How to Find Amplitude of a Sine Function

Instructor: Gerald Lemay

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.

How to Find the Amplitude

In general, we can write a sine function as


sine_function


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.

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

When the equilibrium line is not 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 ω) 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


A sine function
sine_function


The steps for finding the amplitude:

Step 1: Determine the maximum and minimum vertical displacements.

We can draw horizontal lines locating these displacements.


Peak values of sine function
peak_values


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

Step 2: Take the difference of max minus min and divide by 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 Final Result

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


amplitude_equation


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