Phase Shift
TrigonometryPhase shift is the horizontal translation of a sinusoidal graph, shifting it left or right from its standard position.
Formula
\text{phase shift} = -\frac{C}{B} \text{ for } y = A\sin(Bx + C) + D
Definition
Phase shift is how far the sine or cosine graph is slid left or right from its normal position. A positive phase shift moves the graph to the right; a negative one moves it to the left.
Example
$y = \sin(x - \pi/2)$ looks just like $y = \sin(x)$ but slid $\pi/2$ units to the right. $y = \cos(x + \pi)$ is shifted $\pi$ units to the left.
Key Insight
Phase shift does not change the shape of the wave at all, just where it starts along the $x$-axis. It is like sliding the same wave pattern left or right on the page.
Definition
For $y = A\sin(Bx + C) + D$, the phase shift $= -C/B$. A positive phase shift value means rightward translation; negative means leftward. To find the shift, set $Bx + C = 0$ and solve for $x$: the starting point of the cycle is at $x = -C/B$.
Example
$y = 2\sin(3x - \pi)$: $B = 3$, $C = -\pi$. Phase shift $= -(-\pi)/3 = \pi/3$ to the right. The cycle starts at $x = \pi/3$ instead of $x = 0$.
Key Insight
A common error is confusing the sign. In $y = \sin(x - \pi/4)$, the shift is $+\pi/4$ (right), not $-\pi/4$. The subtraction inside the argument means the graph moves in the positive direction.
Definition
Phase shift corresponds to a translation in the time domain: $f(t - \delta)$ shifts $f$ rightward by $\delta$. In the Fourier domain, a time shift by $\delta$ multiplies the Fourier transform by $e^{-i\omega\delta}$ (the shift theorem). This introduces a frequency-dependent phase rotation, which is fundamental in signal processing, communications, and the study of wave interference and diffraction.
Example
Two speakers emit $\sin(\omega t)$ and $\sin(\omega t + \pi)$. The phase shift is $\pi$, causing destructive interference: $\sin(\omega t) + \sin(\omega t + \pi) = \sin(\omega t) - \sin(\omega t) = 0$. This is the principle behind noise-canceling headphones.
Key Insight
The phase of a signal carries information that is invisible in the magnitude spectrum. In MRI imaging, the phase of the nuclear magnetic resonance signal encodes spatial position, which is why MRI sequences carefully track phase shifts to reconstruct images.