Phase Shift Calculator

Analyze the horizontal shift of trigonometric functions based on the equation y = A sin(Bx + F) or y = A cos(Bx + F).

Enter the coefficients B and F from your trigonometric equation to compute the phase shift.

Enter the value of B. You can use 'pi' for π.

Enter the value of F. You can use 'pi' for π.

Practical Examples

Explore how phase shift is calculated for different trigonometric equations.

Sine Function with Positive Shift

Sine Function with Positive Shift

Calculate the phase shift for y = sin(2x - π).

Equation: y = A·sin(2x + -pi)

Coefficient B: 2

Coefficient F (Phase Constant): -pi

Cosine Function with Negative Shift

Cosine Function with Negative Shift

Calculate the phase shift for y = 3cos(x + π/2).

Equation: y = A·sin(1x + pi/2)

Coefficient B: 1

Coefficient F (Phase Constant): pi/2

No Phase Shift

No Phase Shift

An example where there is no horizontal shift: y = 2sin(4x).

Equation: y = A·sin(4x + 0)

Coefficient B: 4

Coefficient F (Phase Constant): 0

Fractional Pi Shift

Fractional Pi Shift

A function with a fractional value for B: y = cos( (π/4)x + 1 ).

Equation: y = A·sin(pi/4x + 1)

Coefficient B: pi/4

Coefficient F (Phase Constant): 1

Other Titles
Understanding Phase Shift: A Comprehensive Guide
A deep dive into the concept of phase shift, its calculation, and its significance in mathematics and the real world.

What is Phase Shift?

  • Defining horizontal displacement in waves
  • The role of coefficients in the standard equation
  • Visualizing the shift on a graph
In trigonometry, a phase shift (or horizontal shift) is a horizontal translation of a periodic function, such as a sine or cosine wave. It dictates how far, and in which direction, the function is moved from its standard position along the x-axis. Understanding phase shift is crucial for analyzing wave behavior in various fields like physics, engineering, and signal processing.
The Standard Equation
The general form of a sinusoidal function is often written as y = A·sin(B(x - C)) + D or y = A·sin(Bx + F) + D. In the first form, 'C' directly represents the phase shift. However, the second form is more common in many textbooks. In this case, the phase shift is not simply 'F'. It depends on both 'B' and 'F'. The formula to calculate it is: Phase Shift = -F / B.
Direction of the Shift
The sign of the calculated phase shift determines the direction of the horizontal movement. A positive phase shift value indicates a shift to the right. A negative phase shift value indicates a shift to the left. For example, in y = sin(x - π/2), the phase shift is -(-π/2)/1 = +π/2, so the wave is shifted to the right by π/2 units.

Basic Phase Shift Scenarios

  • y = sin(x + π): Phase Shift = -π/1 = -π (Shift Left)
  • y = cos(2x - π): Phase Shift = -(-π)/2 = π/2 (Shift Right)
  • y = sin(3x): Phase Shift = -0/3 = 0 (No Shift)

Step-by-Step Guide to Using the Phase Shift Calculator

  • Entering coefficients correctly
  • Handling the constant π (pi)
  • Interpreting the calculated results
Our calculator is designed to be straightforward. It focuses on the core calculation of phase shift from the standard equation y = A·sin(Bx + F) + D. You only need to provide the 'B' and 'F' coefficients.
Input Fields
1. Coefficient B: This is the coefficient of 'x' inside the trigonometric function. It affects the period of the wave. For this calculator, it cannot be zero, as that would result in division by zero when calculating the phase shift.
2. Coefficient F (Phase Constant): This is the constant added to 'Bx' inside the function. This value, along with B, determines the shift.
Using 'pi' for π
For convenience, you can type the word 'pi' in any of the input fields to represent the mathematical constant π (approximately 3.14159). The calculator will automatically parse expressions like 'pi/2' or '2*pi'.
Understanding the Output
The calculator provides not just the phase shift value, but also the direction (left or right), the period of the function (2π/|B|), and its frequency (the reciprocal of the period).

Calculator Input Examples

  • For y = sin(3x + π/2), enter B=3 and F='pi/2'.
  • For y = 2cos(x - 1), enter B=1 and F=-1.
  • For y = 5sin(πx), enter B='pi' and F=0.

Real-World Applications of Phase Shift

  • Phase shift in AC circuits
  • Signal processing and communications
  • Mechanical and sound waves
Phase shift is not just an abstract mathematical concept; it has profound importance in the physical world.
Electrical Engineering
In alternating current (AC) circuits, the voltage and current are often out of phase with each other, especially in circuits containing capacitors and inductors. The phase shift, or phase angle, between voltage and current is critical for calculating power factor and understanding the circuit's efficiency.
Signal Processing
In telecommunications, signals are often modulated by changing their phase. Techniques like Phase Shift Keying (PSK) encode data onto a carrier wave by altering its phase. Analyzing these shifts is fundamental to decoding the transmitted information.
Physics and Mechanics
When two waves interfere, their relative phase shift determines whether they interfere constructively (amplitudes add up) or destructively (amplitudes cancel out). This applies to sound waves, light waves, and even quantum mechanics.

Application Examples

  • AC Circuit: A current I(t) that lags a voltage V(t) by 30° has a phase shift.
  • Noise Cancellation: Headphones generate sound waves with an opposite phase to cancel out ambient noise.
  • Radio: An FM radio signal encodes audio information by modulating the frequency and phase of a carrier wave.

Common Misconceptions and Correct Methods

  • Confusing phase constant with phase shift
  • Incorrectly identifying the sign/direction
  • Mistakes with factored vs. unfactored forms
Phase Constant (F) vs. Phase Shift
A very common mistake is to assume that the constant 'F' in y = sin(Bx + F) is the phase shift. This is incorrect. The phase shift is -F/B. The coefficient 'B' scales the horizontal axis, and this scaling must be accounted for.
The Sign Convention
It can be counter-intuitive, but a positive sign in the equation, like in sin(x + π/2), leads to a negative phase shift (-π/2), which means a shift to the LEFT. Conversely, a negative sign, as in sin(x - π/2), leads to a positive phase shift (+π/2) and a shift to the RIGHT.
Factored Form: y = A·sin(B(x - C))
Sometimes the equation is presented with B factored out. In the form y = sin(B(x - C)), the value 'C' is the phase shift directly. Note that C = -F/B. For example, sin(2x + π) is the same as sin(2(x + π/2)). Here, B=2, F=π, and C=-π/2. The phase shift is -F/B = -π/2, which is different from C. The shift is given by C, so it is a shift of π/2 to the left. Our calculator uses the y = sin(Bx + F) form.

Clarification Examples

  • In y = sin(2x + π/3), the phase constant F is π/3, but the phase shift is -(π/3)/2 = -π/6.
  • y = cos(x - 2) shifts RIGHT by 2 units because the phase shift is -(-2)/1 = +2.
  • y = sin(3(x + 1)) is equivalent to y = sin(3x + 3). The phase shift is -3/3 = -1 (Left by 1).

Mathematical Derivation and Formulas

  • Deriving the phase shift formula
  • Formula for period and frequency
  • Relationship between sine and cosine via phase shift
Derivation of Phase Shift
To find the horizontal shift, we want to see what value of 'x' makes the argument of the function zero. This point corresponds to the start of a fundamental cycle in the un-shifted function. We set the argument Bx + F to zero and solve for x: Bx + F = 0 => Bx = -F => x = -F/B. This value of x is the horizontal shift, which we call the phase shift.
Formulas for Period and Frequency
The period is the length of one complete cycle of the wave. For sine and cosine, the standard period is 2π. The coefficient B compresses or stretches the wave horizontally. The formula for the period is: Period = 2π / |B|. Frequency is the reciprocal of the period, representing how many cycles occur per unit of time: Frequency = 1 / Period = |B| / 2π.
Sine and Cosine Relationship
Sine and cosine are essentially the same wave, just phase-shifted from each other. Specifically, sin(x) = cos(x - π/2) and cos(x) = sin(x + π/2). This means a cosine wave is just a sine wave shifted to the right by π/2, and a sine wave is a cosine wave shifted to the left by π/2. This relationship is fundamental in trigonometry and wave analysis.

Key Formulas

  • Phase Shift = -F / B
  • Period = 2π / |B|
  • Frequency = |B| / 2π