trigonometry

Phase Shift

Phase shift is a horizontal translation of a periodic function. For y = sin(Bx + C), the phase shift is -C/B (positive = right, negative = left).

Phase shift is the horizontal translation of a periodic function. For y=Asin⁡(Bx+C)+Dy = A\sin(Bx + C) + D, the phase shift is −C/B-C/B:

  • Positive: right.
  • Negative: left.

Sign convention: it's −C/B-C/B, not C/BC/B. Factor out: sin⁡(Bx+C)=sin⁡(B(x+C/B))\sin(Bx + C) = \sin(B(x + C/B)).

Examples:

  • sin⁡(x−π/2)\sin(x - \pi/2): phase shift π/2\pi/2 (right).
  • cos⁡(2x+π)\cos(2x + \pi): phase shift −π/2-\pi/2 (left).

In physics (waves, AC current), phase shift is how much one wave is delayed relative to another. Two sine waves with same period but different phase can interfere constructively (in phase), destructively (180° out of phase), or anything in between — the foundation of acoustics, optics, and signal processing.

Phase shift is one of four sinusoid parameters: amplitude, period, phase shift, vertical shift.