# How to Find the Period of Cosine Functions

## Steps to Solve

The **cosine function** is a trigonometric function that's called periodic. In mathematics, a **periodic function** is a function that repeats itself over and over again forever in both directions. Take a look at the basic cosine function *f*(*x*) = cos(*x*).

If we look at the cosine function from *x* = 0 to *x* = 2Ï€, we have an interval of the graph that's repeated over and over again in both directions, so we can see why the cosine function is a periodic function.

This interval from *x* = 0 to *x* = 2Ï€ of the graph of *f*(*x*) = cos(*x*) is called the **period** of the function. The period of a periodic function is the interval of *x*-values on which the cycle of the graph that's repeated in both directions lies. Therefore, in the case of the basic cosine function, *f*(*x*) = cos(*x*), the period is 2Ï€.

Wouldn't it be easy if that were the end of the story? Easy, yes, but much less interesting! You see, the cosine function takes on many forms, expressed as:

*f*(*x*) = *A*cos(*B**x* + *C*) + *D*

where *A*, *B*, *C*, and *D* are numbers, and the periods of these cosine functions differ. Thankfully, finding the period of these functions is still quite simple. It all depends on the value of *B* in the function *f*(*x*) = *A*cos(*B**x* + *C*) + *D*, where *B* is the coefficient of *x*. This is because the period of this function is 2Ï€ / |*B*|.

To find the period of *f*(*x*) = *A*cos(*B**x* + *C*) + *D*, we follow these steps:

- Identify the coefficient of
*x*as*B*. - Plug
*B*into 2Ï€ / |*B*|. This is the period of the function.

Let's now consider an example. Suppose we want to find the period of the function *g*(*x*) = 3cos(8*x* + 1). The first thing we would do is find the coefficient of *x*, which is 8, and take *B* = 8. Next, we plug *B* = 8 into the period formula. We end up getting:

Period = 2Ï€ / |*B*| = 2Ï€ / |8| = 2Ï€ / 8 = Ï€ / 4

We see that the period of the function *g*(*x*) = 3cos(8*x* + 1) is Ï€ / 4.

## Application

Because of the nature of the cosine function, it can be used to model anything in the real world having simple harmonic motion, where simple harmonic motion is described as moving back and forth in a constant fashion with no friction involved. Some examples of this could be a pendulum on a clock, springs, or alternating currents.

For example, suppose you hang a spring from your ceiling that you are considering using to hang a potted plant from. Without the plant on it, when you compress the spring 2 inches and let it go, its motion can be modeled by the following cosine function:

*y*= 2 cos ((3 Ï€ / 2)*x*)

where *y* is the displacement of the end of the spring, and *x* is the time in seconds.

In this scenario, do you see what the period of the function would represent? The period represents one cycle of the cosine function that repeats itself over and over again. Thus, in this example, the period would represent one cycle of the spring going from its highest, or most compressed position, to its lowest, or most stretched position, and then back to its highest position. Is that what you were thinking? You're definitely getting the hang of this, pun intended!

Okay, so let's find the period of this function. All we have to do is take the function through our steps. First, we identify *B* in the function, which is the coefficient of *x*. In this case, the coefficient of *x* is 3 Ï€ / 2, so *B* = 3 Ï€ / 2. Now, we simply plug it into our period formula.

We see that the period of the function is 4/3. This tells us that it takes 4/3 or 1 1/3 seconds for the spring to go through one cycle of bouncing. You may want to use something a bit more stable to hang your plant from, but you get the idea!

As we can see, the cosine function and its period can show up very easily in the world around us, so it's a good idea to tuck this newly acquired knowledge into our mathematical toolboxes to be used when we need it.

## Lesson Summary

Let's take a few moments to review the important information regarding what we've learned about finding the period of cosine functions. First, we learned that the **cosine function** is a trigonometric function that's called periodic and that, in mathematics, a **periodic function** is a function that repeats itself over and over again forever in both directions. The cosine function is expressed as *f*(*x*) = *A*cos(*B**x* + *C*) + *D* where *A*, *B*, *C*, and *D* are numbers, and the periods of these cosine functions differ.

We also learned the two simple steps to finding the **period**, or the interval between two points on the graph, of the cosine function, which are as follows:

- Identify the coefficient of
*x*as*B*. - Plug
*B*into 2Ï€ / |*B*|. This is the period of the function.

With this knowledge in hand, we should not only be able to solve cosine functions with ease, but also recognize them in the world around us.

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## Shifting or stretching Cosine Functions

In the video lesson we learned that for a cosine function {eq}f(x)=A\cos(Bx+C)+D {/eq}, the period of the cosine function is {eq}\dfrac{2\pi}{|B|} {/eq}. But what do the other numbers ({eq}A, C, D {/eq}) do to the graph of the cosine function?

## Example

Consider the functions: {eq}f(x)=\cos(2x)\\ g(x)=3\cos(2x)\\ h(x)=\cos(2x+\pi)\\ j(x)=\cos(2x)-1 {/eq}

1) What is the period of these functions?

2) Graph all four functions on the same coordinate plane. This can be done using a graphing calculator or online graphing tool, or by hand by picking values for {eq}x
{/eq} and plotting points.

3) How did the graphs of {eq}g(x), h(x), j(x) {/eq} differ from the graph of {eq}f(x) {/eq}?

## Solutions

1) All of the functions are of the form, {eq}y=A\cos(Bx+C)+D {/eq}, and all of the functions have the same {eq}B {/eq} value of {eq}2 {/eq}. The period of a cosine function is given by {eq}\dfrac{2\pi}{|B|} {/eq}, so all four functions have a period of {eq}\dfrac{2\pi}{2}=\pi {/eq}

2)

3) All of the graphs are similar looking - but the graph of {eq}g(x)=3\cos(2x) {/eq} is taller - it appears that multiplying by {eq}3 {/eq} stretched the graph vertically. The graph of {eq}h(x)=\cos(2x+\pi) {/eq} looks just like the graph of {eq}f(x)=\cos(2x) {/eq}, but shifted to the side {eq}\pi {/eq} units. The graph of {eq}j(x)=\cos(2x)-1 {/eq} looks like the graph of {eq}f(x)=\cos(2x) {/eq}, shifted down one unit.

## Explanation

For a function of the form, {eq}f(x)=A\cos(Bx+C)+D {/eq}, the number, {eq}A {/eq}, vertically stretches the graph of {eq}\cos(Bx) {/eq} by a factor of {eq}A {/eq}. The number, {eq}C {/eq}, shifts the graph of {eq}\cos(Bx) {/eq} horizontally {eq}C {/eq} units to the left, if {eq}C>0 {/eq} and {eq}C {/eq} units to the right if {eq}C<0 {/eq}. The number, {eq}D {/eq}, shifts the graph of {eq}\cos(Bx) {/eq} vertically {eq}D {/eq} units up if {eq}D>0 {/eq} and {eq}D {/eq} units down if {eq}D<0 {/eq}

## Questions to Check Understanding

1) What will the graph of {eq}g(x)=4\cos(3x-4)+1 {/eq} look like compared to the graph of {eq}f(x) = \cos(3x) {/eq}?

2) What are the periods of these functions?

3) Graph both functions to check your answer to part 1).

## Solutions

1) The graph of {eq}g(x) {/eq} will be shifted right {eq}4 {/eq} units, vertically stretched by a factor of {eq}4 {/eq}, and shifted up {eq}1 {/eq} unit.

2) Both functions have a period of {eq}\dfrac{2\pi}{3} {/eq}

3)

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