*Gerald Lemay*Show bio

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Lesson Transcript

Instructor:
*Gerald Lemay*
Show bio

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we look at a method for finding the amplitude of a sine function. This method works even when the equilibrium line is not the horizontal axis.
Updated: 04/20/2021

In general, we can write a sine function as:

The function of time, *f*(*t*), equals the amplitude, *A*, times the sine of *at* plus *b*, plus a vertical offset, *c*. If we're given an equation of this form, the amplitude of the sine function is simply *A*. When we don't have the equation and only a plot, we have to be careful. The vertical offset, *c*, can cause some difficulty in finding the amplitude.

The **amplitude** is defined as the vertical distance from the equilibrium line to the maximum of the curve, with the the maximum of the curve being called the **crest**. This vertical distance is the same as the distance from the equilibrium line to the minimum of the curve, with the minimum of the curve being called the **trough**. If there is no vertical offset, as in *c* = 0, the equilibrium line is the horizontal axis and the crest equals the amplitude.

When the equilibrium line isn't the horizontal axis, we can't say the amplitude is the crest. We can either find the new equilibrium line or we can use the peak values of the sine function. In this lesson we will use the peak values to determine the amplitude.

By the way, *a* is the frequency in radians per second (often written as Ï‰) and *b* is related to the horizontal shift. With *t* as the independent variable, we can find the horizontal shift by setting *at* + *b* equal to 0 and solving for *t*. Then, the horizontal shift equals -*b*/*a*. This horizontal shift is usually called the phase shift.

A sinusoid with arbitrary phase shift and frequency looks like this here:

The steps for finding the amplitude are as follows:

We can draw horizontal lines locating these displacements.

The maximum vertical displacement (the crest) is 2. The minimum vertical displacement (the trough) is -2.

max - min = 2 - (-2) = 4 and 4 divided by 2 is 2. Thus, the amplitude, *A* is 2.

In this example, the vertical offset, *c*, was zero.

The amplitude, *A*, of a sine function is given by

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These steps will work even if there's a vertical offset. For example, let's find the amplitude, *A*, for the following graph:

The maximum vertical displacement is 3, while the minimum vertical displacement is -1.

max - min = 3 - (-1) = 4 and 4 divided by 2 is 2. Thus, *A* = 2

Let's look at an example where we have to find the amplitude for the following sine function:

In this example, the entire function is below the horizontal axis. The method still works.

The maximum vertical displacement is -0.5 and the -4.5 is the minimum.

max - min = -0.5 - (-4.5) = 4 and 4 divided by 2 is 2. The amplitude, *A*, is 2

All right, let's take a moment or two to review. We first learned that a sine function can generally be defined as the function of time, *f*'(*t*), equals the amplitude, *A*, times the sine of *at* plus *b*, plus a vertical offset, *c*:

If we are given an equation of this form, the amplitude of the sine function is simply *A*. When we don't have the equation and only a plot, we have to be careful. The vertical offset, *c*, can cause some difficulty in finding the **amplitude**, which is the vertical distance from the equilibrium line to the maximum of the curve. We also learned that the maximum of the curve is called the **crest**, while the minimum of the curve is called the **trough**.

We also learned that in order to find the amplitude of a sine function, we only have to follow these two steps:

- Step 1: Determine the maximum and minimum vertical displacements, and
- Step 2: Take the difference of max minus min and divide by 2

It's really that simple! So, you should have no trouble finding the amplitude of a sine function the next time you see that on a test.

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