Phase Shift Calculator - Sine, Cosine, or Tangent
Use this phase shift calculator to read A, B, C, and D in y = A sin(Bx - C) + D and return the amplitude, period, phase shift, and vertical shift.
Phase Shift Calculator
Results
What Is Phase Shift Calculator?
A phase shift calculator reads the four coefficients A, B, C, and D in y = A * f(Bx - C) + D and returns the horizontal translation, the period, the amplitude, the vertical shift, and the max and min y-values.
- • Precalculus and AP trig homework: Verify the phase shift, period, amplitude, and vertical shift of a sinusoidal function like y = 0.5 sin(2x - 3) + 4 from a trig-transformations problem set.
- • Signal processing and wave analysis: Translate a measured sine wave A sin(Bx - C) + D into a phase value in radians for alignment with a reference signal.
- • Engineering oscillations and AC circuits: Convert the standard form of a sinusoidal oscillator into the phase shift, period, and amplitude used in control-system analysis.
Pick a trig function (sine, cosine, or tangent), type the four numeric coefficients, and the calculator returns the phase shift C / B, the period 2 pi / |B| (or pi / |B| for tangent), the amplitude |A|, the vertical shift D, the max and min, and the full trig equation in real time.
When you have a y value and need to read the corresponding angle back out of a sine, Arcsin Calculator runs the inverse of the same y = A sin(Bx - C) + D form so the phase shift can be traced back to a specific angle.
How Phase Shift Calculator Works
The calculator applies the standard phase shift identity for y = A * f(Bx - C) + D, where f is sine, cosine, or tangent, and formats the output quantities plus the rewritten equation text.
- function: Choice of trig function: sine, cosine, or tangent. Only the period differs between them.
- A: Amplitude coefficient. The amplitude is |A|; the sign of A flips the graph vertically across the centerline y = D.
- B: Frequency coefficient. The period is 2 pi / |B| (sine, cosine) or pi / |B| (tangent). B must be nonzero.
- C: Phase shift coefficient. The horizontal translation is C / B; positive moves right, negative moves left.
- D: Vertical shift. D is the new centerline (midline) of the graph.
The phase shift identity is the rewrite Bx - C = B * (x - C / B), which factors the horizontal shift out of the trig function. The period identity is sin(Bx) = sin(B * (x + 2 pi / B)), showing the function repeats every 2 pi / |B| units of x (or pi / |B| for tangent).
Worked example: y = 0.5 sin(2x - 3) + 4 (the Omni example)
Trig function: sine. A = 0.5, B = 2, C = 3, D = 4.
Amplitude: |A| = 0.5. Period: 2 pi / |B| = pi. Phase shift: C / B = 1.5. Vertical shift: D = 4. Max: 4.5. Min: 3.5.
f(x) = 0.5 * sin(2x - 3) + 4, period pi, phase shift 1.5, amplitude 0.5, range [3.5, 4.5].
The graph is the standard sine curve compressed by a factor of 2, raised by 4, scaled to amplitude 0.5, and shifted 1.5 right.
According to Wikipedia, the sine function has period 2 pi and amplitude 1, and a horizontal translation of a * sin(bx - c) by c / b produces the phase shift
According to Wolfram MathWorld, the sine function sin(x) has period 2 pi and amplitude 1, and the generalized form A * sin(Bx + C) has period 2 pi / |B| and amplitude |A|
Because the period, phase shift, and frequency B all assume x is in radians, Radians to Degrees Calculator is the right tool to convert those values into degrees when the original problem was written with a degree-mode calculator.
Key Concepts Explained
Four small ideas show up every time you read a phase shift equation, and they are the only pieces of information the phase shift calculator ever needs from you.
Phase shift identity (C / B)
A phase shift of C / B to the right means rewriting Bx - C as B * (x - C / B) and reading C / B as the horizontal translation.
Period identity (2 pi / |B|)
The period is the distance along x after which the trig function repeats. For sin(Bx) and cos(Bx) it is 2 pi / |B|, and for tan(Bx) it is pi / |B|.
Amplitude (|A|)
The amplitude is how far the values run from the centerline. A negative A reflects the graph across y = D but does not change the amplitude.
Vertical shift (D) and midline
D sets the new centerline. The midline is the horizontal line y = D, with max D + |A| and min D - |A|. For tangent, max and min are null.
These four concepts are why a phase shift equation is fully described by four coefficients, and the calculator mirrors them as labeled rows in the results panel.
The frequency coefficient B and the period 2 pi / |B| are the two sides of the same sinusoidal relationship, so Frequency Calculator converts the period this calculator returns into Hz, RPM, or another unit of frequency.
How to Use This Calculator
Using the calculator is a four-step flow: pick a trig function, type the four coefficients, read the result on the right.
- 1 Pick a trig function: Use the drop-down to choose sine, cosine, or tangent. The calculator updates the displayed equation text and period formula.
- 2 Enter the amplitude A and frequency B: Type the amplitude A and the frequency B. Negative A reflects the graph; negative B keeps the same period.
- 3 Enter the phase shift C and vertical shift D: Type C (subtracted inside the trig function) and D (added after it). The phase shift is C / B and the vertical shift is D.
- 4 Read the phase shift, period, amplitude, vertical shift, and full equation: The results panel shows the phase shift C / B, the period 2 pi / |B|, the amplitude |A|, the vertical shift D, the max and min, and the full trig equation. Values update in real time.
- 5 Reset to try a new example: Press Reset to restore the default y = 0.5 sin(2x - 3) + 4 example, the fastest way to compare trig functions.
For a 440 Hz tone y = 1.2 sin(2 pi * 440 * x - 0.7), enter A = 1.2, B = 2764.6019, C = 0.7, D = 0. The phase shift is about 0.000253 radians, the period about 0.002273 seconds, and the amplitude 1.2.
When the input function is tangent and you need to back out the angle from a y value at a known x, Arctan Calculator runs the inverse of A * tan(Bx - C) + D so the phase shift C / B still lines up with the angle in the equation.
Benefits of Using This Calculator
These benefits come from real trig homework, signal-processing, and AC-circuit work, not from treating the phase shift as a toy problem.
- • Four coefficients in, six results out: The phase shift calculator returns the phase shift, period, amplitude, vertical shift, max, and min at the same time, replacing six hand calculations.
- • Phase shift identity built in: The C / B rule is applied automatically, so you cannot get the phase shift wrong by dropping the sign or dividing wrong.
- • Tangent mode is explicit: When the trig function is tangent, the period becomes pi / |B| and the max and min are reported as null because the function is unbounded.
- • Equation text is ready to paste: The full f(x) = A * sin(Bx - C) + D is rendered with the actual coefficient values, so the result can be pasted into a worksheet.
- • Accepts negative A, B, and C: Negative A reflects the graph, negative B keeps the same period, and negative C produces a leftward phase shift.
The biggest practical benefit is that a phase shift equation becomes a single reading instead of a multi-step hand calculation.
If the next problem in the same trig-transformations unit is a quadratic in vertex form y = a(x - h)^2 + k, Parabola Calculator returns the same kind of horizontal and vertical shift, just for a polynomial rather than a sinusoid, and the same coefficient-by-coefficient workflow applies.
Factors That Affect Your Results
These factors decide whether the calculator output matches what you would draw on a graph.
Sign and magnitude of A
The amplitude is |A|; the sign of A flips the graph across the centerline y = D.
Sign and magnitude of B
The period is 2 pi / |B| for sine and cosine and pi / |B| for tangent. A large |B| compresses the graph; a small |B| stretches it.
Magnitude and sign of C
The phase shift is C / B. A positive C (with positive B) moves the graph right, a negative C moves it left.
Vertical shift D
D is the new centerline. Changing D slides the whole graph up or down without affecting the period, phase shift, or amplitude.
Choice of trig function
Sine and cosine have period 2 pi / |B| and are bounded by D +/- |A|. Tangent has period pi / |B| and is unbounded.
- • B must be nonzero. A function of the form A * f(-C) + D is a constant, and the period, phase shift, and amplitude are not defined. The calculator flags B = 0 in the equation text.
- • The phase shift identity assumes x is in radians, not degrees. If your coefficients are in degrees, convert B and C to radians (multiply by pi / 180) before entering them.
Treat the calculator output as an exact analytic value whenever the coefficients are real and B is nonzero.
According to Wikipedia, the phase of a sinusoidal wave is the horizontal shift from a reference waveform, and the phase shift is the difference between the two phases in radians or fractions of a period
Frequently Asked Questions
Q: How do I calculate the phase shift?
A: For a function of the form A * sin(Bx - C) + D (or cos or tan), the phase shift is C / B. If the result is positive the graph is shifted to the right; if it is negative the graph is shifted to the left. The period is 2 pi / |B| for sine and cosine, or pi / |B| for tangent.
Q: How do I find the phase shift from a graph?
A: Identify the closest sine or cosine to the shape, then look to the right of the y-axis for the first peak (positive coefficient) or trough (negative coefficient). Measure the distance from the y-axis to that point, and subtract pi / 2 (for a peak) or 3 pi / 2 (for a trough) if the function is a sine. The remaining distance is the phase shift.
Q: How do I find the amplitude, period, and phase shift?
A: For f(x) = A * sin(Bx - C) + D, the amplitude is |A|, the period is 2 pi / |B| (or pi / |B| for tangent), and the phase shift is C / B. The sign of C / B tells you the direction: positive is right, negative is left. D is the vertical shift.
Q: How do I graph trig functions with phase shift?
A: Pick the trig function (sine, cosine, or tangent), then move the starting point (0, 0) by the phase shift C / B. A positive phase shift moves the starting point to the right; a negative phase shift moves it to the left. Draw the unshifted function from the new starting point, scaled by the period 2 pi / |B| and the amplitude |A|, and shifted vertically by D.
Q: Are horizontal shift and phase shift the same?
A: When the function is trigonometric, yes. A horizontal shift of a sine, cosine, or tangent graph is usually called a phase shift. The two terms describe the same translation, and the value is read as C / B for the form A * f(Bx - C) + D.
Q: What is the phase shift of a tangent function?
A: For f(x) = A * tan(Bx - C) + D, the phase shift is still C / B, but the period is pi / |B| instead of 2 pi / |B|. The amplitude and the max and min are undefined because the tangent function is unbounded, but the phase shift, vertical shift, and period are still well-defined and reported by the calculator.