex]Since the cosine function has an extreme point for[latex]\,x=0,\,[/latex]let us write our equation in terms of a cosine function.

Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below[latex]\,y=0.5.\,[/latex]This value, which is the midline, is[latex]\,D\,[/latex]in the equation, so[latex]\,D=0.5.[/latex]

The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So[latex]\,|A|=0.5.\,[/latex]Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so[latex]\,|A|=\frac{1}{2}=0.5.\,[/latex]Also, the graph is reflected about the x-axis so that[latex]\,A=-0.5.[/latex]

The graph is not horizontally stretched or compressed, so[latex]\,B=1;\,[/latex]and the graph is not shifted horizontally, so[latex]\,C=0.[/latex]

Putting this all together,

[latex]g\left(x\right)=-0.5\mathrm{cos}\left(x\right)+0.5[/latex]