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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will show how to find the period of cosine functions. We will then take it a step further and look at the amplitude, phase shift, and vertical shift of a cosine function and how to find all of these characteristics of cosine functions.

The **cosine function** is a trigonometric function that is 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* = 2π, we have an interval of the graph that is 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* = 2π 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 is repeated in both directions lies. Therefore, in the case of the basic cosine function, *f*(*x*) = cos(*x*), the period is 2π.

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 2π / |*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 2π / |*B*|. This is the period of the function.

Let's 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.

- Period = 2π / |
*B*| = 2π / |8| = 2π / 8 = π / 4

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

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

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

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 π / 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 π / 2, so *B* = 3 π / 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 see, the cosine function and its period can show up very easily in the world around us, so its a good idea to tuck this newly acquired knowledge into our mathematical toolboxes to be used when we need it!

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High School Precalculus: Help and Review32 chapters | 297 lessons

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