Two identical ocean waves can add into a monster or cancel into flat calm, and the difference is phase — where each wave is in its cycle when they meet. For a sine wave written A·sin(Bx + C), the phase shift is −C ÷ B: it tells you how far the whole wave slides left or right compared with plain sin(x). A phase shift of π/2 turns a sine wave into a cosine wave, which is why electricians say current 'leads' or 'lags' voltage by 90° in capacitors and inductors.

Sound engineers chase phase when miking a drum kit — two mics half a wavelength apart record the same hit out of phase and hollow out the sound. AC power analysis, signal processing, and even the timing of traffic lights lean on phase arithmetic. This calculator isolates the division at the heart of it: type the negative of C in the Variable A box and 1 ÷ B in the Variable B box. For y = sin(2x + π), enter −3.1416 in A and 0.5 in B; the Result box shows −1.5708, so the wave shifts left by π/2. Positive results shift right, negative shift left — but check your textbook's sign convention, since some define it as C ÷ B instead.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Phase Shift Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.