Phase Shift Calculator

How to Calculate Phase Shift for Trigonometric Functions

In mathematics, physics, and electrical engineering, the phase shift describes the horizontal translation of a periodic wave (such as a sine or cosine function) relative to its standard baseline position.

The general form of a sinusoidal wave is written as:

$$y = A \sin(Bx - C) + D \quad \text{or} \quad y = A \cos(Bx - C) + D$$

By factoring out the frequency coefficient $B$, the equation transforms into its factored form:

$$y = A \sin\left(B\left(x - \frac{C}{B}\right)\right) + D$$

The horizontal phase shift $h$ is given by:

$$h = \frac{C}{B}$$

Understanding the Four Wave Parameters

• Amplitude ($|A|$): The peak height of the wave from its center midline to its crest.
• Frequency Coefficient ($B$): Controls how rapidly the wave oscillates. The wave period is $T = \frac{2\pi}{|B|}$.
• Phase Constant ($C$): Shifts the wave horizontally. When $C / B > 0$, the wave shifts to the right (lagging). When $C / B < 0$, it shifts to the left (leading).
• Vertical Shift ($D$): Moves the entire wave up or down, defining the midline at $y = D$.

Phase Shift vs. Phase Angle

It is important to distinguish between phase constant ($C$) and phase shift ($h = C / B$):

• Phase Constant ($C$): An angle (usually in radians or degrees) inside the trigonometric argument.
• Phase Shift ($h$): A distance along the $x$-axis representing the actual physical horizontal translation.

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