The cosine function is a trigonometric function that is periodic. A periodic function is a function that repeats itself over and over in both directions. The period of the cosine function is 2π, therefore, the value of the function is equivalent every 2π units. For example, we know that we have cos(π) = 1.
Every time we add 2π to the x values of the function, we have cos(π+2π). This is equivalent to cos(3π). We have the result cos(π)=1 and since the function is periodic, we also have the result cos(3π) = 1.
Period of the basic cosine function
The cosine function in its most basic form is $latex y=\cos(x)$. This function can be evaluated for any real value, so we can use all real values of x. This means that the function extends indefinitely to the right and to the left.
Using a graph of the cosine function, we can determine its period by looking at the distance between “equivalent” points. That is, the period of the function $latex y = \cos(x)$ is the distance on the x-axis between repeating patterns.
We can easily see that the graph repeats after 2π. Therefore, we conclude that the period of the function is 2π. The reason we have this period is that in the unit circle, 2π equals one complete revolution around the circle.
This means that if we have a value greater than 2π, we would simply be repeating the loop around the unit circle and we would obtain values equivalent to the angles between 0 and 2π.
Period of other variations of the cosine function
The period of the cosine function in its basic form, $latex y = \cos(x)$, is 2π. This period can be modified by multiplying the variable x by a constant.
We can reduce the period of the function by multiplying x by a number greater than 1. This will cause the function to be “sped up” and the period to get smaller. This means that the function will occur more quickly and it will take less for it to start repeating itself.
For example, in the function $latex y = \cos (2x)$, the period is π, which is half the period of the original function.
When we multiply the variable x by a fractional number that is greater than 0 and less than 1, we will make the function reduce its “speed” and make it have a larger period. This means that the function will take longer to start repeating itself.
For example, in the function $latex y = \cos(\frac{x}{2})$, the period is 4π, which is twice the period of the original function.
How to determine the period of a cosine function?
We can determine the period of a cosine function by using the coefficient of the variable x. This coefficient is usually represented by the letter B. Therefore, the standard form of the cosine function is $latex y = \sin(Bx)$. Using this form, we can obtain the following formula:
| $latex \text{Period}=\frac{2\pi}{|B|}$ |
This means that to obtain the period, we simply have to divide 2π by |B|, where, |B| is the absolute value of B. To find the absolute value, we just have to take the positive version of the number. For example, if we have -2, its absolute value is 2.
We can use this formula even when we have other variations of the cosine function. For example, if we have the function $latex y = 2 \cos(2x+5)$, we only take the coefficient of the variable x:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{2}$
$latex \text{Period}=\pi$
Period of the cosine function – Examples with answers
The following examples are solved using the formula for the period of cosine functions. Each example has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the solution.
EXAMPLE 1
- If we have the function $latex y = \cos(3x)$, what is its period?
Solution: We can identify the value $latex |B|=5$. Using this value in the formula for the period, we have:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{5}$
The period of the function is $latex \frac{2}{5}\pi$.
EXAMPLE 2
- What is the period of the cosine function $latex y=2 \cos(4x)-3$?
Solution: The function has a more complex form than the previous one, but we only need the coefficient of x. Therefore, we recognize the value $latex |B|=4$ and use it in the period formula:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{4}$
$latex \text{Period}=\frac{\pi}{2}$
The period of this function is $latex \frac{\pi}{2}$.
EXAMPLE 3
- What is the period of the function $latex y = \frac{1}{3}(- \frac{1}{5} x-2)$?
Solution: Again, we just use the coefficient of x to find the period. In this case, the coefficient is negative, so we only take its positive value. Therefore, we use the value $latex |B| = \frac{1}{5}$ in the period formula:
$latex \text{Period}=\frac{2\pi}{|B|}$
$latex \text{Period}=\frac{2\pi}{\frac{1}{5}}$
$latex \text{Period}=10\pi$
The period of the function is $latex 10\pi$.
Period of the cosine – Practice problems
Use what you have learned to solve the following exercises for period of cosine functions. If you need help with this, you can look at the solved examples above.
See also
Interested in learning more about the cosine of an angle? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
