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Amplitude and Period Calculator

Choose a sine or cosine function, enter the coefficients and optional x-value, then click "Calculate" to determine the amplitude, period, phase shift, vertical shift, and function value.

Trigometric Function is f:

f(x) = A × sin(Bx-C) + D

x₀

Optional

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Amplitude and Period Calculator:

This calculator determines the amplitude, period, phase shift, and vertical shift for a periodic sinusoidal function, such as sine (f(x)=A·sin(Bx−C)+D) and cosine (f(x)=A·cos(Bx−C)+D).

Understanding amplitude and period is important because they help model the patterns of a sinusoidal function over time.

Sinusoidal Characteristics:

2.1. Amplitude:

amplitude of a sinusoidal

Amplitude is half the distance between the crest and trough of a sinusoidal wave. For standard sine and cosine functions, the amplitude is 1 because the centerline is at 0 and the range of the function is (-1, 1).

2.2. Period:

period of a sinusoidal function

The period is the length of one complete cycle of a periodic wave. For sine and cosine, the fundamental period is 2π since the functions repeat their pattern after this interval.

sin(0) = sin(2π) = sin(4π) = …
cos(0) = cos(2π) = cos(4π) = …

2.3. Phase (Horizontal) Shift:

phase shift of a sinusoidal function

Phase shift is the horizontal movement of a wave left or right. It does not affect the shape, amplitude, or period, but shifts the entire wave along the x-axis.

2.4. Vertical Shift:

vertical shift of a sinusoidal function

Vertical shift moves the entire function up or down along the y-axis. Like phase shift, it does not affect amplitude, period, or overall shape.

How to Calculate Amplitude and Period?

3.1. Using Sine & Cosine Equations:

Use the general form:

y = A sin(Bx + C) + D or y = A cos(Bx + C) + D

Formulas:

Amplitude = A
Period = 2π / |B|
Phase Shift = -C / B
Vertical Shift = D

3.2. Using a Graph

If you have a graph, analyze it as follows:

Amplitude:

Identify the mean line and peak of the wave
Calculate the vertical distance between these points
This value is the amplitude

Period:

Identify two consecutive peaks of the wave
Calculate the horizontal distance between these peaks
This distance is the period

Solved Problem (Amplitude & Period):

Example:

Find the amplitude, period, phase shift, and vertical shift for:

y = 3 sin(5x + 1) + 9

Step 1: Amplitude

Amplitude = A = 3

Step 2: Period

Period = 2π / |B| = 2π / 5 ≈ 1.256

Step 3: Phase Shift

Phase Shift = -C / B = -1 / 5 = -0.2

Step 4: Vertical Shift

Vertical Shift = D = 9

For instant calculations and detailed explanations, use our Amplitude and Period Calculator to analyze sine and cosine functions.

References:

Wikipedia: Amplitude, Peak amplitude & semi-amplitude, Peak-to-peak amplitude, Pulse amplitude, Amplitude normalization.
Khan Academy: Midline, amplitude, and period.
Lumen Learning: Amplitude and wavelength.