Formula and Method for Phase Shift
A sinusoidal function is usually written in the standard form y = A sin(Bx − C) + D (or the cosine version, y = A cos(Bx − C) + D). Each coefficient controls one visual feature of the graph: A sets the amplitude, B sets how quickly the wave repeats, C together with B sets the phase shift, and D sets the vertical shift. The phase shift is the horizontal distance the parent sine or cosine curve has been slid left or right, and it equals C / B — not C by itself.
How the calculation works
Enter A, B, C, and D from your equation, choose whether C is measured in radians or degrees, and pick sine or cosine (the phase-shift math is identical for both). The calculator finds the amplitude as |A|, the period as 2π / |B| (or 360° / |B| in degrees), and the phase shift as C / B. A positive phase shift means the graph is shifted to the right of the parent curve; a negative value means it is shifted to the left. This falls out of setting the argument equal to zero: Bx − C = 0 when x = C/B, which is where the shifted wave begins its cycle. The vertical shift D is simply read off — it is the y-value of the curve's midline.
Common mistakes
- Forgetting to divide by B: in y = sin(2x − π), the phase shift is π/2, not π — always divide C by B, don't read C alone.
- Sign errors: if the equation is written as y = A sin(Bx + C) + D, rewrite it as Bx − (−C) first; the phase shift is then −C/B, the opposite sign of what a careless reading would suggest.
- Mixing radians and degrees: a period of 2π only applies when x is in radians; if the problem uses degrees, the period is 360°/|B| instead — keep the angle unit consistent throughout.
Real-world applications
- Electrical engineering uses phase shift to describe the lag or lead between AC voltage and current waveforms of the same frequency.
- Sound and signal processing use phase shift to align or compare waveforms, such as syncing audio channels or detecting Doppler-shifted signals.
- Physics and engineering use it to model oscillations, tides, and periodic motion that start partway through a cycle rather than at a peak or zero crossing.
- Trigonometry and precalculus courses use phase shift, alongside amplitude and period, to graph transformed sine and cosine curves from their equations.