The amplitude along with the period are characteristics of all cosine functions. The amplitude of a function is defined as the distance from the central axis to the maximum or minimum value of the function. In the case of the sine and cosine functions, the central axis is called the sinusoidal axis. The amplitude can also be defined as half the distance between the maximum value and the minimum value of the function. In the case of sound waves, amplitude represents a measure of how loud a sound is.

Here, we will learn how to determine the amplitude of cosine functions and solve some practice problems.

## How to determine the amplitude of cosine functions?

We can determine the amplitude of cosine functions by comparing the function to its general form. The general form of a cosine function is:

$latex f(x)=\pm A~\cos(B(x+C))+D$

In general form, the coefficient A is the amplitude of the cosine. If there is no number in front of the cosine function, we know that the amplitude is 1. The amplitude can be better understood using the graph of a cosine function. The following represents the graph of the function $latex y = 2 ~ \cos(x)$. The amplitude of this function is 2.

The amplitude is measured as a distance, so we use the absolute value of the maximum value or minimum value of the function. For example, in the case of the function $latex y= -2 ~ \cos(x)$, the graph would have a reflection with respect to the *x*-axis. However, this function would still have the same amplitude.

In this function, the sinusoidal axis is located on the *x-axis*. The sinusoidal axis is located exactly midway between the peaks and troughs of the function. If the function were translated vertically, the sinusoidal axis would be translated by the same amount, maintaining its initial position with respect to the peaks and troughs of the function.

Knowing the value of the amplitude of the function, it is possible to determine what the graph of the function will look like. As the amplitude of the function gets larger, its graph looks taller. Similarly, as the amplitude of the function gets smaller, its graph looks lower.

## Amplitude of the cosine function – Examples with answers

What has been learned about the amplitude of cosine functions is applied to solve the following examples. Each example has its respective solution, but try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

If we have the function $latex y=4 ~ \cos(2x)$, what is its amplitude?

##### Solution

We use the general form $latex y=A~\cos(B(x+C))+D$ and we find the value of *A* to determine the amplitude. If we compare the general form with the given function, we have:

$latex A=4$

This means that the amplitude is equal to 4.

**EXAMPLE 2**

What is the amplitude of the cosine function $latex y = -11 ~ \cos(3x) +4$?

##### Solution

We compare this function with the general form $latex y = A ~ \cos(B (x + C)) + D$. By doing this, we can find the following value:

$latex A=-11$

We know that the amplitude is measured using the absolute value, so the amplitude is equal to 11.

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**EXAMPLE 3**

If we have the function $latex y = \frac{1}{3} \cos(- \frac{1}{2} x-3)$, what is its amplitude?

##### Solution

Again, we compare the given function with the general form $latex y = A ~ \cos (B (x + C)) + D$. Therefore, we have the value:

$latex A=\frac{1}{3}$

The amplitude is equal to $latex \frac{1}{3}$. We can see that the amplitude can also be a fractional number and less than 1.

**EXAMPLE 4**

What is the amplitude of the function $latex y = 3 \cos(\frac{2}{3}(5x-2))$?

##### Solution

This function has a factor in front of it. The whole function is being multiplied by 3. Comparing this function with the general form $latex y = A ~ \cos (B (x + C)) + D$, we have:

$latex A=3(\frac{2}{3})$

$latex A=2$

The amplitude of the function is 2.

## Amplitude of the cosine – Practice problems

Solve the following problems using what you have learned about the amplitude 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:

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