Phase Shift Calculator

Phase shift is the horizontal displacement of a sinusoidal function from its standard position. For y = a·sin(bx − c) + d, the phase shift equals c / b. A positive value moves the graph to the right, and a negative value moves it to the left.

Factor out b from the argument so the equation reads y = a·sin[b(x − h)] + d. The value h = c / b is the phase shift. For example, y = sin(2x − π) rewrites as y = sin[2(x − π/2)], giving a phase shift of π/2 radians (90°).

Phase shift is the horizontal displacement in the units of x (radians or degrees). Phase angle is the raw constant c before dividing by b. Phase shift = phase angle / b.

Yes. The formula c / b works identically for y = a·cos(bx − c) + d. Cosine simply starts at its peak rather than at zero, but the horizontal shift calculation is unchanged.

Use |b| for the period (Period = 2π / |b|). The phase shift c / b will flip sign. For clearer graphing, factor out the negative and interpret the shift from the rewritten form.

Locate where the sine wave crosses its midline going upward (or where cosine reaches its peak). The horizontal distance from the origin to that point is the phase shift. Compare to the standard unshifted curve to determine direction.

Phase shift measures horizontal displacement; period measures one complete cycle length. Dividing phase shift by period tells you what fraction of a cycle the wave has shifted.

Yes, but because sinusoidal functions repeat every period, shifting by exactly one period yields an identical graph. You can always reduce the phase shift modulo the period.

In electrical engineering, physics, and acoustics, phase shifts represent time delays in AC circuits, sound waves, and electromagnetic signals. A 180° shift inverts a signal; a 90° shift moves it to its quadrature component.

The constant d moves the entire graph vertically. It defines the midline. The wave oscillates between d + |a| and d − |a|. When d = 0 the oscillation is centered on the x-axis.

Factoring converts y = a·sin(bx − c) + d into y = a·sin[b(x − c/b)] + d, making the horizontal shift c/b immediately visible. Without factoring, it is easy to misidentify the shift as just c.

In AC circuits, the phase shift between voltage and current waveforms determines power factor. An inductive load causes current to lag voltage; a capacitive load causes current to lead. Phase shift calculators help size correction capacitors.

Rewrite as y = sin[1·(x + π/3)]. Here b = 1, c = −π, but more directly the shift is −π/3 radians (−60°), meaning the graph shifts 60° to the left.

Yes. For example, y = 2·sin(4x − 2π) and y = 5·sin(4x − 6π) both have phase shift = 2π / 4 = π/2 radians. The amplitude and vertical shift differ, but the horizontal displacement is the same.

Amplitude (|a|) does not affect phase shift at all. Phase shift depends only on c and b. Changing the amplitude stretches or compresses the wave vertically without altering its horizontal position.

Any function A·sin(x) + B·cos(x) can be rewritten as R·sin(x + φ) where R = √(A² + B²) and φ = arctan(B/A). The phase shift is then −φ radians.

Multiply radians by 180/π to get degrees. Multiply degrees by π/180 to get radians. For example, π/4 radians × (180/π) = 45°.

The negative sign reflects the graph across the x-axis, which is equivalent to a phase shift of π radians (180°). So y = −sin(x) = sin(x + π) = sin(x − π).

Phase shift is measured in angular units (degrees or radians). Time delay is the physical time difference: time delay = phase shift / (2π × frequency). They describe the same phenomenon in different units.

This calculator handles single-term sinusoidal functions y = a·sin(bx − c) + d or their cosine equivalents. For sums of multiple sinusoidal terms, use Fourier analysis tools.

A negative phase shift means the graph moves to the left. This happens when c and b have opposite signs, making c/b negative. The function reaches its key points earlier than the standard wave.

The formula c/b gives the same numerical phase shift regardless of whether you use sin or cos. However, the visual starting point differs: sine starts at the midline; cosine starts at the peak. The shift is relative to whichever base you choose.

Convert using the identity A·sin(θ) + B·cos(θ) = R·sin(θ + φ). Here R = √2 and φ = 45°, so the combined wave is √2·sin(2x + π/4), giving a phase shift of −π/8 radians relative to sin(2x).

Audio engineers use phase shift to align multiple microphones, design crossover networks, and create spatial effects. Phase cancellation from misaligned speakers causes frequency nulls — this calculator helps visualize the underlying math.

The Phase Shift Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Phase Shift Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.