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How to Find the Period of Cosine Functions

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.

Steps to Solve

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).


Periodic Cosine Function
cosper1


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) = Acos(Bx + 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) = Acos(Bx + C) + D, where B is the coefficient of x. This is because the period of this function is 2π / |B|.


percos1


To find the period of f(x) = Acos(Bx + C) + D, we follow these steps:

  1. Identify the coefficient of x as B.
  2. 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(8x + 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(8x + 1) is π / 4.

Solution

To find the period of cosine functions of the form f(x) = Acos(Bx + C) + D, we follow these steps:

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

Application

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.


Problem
cosper11


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