TRIGONOMETRYTrigonometryMathematics Calculator
〰️

Phase Shift and Amplitude

y = A·sin(Bx+C)+D. Amplitude A, period 2π/B, phase shift -C/B, vertical shift D. Essential for waves, AC circuits, and signal processing.

Concept Fundamentals
|A|
Amplitude
2π/B
Period
-C/B
Phase
D (vertical)
Shift

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Phase shift -C/B: positive C shifts the graph left, negative C shifts right. sin(x + π/2) = cos(x) — cosine is sine with a 90° phase shift. In AC circuits, voltage and current can be out of phase — the phase difference matters for power.

Key quantities
|A|
Amplitude
Key relation
2π/B
Period
Key relation
-C/B
Phase
Key relation
D (vertical)
Shift
Key relation

Ready to run the numbers?

Why: Phase shift determines when a wave starts; amplitude is its height. Used in sound, light, AC circuits, and any periodic signal.

How: For y = A·sin(Bx+C)+D: A = amplitude, period = 2π/B, phase shift = -C/B (horizontal), D = vertical shift. Frequency f = B/(2π) in cycles per unit.

Phase shift -C/B: positive C shifts the graph left, negative C shifts right.sin(x + π/2) = cos(x) — cosine is sine with a 90° phase shift.
Sources:Wolfram MathWorld — PhaseKhan Academy — Trig Graphs

Run the calculator when you are ready.

Start CalculatingEnter A, B, C, D to analyze the sinusoidal function

Examples — Click to Load

radians
phase-shift.sh
CALCULATED
$ analyze --wave y=A·sin(Bx+C)+D
Equation
y = sin(x)
Period
6.283185
Phase Shift
0°
Amplitude
1
Frequency
0.159155 Hz
Vert Shift
0
Type
sin
Share:
Phase Shift Result
y = sin(x)
Period = 6.283185 | Phase = 0°
Amplitude 1Vert shift 0
numbervibe.com/calculators/mathematics/trigonometry/phase-shift-calculator

Wave Parameters

Amplitude / Period / Shift

Wave Composition

Calculation Breakdown

FORMULA
Standard Form
y = A·sin(Bx+C)+D
A= ext{amplitude}, B= ext{freq}, C= ext{phase}, D= ext{vertical}
PARAMETERS
Amplitude |A|
1
ext{Vertical} ext{stretch}
PARAMETERS
Period
6.283185
2\text{pi} /B
PRIMARY RESULT
Phase Shift
0°
-C/B cdot (180/\text{pi} )
PARAMETERS
Vertical Shift
0
D ext{moves} ext{graph} ext{up}/ ext{down}
OUTPUT
Equation
y = sin(x)
ext{Full} ext{equation}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

〰️

Amplitude |A| is the maximum displacement from the midline. Period 2π/B is one full cycle.

— Wolfram MathWorld

↔️

Phase shift -C/B moves the graph horizontally. Positive C → left shift.

— Khan Academy

Key Takeaways

  • • y = A·sin(Bx+C)+D — A=amplitude, B=frequency, C=phase (radians), D=vertical shift
  • • Period = 2π/B; frequency in Hz = B/(2π)
  • • Phase shift = -C/B (in x-units); in degrees: -C/B · 180/π
  • • Positive C shifts the wave left; negative C shifts right
  • • Essential for sound waves, AC circuits, signal processing

Did You Know?

🎵Noise-canceling headphones use 180° phase shift — the anti-wave cancels ambient sound by destructive interferenceSource: Audio Engineering
⚡AC power uses phase: voltage and current can be out of phase. Real power = V·I·cos(φ) where φ is the phase differenceSource: Electrical Engineering
📡Fourier analysis decomposes any periodic signal into sine waves with different amplitudes, frequencies, and phase shiftsSource: Signal Processing
🌊Ocean waves, sound, light, and radio waves all follow y = A·sin(Bx+C)+D. Phase determines how waves combineSource: Physics textbooks
📱Your phone uses phase shift in modems, audio codecs, and touch screens — millions of calculations per secondSource: IEEE
🎸Guitar pedals and synthesizers manipulate phase to create chorus, flanger, and phaser effectsSource: Music Technology

How It Works

Amplitude (A)

|A| is the vertical stretch. The wave oscillates between +A and -A (before vertical shift). Negative A flips the wave.

Period and Phase

Period T = 2π/B. Phase shift in x-units is -C/B. Write as A·sin(B(x - h)) + D to see h = -C/B directly. Sine and cosine differ by π/2 phase.

Vertical Shift (D)

D moves the entire wave up or down. The midline is y = D. Range becomes [D - |A|, D + |A|]. See Sine Calculator.

Expert Tips

Factored Form

Write y = A·sin(B(x - h)) + D. Then h = -C/B is the phase shift. Easier to read from the graph.

Phase is Cyclic

Adding 2π to C gives the same wave. Phase shift is defined modulo the period. Use Unit Circle.

AC Circuits

In AC, phase difference between V and I affects power factor. Inductors and capacitors shift phase by 90°.

Sine vs Cosine

sin(x + π/2) = cos(x). They are the same wave with a 90° phase difference. Choose based on initial conditions.

Comparison: Wave Parameters

ParameterEffectFormulaUnits
A (Amplitude)Vertical stretch|A|Same as y
B (Frequency)Horizontal squeeze2π/B = periodrad/unit x
C (Phase)Horizontal shift-C/Bradians
D (Vertical)Up/down shiftDSame as y
PeriodOne full cycle2π/Bx-units
Phase (deg)Shift in degrees-C/B · 180/πdegrees
Freq (Hz)Cycles per unitB/(2π)Hz
MidlineCenter of wavey = Dy-value

Frequently Asked Questions

How does phase shift affect the wave?

Phase shift moves the wave horizontally. Positive C shifts left; negative C shifts right. The displacement in x-units is -C/B.

What is the difference between phase shift and vertical shift?

Phase shift moves the wave left/right (horizontal). Vertical shift (D) moves it up/down. They are independent.

Why is the phase shift -C/B and not C?

In sin(Bx+C), when Bx+C = 0, x = -C/B. So the "start" of the wave is at x = -C/B, which is a shift of -C/B units.

How do I convert phase shift from radians to degrees?

Multiply by 180/π. So phase_deg = (-C/B) · (180/π). A full period (2π rad) = 360°.

What happens when B is negative?

B should be positive for standard form. If B < 0, the wave reverses direction. Use |B| and adjust C.

Applications in signal processing?

Phase shifts are used in filters, modulation (AM/FM), coherent detection, and beamforming. Phase relationships determine interference.

How does amplitude affect the wave?

Amplitude |A| is the maximum displacement from the midline. It scales the wave vertically. Negative A reflects the wave.

What is the relationship between sine and cosine?

cos(x) = sin(x + π/2). They are the same wave with a 90° (π/2) phase difference. Choose based on where the wave "starts".

Phase Shift by the Numbers

4
Parameters (A,B,C,D)
2π/B
Period Formula
-C/B
Phase Shift
360°
Full Phase Cycle

Disclaimer: Results for educational use. For signal processing or AC circuit design, verify with domain-specific tools.

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