How to Graph cos(x)

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  • 0:04 Steps to Solve Cos(x)
  • 2:54 Cosine Function from…
  • 3:56 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

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

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

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)

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

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)

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