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Phase Shift Calculator

Calculate, understand, and apply phase shifts for trigonometry, wave functions, physics, engineering, and sinusoidal equations. Enter A, B, C, and D in the live panel below, then read the full guide for formulas, examples, and applications.

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Phase Shift Calculator

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The trigonometric function in f

Result

Phase shift C / B =

Amplitude A =

Period 2π / B =

Vertical shift D =

B cannot be zero when finding phase shift or period.

Quick checks below the panel

Try A = 1, B = 2, C = π, D = 0 in sine mode: phase shift = π/2, period = π.
Switch to cosine with A = 3, B = 1, C = π/2, D = -2 to compare vertical shift and amplitude readouts.
See more worked examples in the Phase Shift Examples section.

What Is a Phase Shift?

A phase shift is the horizontal displacement of a sinusoidal graph. It tells you how far a sine or cosine wave moves left or right along the x-axis before the pattern repeats.

In standard form f(x) = A sin(Bx - C) + D, the phase shift equals C divided by B when B is not zero. That value describes where the wave starts relative to the parent function.

Phase shift appears in trigonometry homework, wave motion in physics, and timing offsets in signal processing. It is different from amplitude (height), period (length of one cycle), and vertical shift (midline height).

Definition: horizontal translation of a periodic graph, read from C and B in standard form.
Meaning: wave displacement along the input axis; often described as an angle or x-units.
Horizontal transformation: the graph slides without changing its shape when only C changes.
Real-world applications: matching alternating current timing, aligning audio tracks, and syncing oscillations in lab data.

Phase Shift Formula

General form: y = A sin(B(x - C)) + D → phase shift = C

Expanded form: f(x) = A sin(Bx - C) + D → phase shift = C / B

Cosine: f(x) = A cos(Bx - C) + D → phase shift = C / B

Amplitude = A

Period = 2π / B

Vertical shift = D

Many textbooks write B(x - C) inside the sine. If you expand that expression, the horizontal shift still comes from C, but the coefficient on x inside Bx - C must be matched carefully before you divide.

This calculator uses f(x) = A sin(Bx - C) + D (or cosine). Enter the numbers exactly as they appear in that form so phase shift = C / B matches your worksheet.

Graph interpretation: the phase shift marks where the first key point of the cycle occurs compared with sin(x) or cos(x). Sketch one cycle to confirm direction (left or right) using your course sign convention.

How to Calculate Phase Shift

Follow these steps by hand, in degrees or radians, or enter the same values in the calculator panel above for instant readouts.

Write the equation in standard form. Identify A, B, C, and D in f(x) = A sin(Bx - C) + D or the cosine version. Factor if the problem gives sin(B(x - h)) + k first.
Identify transformation values. Amplitude is A, vertical shift is D, and period uses B through 2π/B. Keep angle units consistent (radians or degrees) for C.
Extract the horizontal shift. Phase shift = C / B. Simplify the fraction. Example: C = π and B = 2 gives π/2 radians.
Convert degrees if needed. If C is in degrees, the shift can be reported in degrees. For radians, multiply degree measures by π/180 before dividing when mixing forms.
Verify with the calculator. Type A, B, C, and D into the panel, choose sine or cosine, and compare phase shift, amplitude, period, and vertical shift rows.

Phase Shift Examples

These examples cover sine, cosine, and applied wave settings. For tangent phase shifts (different period rules), see our trigonometry guide. More walkthroughs live in the examples article.

Sine function example

f(x) = 2 sin(3x - π/2) + 1

Result: A = 2, B = 3, C = π/2, D = 1. Phase shift = π/6, period = 2π/3, amplitude = 2, vertical shift = 1.

Cosine function example

f(x) = -1 cos(2x - π) + 4

Result: Phase shift = π/2, amplitude = -1, period = π, vertical shift = 4.

Alternate form B(x - C)

y = 5 sin(2(x - 3)) + 1 expands to sin(2x - 6) + 1 with B = 2 and C = 6.

Result: Phase shift = 6/2 = 3 in the Bx - C form used by the calculator.

Waveform timing example

Two signals y = sin(x) and y = sin(x - π/4) are offset by a quarter-cycle.

Result: Phase shift = π/4 radians between the waves, useful when aligning peaks in signal analysis.

Phase Shift in Trigonometry

Sinusoidal graphs combine amplitude, period, phase shift, and vertical shift. Phase shift is the horizontal piece: it moves every point on the graph by the same x-distance.

On the unit circle, a phase shift in the input angle rotates where the wave begins. Identities such as sin(x - φ) relate to cos(x) with an extra shift, which helps rewrite equations.

For deeper coverage, read Phase Shift in Trigonometry.

Function transformations: Phase shift is one of four parameters that reshape sin(x) or cos(x).
Unit circle: Input offsets correspond to starting angles on the circle.
Graph movements: Left or right slides depend on the sign of C and your textbook convention.
Identity relationships: cos(x) = sin(x + π/2) links sine and cosine shifts.

Phase Shift in Physics

Waves in physics are often written as sinusoids. Phase shift describes how far one oscillation is ahead of or behind another at the same frequency.

In simple harmonic motion, matching phase helps predict when a mass-spring system or pendulum reaches maximum displacement relative to a reference wave.

Explore applications in Phase Shift in Physics.

Wave motion: Compare crest timing between two traveling waves.
Oscillations: Shifted sine models describe motors, springs, and AC voltage.
Harmonic motion: Phase constants appear in x(t) = A cos(ωt + φ).
Frequency relationships: Same frequency with different phase still share period 2π/B.

Phase Shift in Signal Processing

Engineers describe timing offsets between periodic signals as phase. A shift in degrees or radians shows how much one waveform leads or lags another.

Audio alignment, electrical AC analysis, and communication timing all use the same math as trigonometry phase shift, with units tied to frequency.

Learn more in Phase Shift in Signal Processing.

Audio signals: Delay and phase correction align tracks and speakers.
Electrical waveforms: Voltage and current may be out of phase in AC circuits.
Communication systems: Carrier phase affects how receivers decode symbols.
Digital applications: Sampled sinusoids still use the same C/B relationship in discrete time.

Phase Shift vs Frequency Shift

Phase shift moves a graph horizontally at a fixed frequency. Frequency shift changes how fast the wave oscillates, which alters period and pitch.

Do not confuse C/B with B itself. B controls period (2π/B) and scaling of the input. C controls where the cycle starts.

Read the full comparison in Phase Shift vs Frequency Shift.

Phase shift: horizontal translation; formula C/B in f(x) = A sin(Bx - C) + D.
Frequency shift: change in B or ω that stretches or compresses the wave in time.
Graph view: phase slides the wave; frequency rescales its width.
Applications: phase aligns peaks; frequency tuning changes tone or carrier rate.

Common Phase Shift Mistakes

Most errors come from mixing the two standard forms or forgetting to divide by B. Always rewrite the equation before you report a shift.

Using C as the shift when the form is Bx - C instead of B(x - C).
Forgetting to divide by B after identifying C in the expanded form.
Mixing degrees and radians in the same calculation.
Confusing phase shift with vertical shift D or amplitude A.
Assuming tangent phase rules match sine without checking period formulas.

Graphing Phase Shift Functions

Sketching confirms calculator output. For a full plotting walkthrough, see Graphing Phase Shift Functions.

Mark amplitude and midline. Draw horizontal lines at y = D and y = D ± A.
Find period. One cycle spans 2π/B on the x-axis. Mark that interval.
Apply phase shift. Start the first key point at x = C/B (for the Bx - C form) relative to the parent graph.
Plot one cycle. Use quarter-period points (max, midline, min, midline) and repeat the pattern.
Compare with the calculator. Enter the same A, B, C, D and verify phase shift and period readouts match your sketch.

FAQs About Phase Shifts

What is the phase shift of sin(Bx - C)?

In f(x) = A sin(Bx - C) + D, phase shift = C / B. That is the horizontal shift in x-units when B is not zero.

How is y = A sin(B(x - C)) + D different?

That form uses C directly as the shift before factoring. Expanded, it becomes sin(Bx - BC) + D, so the Bx - C form uses C_input = BC.

How do you find period from B?

Period = 2π / B. The calculator shows this as a coefficient times π radians.

Does cosine use the same phase shift rule?

Yes. For f(x) = A cos(Bx - C) + D, phase shift = C / B with the same amplitude, period, and vertical shift rules.

What is phase shift in physics and signals?

It describes timing offset between periodic waves at the same frequency, such as voltage and current in AC circuits or aligned audio channels.

Is phase shift the same as frequency shift?

No. Phase shift slides the graph horizontally. Frequency shift changes B or ω and alters how quickly the wave repeats.

Why must B not be zero?

Both phase shift and period require division by B. When B = 0 the function is no longer a periodic wave in x.

Can I use degrees in the calculator?

Enter C as a decimal. For π/2 use 1.5708 or symbolic fractions if your workflow allows. Keep units consistent when comparing to textbook answers.