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Phase Shift Calculator

Calculate the phase shift of a trigonometric function y = A sin(Bx − C) + D.

Reviewed by Christopher FloiedPublished Updated

This free online phase shift calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Minimum: 0

Coefficient of x inside the trig function

The value subtracted from Bx

Results

Phase Shift

1.5 units right

Period

3.1416 units

Amplitude

1

Vertical Shift

0

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Phase Shift Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Phase Shift Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Phase Shift Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Phase Shift Calculator

The phase shift of a trigonometric function describes the horizontal displacement of the wave from its standard position. For the general sinusoidal function y = A sin(Bx − C) + D, the phase shift is C/B units to the right (or −C/B to the left if negative). Understanding phase shift is crucial in physics, where it describes the timing difference between waves, in electrical engineering for analyzing AC circuits, in signal processing for filtering and modulation, and in music for understanding sound wave interference. The amplitude A controls the height, B controls the frequency (and thus the period, 2π/B), C/B controls the horizontal shift, and D controls the vertical shift. This calculator determines the phase shift and period from the parameters of the sinusoidal function.

The Math Behind It

The function y = A sin(Bx − C) + D can be rewritten as y = A sin(B(x − C/B)) + D, which reveals that the graph is a sine curve shifted C/B units to the right. The four parameters control different aspects: |A| is the amplitude (half the peak-to-peak distance), 2π/B is the period (horizontal length of one cycle), C/B is the phase shift (horizontal displacement), and D is the midline (vertical displacement). The frequency is f = B/(2π) cycles per unit. In physics, the angular frequency ω = B relates to the frequency by ω = 2πf. Phase shift is often expressed in degrees or radians relative to a reference wave: if two waves have the same frequency but different phase shifts, their interference pattern depends on the phase difference. Constructive interference occurs when the phase difference is 0° (or multiples of 360°), and destructive interference occurs at 180° phase difference. In AC circuits, the voltage and current can have different phase shifts, and the power factor is cos(phase difference).

Formula Reference

Phase Shift

Phase Shift = C / B

Variables: From y = A sin(Bx − C) + D

Period

Period = 2π / B

Variables: B = frequency coefficient

Worked Examples

Example 1: Analyzing y = 3 sin(2x − π) + 1

Identify amplitude, period, phase shift, and vertical shift.

Step 1:A = 3 (amplitude)
Step 2:B = 2, so period = 2π/2 = π
Step 3:C = π, so phase shift = π/2 to the right
Step 4:D = 1 (vertical shift up by 1)

Amplitude: 3, Period: π, Phase Shift: π/2 right, Vertical Shift: 1 up

Common Mistakes & Tips

  • !Confusing C with the phase shift — the phase shift is C/B, not C alone.
  • !Forgetting to factor out B before identifying the phase shift.
  • !Confusing phase shift direction: positive C/B shifts right, negative shifts left.

Related Concepts

Sine Calculator

Compute sine values.

Cosine Calculator

Compute cosine values.

Frequently Asked Questions

What is the difference between phase shift and phase angle?

Phase shift is the horizontal displacement (C/B in units of x), while the phase angle is C itself (in radians or degrees). They are related by phase shift = phase angle / B.

Does phase shift affect the shape of the wave?

No. Phase shift only moves the wave horizontally. The shape, amplitude, and period remain unchanged.

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