How to Graph cos(x)

How to Graph cos(x)
Coming up next: How to Graph 1-cos(x)

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:04 Steps to Solve Cos(x)
  • 2:54 Cosine Function from…
  • 3:56 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

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

The cosine function shows up often in mathematics, so being familiar with it is very useful. We will learn how to graph cos(x) using its various properties, and we'll look at how to use the graph of the sine function to create the graph of cos(x).

Steps to Solve Cos(x)

To graph y = cos(x), we need to be familiar with the properties of the cosine function. We can use these properties to create the graph of y = cos(x).

Let's first take a look at the properties of the cosine function.

  • The domain (which includes the x-values we can plug into y = cos(x) and have a defined function) of y = cos(x) is all real numbers.
  • The range (which is the y-values the function takes on) of y = cos(x) is all real numbers greater than or equal to -1 and less than or equal to 1.
  • A periodic function is a function that takes on the same values at regular intervals. In other words, it's a function that repeats itself after a specific period of time. The cosine function is periodic.
  • The period of a periodic function is the length of the interval of x-values before the function repeats itself. The period of the cosine function is 2pi.

By analyzing these properties along with plotting a few strategic points on the graph, we can graph y = cos(x).

Because the cosine function is periodic with period 2pi, we know that it completes one cycle from x = 0 to x = 2pi. We also know that the domain function of the cosine function is all real numbers. Using these two facts, we can graph one cycle of the cosine function between x = 0 and x = 2pi. Then we can extend it in both directions, since we know it will repeat itself forever along the x-axis.

We're also given that the range, or the y-values of the cosine function, is between -1 and 1, so we know the graph won't go above y = 1 or below y = -1. Thus, the entire graph will lie between y = -1 and y = 1. This information gives us an idea of where we want to sketch one cycle of y = cos(x) before extending it in both directions, which you can see in the square in the graph here.


Area Where One Period Will be Graphed
cos1


Let's strategically plot some points by plugging values of x into y = cos(x) and finding corresponding y-values. This will give us some points to plot and then connect with a smooth continuous curve. We want to use values of x so that cos(x) is easy to calculate, and we want those values of x to fall between 0 and 2pi. The table that's been on your screen shows cos(x) evaluated at some values of x that fall between 0 and 2pi and result in nice values of y.

x y = cos(x)
0 1
pi/3 1/2
pi/2 0
2pi/3 -1/2
pi -1
4pi/3 -1/2
3pi/2 0
5pi/3 1/2
2pi 1

Next, we can plot these points on our graph.


Plotted Points
cos2


Now we're getting somewhere! The next step is to connect the dots with a smooth continuous curve.


One Cycle of the Graph of cos(x)
cos3


We now have one period of the graph of y = cos(x). The last thing we need to do is extend the graph in both directions.

Once we extend the graph in both directions, we have the graph of y = cos(x).


Graph of cos x
cos4


Cosine Function from Sine Function

Another trigonometric function is the sine function. The image below is showing the graph of the sine function.


Graph of sin(x)
cos5


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

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 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

Transferring credit to the school of your choice

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.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support