Phase Shift
Phase 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 right, a negative one moves it left. For $y = A\sin(Bx + C) + D$, the phase shift $= -C/B$: setting $Bx + C = 0$ and solving for $x$ gives the starting point of the cycle. In the Fourier domain, a time shift by $\delta$ corresponds to a translation $f(t - \delta)$ and multiplies the Fourier transform by $e^{-i\omega\delta}$ (the shift theorem), introducing a frequency-dependent phase rotation fundamental to signal processing and wave interference.
Example
$y = \sin(x - \pi/2)$ looks like $y = \sin(x)$ slid $\pi/2$ units to the right, while $y = \cos(x + \pi)$ is shifted $\pi$ units to the left. For $y = 2\sin(3x - \pi)$: $B = 3$, $C = -\pi$, so the phase shift $= \pi/3$ to the right, meaning the cycle starts at $x = \pi/3$ instead of $x = 0$. Two speakers emitting $\sin(\omega t)$ and $\sin(\omega t + \pi)$ have a phase shift of $\pi$, causing destructive interference, $\sin(\omega t) + \sin(\omega t + \pi) = 0$, the principle behind noise-canceling headphones.
Key Insight
Phase shift does not change the shape of the wave at all, just where it starts along the $x$-axis, like sliding the same pattern left or right on the page; a common error is confusing the sign, since in $y = \sin(x - \pi/4)$ the shift is $+\pi/4$ (right), not $-\pi/4$. The phase of a signal carries information 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.