A phase shift calculator pulls the four descriptive numbers out of a sinusoid: how tall it is, how long one cycle takes, how far sideways it has slid, and how far up or down the whole thing sits. Three of those are straightforward. The fourth — the sideways slide — is where nearly everyone loses a sign, and it is the reason this page exists.
Arb Digital built this tool around the convention rather than around the arithmetic. The arithmetic is one division. The convention is that the standard form is written y = A·sin(B(x − C)) + D, with a minus sign already inside the bracket, so a positive C means a shift to the right. Miss that and every shift you calculate points the wrong way, and the graph you draw is a mirror of the one you wanted.
The Sign Convention, Stated Plainly
Write the equation as y = A·sin(B(x − C)) + D. Then:
- C positive means shift right by C. y = sin(x − 2) is the sine curve moved 2 units to the right.
- C negative means shift left by |C|. y = sin(x + 2) is sin(x − (−2)), so C = −2 and the curve moves 2 units left.
- The shift is measured in x-units, not in radians of angle unless B happens to be 1. This is the part that catches people out when B is not 1.
The intuition that makes it stick: the graph reaches whatever it used to do at x = 0 only when the bracket is zero, which now happens at x = C. Everything the curve did has been postponed until x reaches C, so a positive C delays the curve — and delaying it means moving it right. Read the minus sign as "the curve waits until x catches up with C".
Why the Expanded Form Misleads
Textbooks and exam papers usually give you the expanded form, y = A·sin(Bx − c) + D, and the phase shift is not c. It is c ÷ B. The two forms are related by factoring B out of the bracket: Bx − c = B(x − c/B).
Take y = 3 sin(2x − π/2) + 1, which is the default loaded above. The temptation is to say the shift is π/2 to the right. It is not. Factoring gives 3 sin(2(x − π/4)) + 1, so the shift is π/4 ≈ 0.7854 to the right — half of what the raw number suggested. The larger B is, the bigger the discrepancy, because B compresses the x-axis and the shift is compressed with it.
This is the single most common error in the topic, and it has a memorable form: divide by B before you decide anything. The tool's "convert to the other form" button does that conversion for you in both directions so you can see the two ways of writing the same wave next to each other. Our trigonometric functions calculator evaluates individual sine and cosine values if you want to verify a point on the curve by hand.
The Formulas: How Each Value Is Calculated
From y = A·sin(B(x − C)) + D, every descriptor is a one-line read:
- Amplitude = |A|. The absolute value matters: a negative A does not change the height, it flips the curve upside down.
- Period = 2π ÷ |B| in radians, or 360° ÷ |B| in degrees. With B = 2 the period is π, so the wave completes two cycles in the space the plain sine takes for one.
- Phase shift = C in the factored form, or c ÷ B in the expanded form.
- Vertical shift = D, which is also the midline. The maximum is D + |A| and the minimum is D − |A|.
Checking the default: A = 3, so amplitude 3. B = 2, so period 2π ÷ 2 = π ≈ 3.1416. c = π/2 in the expanded form, so shift π/4 ≈ 0.7854 right. D = 1, so the midline is y = 1 and the range runs from −2 to 4. Frequency, the reciprocal of period, is 1 ÷ π ≈ 0.3183 cycles per unit. The definitions and periodicity underlying all of this are set out in the DLMF section on trigonometric functions.
A Negative B Is a Reflection, Not a Backwards Wave
If B comes out negative, do not simply drop the sign. Because sine is an odd function, sin(−u) = −sin(u), so y = 2 sin(−3x) is identical to y = −2 sin(3x): the sign moves out to the amplitude and the curve is reflected in its midline. Cosine is even, so cos(−u) = cos(u) and a negative B changes nothing at all for a cosine.
The practical rule is to normalise first: rewrite any equation so that B is positive, absorbing the sign into A for sine and discarding it for cosine, and only then read off the phase shift. This tool performs that normalisation and tells you when it has done so, because a phase shift computed from a negative B without normalising will have the wrong sign.
Sine and Cosine Are the Same Wave, Shifted
Every fact on this page applies identically to cosine, but with one extra thing worth knowing: cos(x) = sin(x + π/2). The cosine curve is just the sine curve shifted π/2 to the left. That means any wave has two equally valid descriptions, one written with sine and one with cosine, and their phase shifts differ by exactly a quarter period.
This is why two people can analyse the same graph, get different phase shifts, and both be right. Physics and engineering usually prefer cosine, because a cosine with zero phase starts at its maximum, which is a natural reference for an oscillation released from rest. Mathematics courses usually prefer sine, because sine with zero phase starts at zero. Always state which base function you used when you quote a phase shift; without it, the number is ambiguous by a quarter period.
A related ambiguity is that phase shifts are only defined modulo one period. Shifting by C and shifting by C plus a whole period produce exactly the same curve, so any phase shift can be reported as an infinite family of values. Convention is to give the one closest to zero, and this tool reports both the raw value and that reduced equivalent.
Degrees, Radians and Where the Mix-Up Happens
The formulas change shape depending on the unit. In radians the period is 2π ÷ |B| and a quarter period is π ÷ (2|B|). In degrees the period is 360 ÷ |B|. Mixing them produces answers wrong by a factor of about 57.3, which is large enough to be obvious — the dangerous case is the smaller mix-up, where the phase is entered in degrees and the period computed in radians.
Engineering adds a further wrinkle: phase there is normally quoted as an angle rather than a horizontal distance. A signal "lagging by 90°" is shifted by a quarter of a period in time, which in x-units is period ÷ 4, not 90 of anything. The conversion is phase angle in degrees = (shift ÷ period) × 360. This tool reports the shift both ways so you can quote whichever your field expects. Our angle converter handles degree and radian conversion on its own, and the unit circle calculator is useful for checking where a given angle lands.
Where Phase Shift Actually Matters
In physics, two waves that meet in phase reinforce each other and two waves half a period out of phase cancel. That is the whole mechanism behind interference patterns, noise-cancelling headphones and antenna arrays, and it turns on the phase shift between the two waves rather than on either one alone. Our wavelength calculator covers the related spatial quantity.
In alternating-current circuits the phase shift between voltage and current determines the power factor: purely resistive loads have zero shift and transfer power efficiently, while inductive loads push the current behind the voltage and waste capacity. In signal processing, a filter that changes phase differently at different frequencies distorts a waveform even when it leaves every amplitude untouched — which is why linear-phase filters are prized. The differential equations behind driven oscillators, where the steady-state response lags the driving force by a phase that depends on frequency, are developed in MIT's Differential Equations course.
In data work the same idea appears as seasonality: two seasonal series that peak in different months are phase-shifted versions of each other, and aligning them before comparison is the equivalent of removing the shift. Our circle equation calculator sits nearby, since the sinusoid is literally the shadow of a point moving around a circle.
Arb Digital publishes hundreds of free calculators covering trigonometry, algebra, geometry and statistics — all free to use, with no sign-up and no limit on how often you run them.
Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Reading c as the phase shift. In y = A·sin(Bx − c) + D the shift is c ÷ B. Factor B out before you read anything.
- Getting the direction backwards. The standard form has a minus sign built in, so positive C means right. A plus sign inside the bracket means the shift is to the left.
- Taking the amplitude as A rather than |A|. A negative A flips the curve; the amplitude is still positive.
- Ignoring a negative B. Normalise it away first using sin(−u) = −sin(u), or the phase shift you compute will have the wrong sign.
- Quoting a phase shift without naming the base function. Sine and cosine descriptions of the same wave differ by a quarter period.
Related Free Tools From Arb Digital
Use the trigonometric functions calculator to evaluate points on the curve, the angle converter to move between degrees and radians, the unit circle calculator to place an angle, and the wavelength calculator for the spatial side of wave problems. More are listed in the free tools hub.
Frequently Asked Questions
For y equals A times sine of B times the quantity x minus C, plus D, the phase shift is C. For the expanded form y equals A times sine of Bx minus c, plus D, the phase shift is c divided by B. Factoring B out of the bracket converts one form into the other.
Right. The standard form already contains a minus sign inside the bracket, so a positive C delays the curve until x reaches C. If the equation instead shows a plus sign inside the bracket, the shift is to the left by that amount.
Because B compresses or stretches the horizontal axis, and the shift is compressed with it. Factoring gives Bx minus c equals B times the quantity x minus c over B, so the true horizontal displacement is c divided by B rather than c itself.
Normalise it before reading anything off. Sine is odd, so a negative B can be turned into a positive one by moving the sign onto the amplitude, which reflects the curve. Cosine is even, so a negative B changes nothing. Reading a phase shift from a negative B without normalising gives the wrong sign.
No. The amplitude is the absolute value of A. A negative A still describes a curve of the same height, but reflected in its midline, so the sign belongs to the shape of the curve rather than to the amplitude.
Because cosine is the sine curve already shifted a quarter period to the left. Any wave can be written either way, and the two descriptions differ by exactly a quarter of a period, so a phase shift is only meaningful when you state which base function it refers to.
Divide the shift by the period and multiply by three hundred and sixty. A shift of a quarter period is ninety degrees. Engineering quotes phase as an angle, while algebra courses quote it as a horizontal distance, and the two are not interchangeable without this conversion.
Yes. Shifting by any whole number of periods leaves the curve unchanged, so every phase shift belongs to an infinite family of equivalent values. The convention is to report the one closest to zero, which is what makes two correct answers look different at first glance.
This calculator is provided for study and for checking your own working. It is not a substitute for showing the algebraic steps your course or assessment requires.
