Back To Course

High School Precalculus: Help and Review32 chapters | 297 lessons

Watch short & fun videos
**
Start Your Free Trial Today
**

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!

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 24 of the course:

Back To Course

High School Precalculus: Help and Review32 chapters | 297 lessons

- Solving Oblique Triangles Using the Law of Cosines 6:48
- Solving Real World Problems Using the Law of Cosines 6:37
- Using the Law of Sines to Solve a Triangle 7:02
- Solving Real World Problems Using the Law of Sines 7:09
- Proving the Addition & Subtraction Formulas for Sine, Cosine & Tangent 6:35
- How to Find the Period of Cosine Functions
- Go to Trigonometric Applications in Precalculus: Help and Review

- Communications 105: Intro to Mass Communications II
- Chemistry 303: Organic Chemistry I
- View High School: Biology
- Introduction to Purpose-Driven Sales
- Computer Science 311: Artificial Intelligence
- Noble Sales Techniques
- Working with Management in Sales
- Building Your Sales Network
- Cultural Impact of Television
- Sales Calls & Presentations
- FTCE Prep Product Comparison
- TExES Prep Product Comparison
- Study.com ASVAB Scholarship: Application Form & Information
- Study.com GED Scholarship: Application Form & Information
- Study.com GACE Scholarship: Application Form & Information
- Study.com CSET/CBEST Scholarship: Application Form & Information
- Study.com NES Scholarship: Application Form & Information

- Using Cross-Tabulation Reports in Marketing Research
- The Child-Saving Movement: History, Goals & Outcomes
- Designing Marketing Research Questionnaires
- Statistical Software for Marketing Research
- Unsupervised Learning in Machine Learning
- Practical Application for Data Structures: Stacks, Queues & Linked Lists
- Cultural Differences in Oral & Written Discourse
- Developing a Mission Statement & Philosophy for a Library Media Program
- Quiz & Worksheet - HR Negotiation & Bargaining
- Quiz & Worksheet - Bias in Personality Research
- Quiz & Worksheet - Measurement Unit Education
- Quiz & Worksheet - Context Effects & Consumer Choice
- Quiz & Worksheet - Target Market Analysis
- International Law & Global Issues Flashcards
- Foreign Policy, Defense Policy & Government Flashcards

- McDougal Littell Algebra 2: Online Textbook Help
- Worth Publishers Psychology: Online Textbook Help
- Essay Writing: Help & Tutorial
- Precalculus Algebra for Teachers: Professional Development
- Middle School World History Curriculum Resource & Lesson Plans
- Between the World Wars (1919-1939) Lesson Plans
- Unrest in Vietnam During the Eisenhower Years: Help and Review
- Quiz & Worksheet - Equipment for Physics Activities
- Quiz & Worksheet - Life in Different Canadian Locations
- Quiz & Worksheet - Online Marketing for B2C vs. B2B Models
- Quiz & Worksheet - Effects of Age on Long-Term Memory
- Quiz & Worksheet - Adult Development & Aging Controversies

- Distribution of Plants & Animals in South Asia
- North American Countries & Capitals
- Sequencing Activities for 3rd Grade
- Response to Intervention Training
- TExES Core Subjects 4-8 Exam Dates & Registration
- How Many Times Can You Take the NCLEX?
- 4th Grade Word Walls
- Writing Center Ideas for Preschool
- Shape Games for Kids
- MTEL Content Test Requirements
- Georgia Common Core Performance Standards
- How to Answer NCLEX Questions

Browse by subject