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.

Here, we will learn how to determine the amplitude of trigonometric functions, especially sine and 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.

## How to find the amplitude of sine functions?

The general form of a sine function is:

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

In this form, the coefficient A is the “height” of the sine. If we do not have any number present, then the amplitude is assumed to be 1. We can define the amplitude using a graph. The following is the graph of the function $latex y = 2 ~ \sin(x)$, which has an amplitude of 2:

We observe that the amplitude is 2 instead of 4. In this case, the amplitude corresponds to the absolute value of the maximum value or minimum value of the function. If we had the function $latex y = -2 ~ \sin (x)$, the graph would be reflected with respect to the *x*-axis, but its amplitude would remain the same.

The sinusoidal axis is the horizontal line between the peaks and the troughs. In this function, the sinusoidal axis is simply the *x*-axis. However, if the graph were translated vertically, the sinusoidal axis would no longer be on the *x*-axis but would be located exactly in the middle of the peaks and troughs.

The larger the amplitude of the function, the taller its graph will appear. On the other hand, the smaller the amplitude of the function, the lower its graph will appear.

## Amplitude of the trigonometric functions – Examples with answers

**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 function $latex y = 3 ~ \sin(2x)$?

##### Solution

To determine the amplitude of the function, we have to compare it with the general form $latex y = A ~ \sin(B (x + C)) + D$. Comparing the functions, we see that we have:

$latex A=3$

This means that the amplitude is equal to 3.

**EXAMPLE **3

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.

**EXAMPLE **4

If we have the sine function $latex y = -4 ~ \sin(4x) +1$, what is its amplitude?

##### Solution

We use the general form $latex y = A ~ \sin(B(x+C))+D$ and compare it with the given function. When comparing them, we see that we have:

$latex A=-4$

We know that the amplitude is the absolute value of this parameter, so the amplitude is equal to 4.

**EXAMPLE **5

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 **6

What is the amplitude of the function $latex y = \frac{1}{3} \sin(- \frac{1}{4} x-4)$?

##### Solution

Again, we have to compare the given function with the general form $latex y = A ~ \sin(B(x + C)) + D$. By doing this, we have:

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

The amplitude is equal to $latex \frac{1}{3}$. Therefore, the amplitude does not necessarily have to be an integer value.

**EXAMPLE **7

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.

**EXAMPLE **8

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

##### Solution

In this case, we see that the entire function is being multiplied by 2. This means that when we compare the function with the general form $latex y = A ~ \sin (B (x + C)) + D$, we have:

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

$latex A=3$

The amplitude of the function is 3.

## Amplitude of trigonometric functions – Practice problems

#### What is the amplitude of the function $latex y=2.1(-2\cos(\frac{1}{2}x)-5)$?

Write the answer in the input box.

## See also

Interested in learning more about trigonometric functions? Take a look at these pages:

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