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how to find amplitude

How to Find Amplitude (with Simple Examples)

Quick answer: The amplitude of a wave or trig function is its maximum distance from the middle (equilibrium) position. From an equation like $$y = A \sin(\dots)$$ or $$y = A \cos(\dots)$$, the amplitude is $$\lvert A \rvert$$. From a graph, it’s the vertical distance from the midline to a peak or trough.

What Is Amplitude?

  • Amplitude = maximum displacement from the equilibrium (middle) position.
  • It is always taken as a positive value.
  • In a sine or cosine wave, it tells you how “tall” the wave is.
Think of a mass on a spring moving up and down: the amplitude is how far it moves from the center position to its highest (or lowest) point.

Method 1: From a Trig/Wave Equation

Standard sine or cosine form

For functions like:
  • $$y = A \sin(\omega t + \phi)$$
  • $$y = A \cos(\omega t + \phi)$$
The amplitude is:
  • $$\text{Amplitude} = \lvert A \rvert$$
Here:
  • $$A$$ = amplitude
  • $$\omega$$ = angular frequency
  • $$\phi$$ = phase angle
[3][9][1] Example: If y=5sin⁡(10πt−0.1πx)y=5\sin(10\pi t-0.1\pi x)y=5sin(10πt−0.1πx), compare to y=Asin⁡(ωt+ϕ)y=A\sin(\omega t+\phi)y=Asin(ωt+ϕ). You see A=5A=5A=5, so amplitude = 5.

Another example:
If y=−3cos⁡(2x+π)y=-3\cos(2x+\pi)y=−3cos(2x+π), amplitude = ∣−3∣=3\lvert -3\rvert =3∣−3∣=3.

Method 2: From Max and Min Values

If you know the maximum and minimum values of a sine/cosine function (or any periodic signal), you can use:

  • Amplitude = (max value − min value) / 2
(Some sources also state amplitude as (max+min)/2(\text{max}+\text{min})/2(max+min)/2 in a particular sign convention, but the distance from the midline to a peak is effectively (max−min)/2(\text{max}−\text{min})/2(max−min)/2, and we take it as positive.)

Example:
Suppose a wave ranges from −2 to 6.

  • Max = 6
  • Min = −2

  • Amplitude = (6 − (−2)) / 2 = 8 / 2 = 4

Method 3: From a Graph

To find amplitude directly from a graph of a wave:
  1. Find the equilibrium (midline) – often the x-axis (y = 0) or a horizontal center line.
  2. Find the y-value of a highest point (peak) or lowest point (trough).
  3. Compute the vertical distance between that point and the midline.
  4. Take the positive value of that distance – that’s the amplitude.
Example: If the midline is at y = 0 and the highest peak is at y = 3, the amplitude is 3.

If the midline is at y = 1 and the lowest trough is at y = −3, then amplitude = distance from 1 down to −3 = 4.

Method 4: For Simple Harmonic Motion

For a mass on a spring, a pendulum (small angles), or any simple harmonic motion described as:
  • $$x(t) = A\sin(\omega t + \phi)$$ or $$x(t) = A\cos(\omega t + \phi)$$
The amplitude is again ∣A∣\lvert A\rvert ∣A∣, the maximum displacement from the equilibrium position.

Example story:
Imagine a child on a swing whose seat moves between 0.4 m forward and 0.4 m backward from the rest position. The amplitude of the motion is 0.4 m.

Special Case: Two Waves Adding (Superposition)

When two identical waves of amplitude aaa combine, the resultant amplitude depends on the phase difference between them. A common relation used in basic physics is:
  • $$A_{\text{resultant}} = 2a \cos(\phi / 2)$$
Where:
  • $$a$$ = amplitude of each wave
  • $$\phi$$ = phase difference
Example: If each wave has amplitude 3 and the phase difference is 60∘60^\circ 60∘, the resultant amplitude is 2⋅3⋅cos⁡(30∘)=6⋅(3/2)=332·3·\cos(30^\circ)=6·(\sqrt{3}/2)=3\sqrt{3}2⋅3⋅cos(30∘)=6⋅(3​/2)=33​.

Mini FAQ & Quick Reference

  • Q: What’s the fastest way to find amplitude from $$y = A \sin(\dots)$$? A: Take the absolute value of the coefficient in front of sin or cos.
  • Q: What if the graph looks messy? A: Find one clear peak or trough and measure straight up or down to the midline. That distance is the amplitude.
    • [10][7]
  • Q: What if I only know max and min values? A: Use $$(\text{max} − \text{min}) / 2$$ to get amplitude.
    • [3][5]

SEO-style meta description

Meta description: Learn how to find amplitude from equations, graphs, and max–min values. Clear formulas, step-by-step methods, and simple examples to help you understand amplitude quickly.

Note: Information gathered from public educational resources and forums online.

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TL;DR: To find amplitude, look for the maximum distance from the midline: from equations, it’s the absolute value of the coefficient; from a graph, it’s the vertical distance from midline to a peak or trough; from max–min data, use $$(\text{max} − \text{min})/2$$.