Periodic Functions

# Graphs of the Sine and Cosine Functions

### Learning Objectives

In this section, you will:

• Graph variations of  y=sin( x )  and  y=cos( x ).
• Use phase shifts of sine and cosine curves.

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.

Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

### 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 andwhich correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative betweenandwhich 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

In both graphs, the shape of the graph repeats afterwhich means the functions are periodic with a period ofA periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function:for all values ofin the domain ofWhen this occurs, we call the smallest such horizontal shift withthe 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
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

### Characteristics of Sine and Cosine Functions

The sine and cosine functions have several distinct characteristics:

• They are periodic functions with a period of
• The domain of each function isand the range is
• The graph ofis symmetric about the origin, because it is an odd function.
• The graph ofis symmetric about theaxis, 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

#### 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,is related to the period byIfthen the period is less thanand the function undergoes a horizontal compression, whereas ifthen the period is greater thanand the function undergoes a horizontal stretch. For example,$B=1,\,$so the period iswhich we knew. Ifthenso the period isand the graph is compressed. Ifthenso the period isand the graph is stretched. Notice in (Figure) how the period is indirectly related to

### Period of Sinusoidal Functions

If we letandin the general form equations of the sine and cosine functions, we obtain the forms

The period is

### Identifying the Period of a Sine or Cosine Function

Determine the period of the function

Let’s begin by comparing the equation to the general form

In the given equation,so the period will be

### Try It

Determine the period of the function

#### Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variablerelates to the period. Now let’s turn to the variableso we can analyze how it is related to the amplitude, or greatest distance from rest.represents the vertical stretch factor, and its absolute valueis the amplitude. The local maxima will be a distanceabove the horizontal midline of the graph, which is the linebecausein this case, the midline is the x-axis. The local minima will be the same distance below the midline. Ifthe function is stretched. For example, the amplitude ofis twice the amplitude ofIfthe function is compressed. (Figure) compares several sine functions with different amplitudes.

### Amplitude of Sinusoidal Functions

If we letandin the general form equations of the sine and cosine functions, we obtain the forms

The amplitude isand the vertical height from the midline isIn addition, notice in the example that

### Identifying the Amplitude of a Sine or Cosine Function

What is the amplitude of the sinusoidal functionIs the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form

In the given function,so the amplitude isThe function is stretched.[/hidden-answer]

#### Analysis

The negative value ofresults in a reflection across the x-axis of the sine function, as shown in (Figure).

### Try It

What is the amplitude of the sinusoidal functionIs the function stretched or compressed vertically?

compressed

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

Now that we understand howandrelate to the general form equation for the sine and cosine functions, we will explore the variablesandRecall the general form:

The valuefor a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. Ifthe graph shifts to the right. Ifthe graph shifts to the left. The greater the value ofthe more the graph is shifted. (Figure) shows that the graph ofshifts to the right byunits, which is more than we see in the graph ofwhich shifts to the right byunits.

Whilerelates to the horizontal shift,indicates the vertical shift from the midline in the general formula for a sinusoidal function. See (Figure). The functionhas its midline at

Any value ofother than zero shifts the graph up or down. (Figure) compareswithwhich is shifted 2 units up on a graph.

### Variations of Sine and Cosine Functions

Given an equation in the formor$\frac{C}{B}\,$is the phase shift andis the vertical shift.

### Identifying the Phase Shift of a Function

Determine the direction and magnitude of the phase shift for

Let’s begin by comparing the equation to the general form

In the given equation, notice thatandSo the phase shift is

#### Analysis

We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign beforeThereforecan be rewritten asIf the value ofis negative, the shift is to the left.

### Try It

Determine the direction and magnitude of the phase shift for

right

### Identifying the Vertical Shift of a Function

Determine the direction and magnitude of the vertical shift for

Let’s begin by comparing the equation to the general form

In the given equation,so the shift is 3 units downward.

### Try It

Determine the direction and magnitude of the vertical shift for

2 units up

Given a sinusoidal function in the formidentify the midline, amplitude, period, and phase shift.

1. Determine the amplitude as
2. Determine the period as
3. Determine the phase shift as
4. Determine the midline as

### Identifying the Variations of a Sinusoidal Function from an Equation

Determine the midline, amplitude, period, and phase shift of the function

Let’s begin by comparing the equation to the general form

so the amplitude is

Next,so the period is

There is no added constant inside the parentheses, soand the phase shift is

#### Analysis

Inspecting the graph, we can determine that the period isthe midline isand the amplitude is 3. See (Figure).

### Try It

Determine the midline, amplitude, period, and phase shift of the function

midline:amplitude:period:phase shift:

### Identifying the Equation for a Sinusoidal Function from a Graph

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

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

The graph could represent either a sine or a cosine function that is shifted and/or reflected. Whenthe graph has an extreme point,Since the cosine function has an extreme point forlet 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 belowThis value, which is the midline, isin the equation, so

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. SoAnother way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, soAlso, the graph is reflected about the x-axis so that

The graph is not horizontally stretched or compressed, soand the graph is not shifted horizontally, so

Putting this all together,

### Try It

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

### Identifying the Equation for a Sinusoidal Function from a Graph

Determine the equation for the sinusoidal function in (Figure).

With the highest value at 1 and the lowest value atthe midline will be halfway between atSo

The distance from the midline to the highest or lowest value gives an amplitude of

The period of the graph is 6, which can be measured from the peak atto the next peak ator from the distance between the lowest points. Therefore,Using the positive value forwe find that

So far, our equation is eitherorFor the shape and shift, we have more than one option. We could write this as any one of the following:

• a cosine shifted to the right
• a negative cosine shifted to the left
• a sine shifted to the left
• a negative sine shifted to the right

While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes

Again, these functions are equivalent, so both yield the same graph.[/hidden-answer]

### Try It

Write a formula for the function graphed in (Figure).

two possibilities:or

### Graphing Variations of y = sin x and y = cos x

Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.

Instead of focusing on the general form equations

we will letandand work with a simplified form of the equations in the following examples.

Given the functionsketch its graph.

1. Identify the amplitude,
2. Identify the period,
3. Start at the origin, with the function increasing to the right ifis positive or decreasing ifis negative.
4. Atthere is a local maximum foror a minimum forwith
5. The curve returns to the x-axis at
6. There is a local minimum for(maximum for) atwith
7. The curve returns again to the x-axis at

### Graphing a Function and Identifying the Amplitude and Period

Sketch a graph of

Let’s begin by comparing the equation to the form

• Step 1. We can see from the equation thatso the amplitude is 2.
• Step 2. The equation shows thatso the period is
• Step 3. Becauseis negative, the graph descends as we move to the right of the origin.
• Step 4–7. The x-intercepts are at the beginning of one period,the horizontal midpoints are atand at the end of one period at

The quarter points include the minimum atand the maximum atA local minimum will occur 2 units below the midline, atand a local maximum will occur at 2 units above the midline, at(Figure) shows the graph of the function.

### Try It

Sketch a graph ofDetermine the midline, amplitude, period, and phase shift.

midline:amplitude:period:phase shift: or none

Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.

1. Express the function in the general form
2. Identify the amplitude,
3. Identify the period,
4. Identify the phase shift,
5. Draw the graph of shifted to the right or left byand up or down by

### Graphing a Transformed Sinusoid

Sketch a graph of

• Step 1. The function is already written in general form:This graph will have the shape of a sine function, starting at the midline and increasing to the right.
• Step 2.The amplitude is 3.
• Step 3. Sincewe determine the period as follows.

The period is 8.

• Step 4. Sincethe phase shift is

The phase shift is 1 unit.

• Step 5.(Figure) shows the graph of the function.

### Try It

Draw a graph ofDetermine the midline, amplitude, period, and phase shift.

midline:amplitude:period:phase shift:

### Identifying the Properties of a Sinusoidal Function

Givendetermine the amplitude, period, phase shift, and horizontal shift. Then graph the function.

Begin by comparing the equation to the general form and use the steps outlined in (Figure).

• Step 1. The function is already written in general form.
• Step 2. Sincethe amplitude is
• Step 3.so the period isThe period is 4.
• Step 4.so we calculate the phase shift asThe phase shift is
• Step 5.so the midline isand the vertical shift is up 3.

Sinceis negative, the graph of the cosine function has been reflected about the x-axis.

(Figure) shows one cycle of the graph of the function.

### Using Transformations of Sine and Cosine Functions

We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.

### Finding the Vertical Component of Circular Motion

A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y-coordinate of the point as a function of the angle of rotation.

Recall that, for a point on a circle of radius r, the y-coordinate of the point is
so in this case, we get the equation
The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in (Figure).

#### Analysis

Notice that the period of the function is stillas we travel around the circle, we return to the pointforBecause the outputs of the graph will now oscillate betweenandthe amplitude of the sine wave is

### Try It

What is the amplitude of the functionSketch a graph of this function.

7

### Finding the Vertical Component of Circular Motion

A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in (Figure). Sketch a graph of the height above the ground of the pointas the circle is rotated; then find a function that gives the height in terms of the angle of rotation.

Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in (Figure).

Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.

Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.

Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that

### Try It

A weight is attached to a spring that is then hung from a board, as shown in (Figure). As the spring oscillates up and down, the positionof the weight relative to the board ranges fromin. (at timetoin. (at timebelow the board. Assume the position ofis given as a sinusoidal function ofSketch a graph of the function, and then find a cosine function that gives the positionin terms of

### Determining a Rider’s Height on a Ferris Wheel

The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.

With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.

Passengers board 2 m above ground level, so the center of the wheel must be locatedm above ground level. The midline of the oscillation will be at 69.5 m.

The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.

Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.

• Amplitude:so
• Midline:so
• Period:so
• Shape:

An equation for the rider’s height would be

whereis in minutes andis measured in meters.[/hidden-answer]

Access these online resources for additional instruction and practice with graphs of sine and cosine functions.

### Key Equations

 Sinusoidal functions

### Key Concepts

• Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of
• The function is odd, so its graph is symmetric about the origin. The function is even, so its graph is symmetric about the y-axis.
• The graph of a sinusoidal function has the same general shape as a sine or cosine function.
• In the general formula for a sinusoidal function, the period isSee (Figure).
• In the general formula for a sinusoidal function,represents amplitude. Ifthe function is stretched, whereas ifthe function is compressed. See (Figure).
• The valuein the general formula for a sinusoidal function indicates the phase shift. See (Figure).
• The valuein the general formula for a sinusoidal function indicates the vertical shift from the midline. See (Figure).
• Combinations of variations of sinusoidal functions can be detected from an equation. See (Figure).
• The equation for a sinusoidal function can be determined from a graph. See (Figure) and (Figure).
• A function can be graphed by identifying its amplitude and period. See (Figure) and (Figure).
• A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See (Figure).
• Sinusoidal functions can be used to solve real-world problems. See (Figure), (Figure), and (Figure).

### Section Exercises

#### Verbal

Why are the sine and cosine functions called periodic functions?

The sine and cosine functions have the property thatfor a certainThis means that the function values repeat for everyunits on the x-axis.

How does the graph of
compare with the graph of
Explain how you could horizontally translate the graph of
to obtain

For the equationwhat constants affect the range of the function and how do they affect the range?

The absolute value of the constant(amplitude) increases the total range and the constant(vertical shift) shifts the graph vertically.

How does the range of a translated sine function relate to the equation

How can the unit circle be used to construct the graph of

At the point where the terminal side ofintersects the unit circle, you can determine that theequals the y-coordinate of the point.

#### Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period forRound answers to two decimal places if necessary.

amplitude:period:midline:maximum:occurs atminimum:occurs atfor one period, the graph starts at 0 and ends at

amplitude: 4; period:midline:maximumoccurs atminimum:occurs atone full period occurs fromto

amplitude: 1; period:midline:maximum:occurs atminimum:occurs atone full period is graphed fromto

amplitude: 4; period: 2; midline:maximum:occurs atminimum:occurs at

amplitude: 3; period:midline:maximum:occurs atminimum:occurs athorizontal shift:vertical translation 5; one period occurs fromto

amplitude: 5; period:midline:maximum:occurs atminimum:occurs atphase shift:vertical translation:one full period can be graphed onto

For the following exercises, graph one full period of each function, starting atFor each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period forState the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

amplitude: 1 ; period:midline:maximum:occurs atmaximum:occurs atminimum:occurs atphase shift:vertical translation: 1; one full period is fromto

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in (Figure).

amplitude: 2; midline:period: 4; equation:

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).

amplitude: 2; period: 5; midline:equation:

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).

amplitude: 4; period: 2; midline:equation:

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).

amplitude: 2; period: 2; midlineequation:

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).

#### Algebraic

For the following exercises, let

Onsolve

Onsolve

Evaluate

OnFind all values of

Onthe maximum value(s) of the function occur(s) at what x-value(s)?

Onthe minimum value(s) of the function occur(s) at what x-value(s)?

Show thatThis means thatis an odd function and possesses symmetry with respect to ________________.

For the following exercises, let

Onsolve the equation

Onsolve

Onfind the x-intercepts of

Onfind the x-values at which the function has a maximum or minimum value.

Onsolve the equation

#### Technology

GraphonExplain why the graph appears as it does.

GraphonDid the graph appear as predicted in the previous exercise?

The graph appears linear. The linear functions dominate the shape of the graph for large values of

Graphonand verbalize how the graph varies from the graph of

Graphon the windowand explain what the graph shows.

The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.

Graphon the windowand explain what the graph shows.

#### Real-World Applications

A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The functiongives a person’s height in meters above the ground t minutes after the wheel begins to turn.

1. Find the amplitude, midline, and period of
2. Find a formula for the height function
3. How high off the ground is a person after 5 minutes?

1. Amplitude: 12.5; period: 10; midline:
2. 26 ft

### Glossary

amplitude
the vertical height of a function; the constantappearing in the definition of a sinusoidal function
midline
the horizontal linewhereappears in the general form of a sinusoidal function
periodic function
a functionthat satisfiesfor a specific constantand any value of
phase shift
the horizontal displacement of the basic sine or cosine function; the constant
sinusoidal function
any function that can be expressed in the formor