Phase Shift Calculator

Why it is −C/B

The shift is about x, and C is not sitting next to x on its own — it is sitting next to Bx.

Factor the bracket. 3x − 90 is 3(x − 30). Now the bracket reads (x − 30), and the 30 is directly comparable with x, so the shift is 30.

Do that in general: Bx + C = B(x + C/B), so the shift is −C/B. Reading −C off the unfactored equation gives an answer that is wrong by exactly the factor B.

It is invisible when B is 1, which is where everybody meets it first. y = sin(x − 90°) really does shift by 90°, so the division never announces itself — and then the first question with B = 3 gets an answer three times too large.

The habit worth building: factor before answering. In factored form there is nothing left to get wrong.

Which way is positive

A minus inside the bracket moves the graph right. This feels backwards and it is worth understanding rather than memorising.

y = sin(x − 90°) reaches at x = 90° whatever y = sin(x) reached at x = 0°. To get the same output you need a larger input, so every feature happens later — further right.

So the sign inside the bracket and the direction of travel are opposite. The formula −C/B already accounts for it: with C = −90 the shift is +30, and positive means right.

Some sources define the shift as +C/B and say a positive value means left. Both conventions are in use, which is exactly why this page states which one it is using rather than assuming.

The four parameters

A — amplitude. The height from the midline to a peak, reported as |A|. A negative A flips the wave upside down but does not make the amplitude negative; amplitude is a distance.

B — frequency factor. The period is 360°/|B|. Larger B means more cycles in the same span, so the wave is squeezed horizontally. B is not the period; it is what you divide into 360 to get it.

C — phase. Horizontal shift, after dividing by B.

D — vertical shift. Moves the whole wave up or down, and the midline is y = D. The maximum is D + |A| and the minimum is D − |A|.

A and D describe the vertical picture, B and C the horizontal one. Only C needs the other coefficient to interpret it, which is the whole difficulty.

Sine or cosine

The two are the same wave, a quarter cycle apart: cos x = sin(x + 90°).

So any wave can be written either way, with a different phase. y = 3 cos(2x) and y = 3 sin(2x + 90°) are the same curve, and neither form is more correct.

That means the phase shift is not a property of the wave alone — it depends on which function you chose to write it with. When comparing two phase shifts, check they were both written against the same function.

In practice: physics tends to write cosine, since it starts at a maximum, and mathematics tends to write sine. Converting between them shifts the phase by 90°, which is one of the more common sources of a factor-of-nothing disagreement between two correct answers.

Degrees and radians

This page works in degrees, and the period comes out in degrees too.

In radians the only change is the period formula: 2π/|B| instead of 360/|B|. Everything else is identical, since amplitude and vertical shift have no angular unit at all.

The trap is mixing them. An equation written with a phase in radians and read with a degree convention gives a shift wrong by a factor of about 57, and the resulting graph looks nothing like the data.

A quick tell: if the phase constant is a small number like 1.57 or 3.14, it is almost certainly radians. If it is 90 or 180, it is degrees.

Where waves like this turn up

Alternating current. Mains voltage is a sinusoid, and the phase difference between voltage and current is what the power factor measures. A phase shift there has a direct cost on an electricity bill.

Sound and music. A pure tone is a sinusoid; amplitude is loudness and the period sets the pitch. Two identical tones out of phase by half a period cancel entirely, which is how noise-cancelling headphones work.

Tides and daylight. Both are close to sinusoidal over a year, with the phase shift encoding when the peak falls.

Seasonal data. Fitting A sin(Bx + C) + D to monthly figures gives the size of the seasonal swing in A, the average level in D, and when the peak occurs in C — which is often the most interesting of the four.

Sources and methodology

The parameter conventions are standard across curricula and signal processing; these are the references.

• National Council of Teachers of Mathematics — where the transformations of trigonometric graphs are set out
• NIST Time and Frequency Division — on frequency, period and phase as measured quantities
• Trigonometric Functions Calculator — the unshifted function underneath
• How we calculate — methodology

Method. The shift is computed as −C/B and the equation is rewritten in factored form so the reader can see where that division comes from. Amplitude is reported as |A| rather than A, since a negative coefficient flips the wave without making its amplitude negative. The suite checks that the wave really does sit on its midline at the reported shift, across every combination of six B values and twenty-five phases. That engine is verified on every change against 71 hand-written assertions, including that evaluating the wave at the reported phase shift lands exactly on the midline across one hundred and fifty generated equations, which a missing divide-by-B would break immediately. The count and the per-case breakdown are published on the formula verification page.

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Where this goes next:

More in Math, or browse all calculators.

Read the guide

The unshifted functions underneath are on the Trigonometric Functions Calculator, and the identities that let one wave be rewritten as another are on the Sum and Difference Identities Calculator.