How this instrument works
In y = sin(bx + c), the argument factors as b(x + c ⁄ b) — matching the standard translation form b(x − h), where h = −c ⁄ b. That quantity, −c ⁄ b, is the phase shift: the distance the whole curve has been slid along the x-axis, measured in whatever angle unit x itself uses.
The sign catches almost everyone the first time. A positive constant term produces a NEGATIVE shift — the curve moves LEFT, not right — because raising that constant lets the argument reach any target value sooner, so a peak that plain sin(bx) hits later arrives earlier once it is added. Only a negative constant pushes the wave to the right. It is the one spot where the sign does the opposite of the familiar y = f(x − h) intuition.
Two limits are worth knowing. As the coefficient shrinks toward zero, the shift grows without bound, because a vanishing coefficient stretches the period toward infinity and any finite offset stops meaning anything against an infinite wavelength. And whenever the constant term is exactly zero, the shift is exactly zero no matter what the coefficient is — the curve already crosses the origin exactly where a bare sin(bx) does, the reference case every other pairing is measured against.
- Enter the coefficient of x into the "b, in y = sin(bx + c)" field. It scales the wave's period and cannot be zero.
- Enter the constant term into the "c, in y = sin(bx + c)" field, picking degrees, radians, or turns from its unit selector.
- Read "Phase shift": a negative reading means the curve sits left of plain sin(bx); a positive reading means right.
- Switch the unit shown on "Phase shift" to see the same offset in degrees or turns without re-entering b or c.
Worked example — y = sin(2x + π ⁄ 2)
Take the wave y = sin(2x + π ⁄ 2), so b = 2.0 and c = 1.5707963267948966, which is π ⁄ 2 entered in radians. The formula gives shift = −c ⁄ b = −1.5707963267948966 ⁄ 2.0 = −0.7853981633974483 radians — exactly −π ⁄ 4, close to −45°.
The negative sign says this curve sits a quarter of a period to the left of plain sin(2x). Plain sin(2x) peaks where 2x = π ⁄ 2, at x = π ⁄ 4; the shifted wave sin(2x + π ⁄ 2) reaches that same peak where 2x + π ⁄ 2 = π ⁄ 2, at x = 0 — earlier along the axis, matching a shift of −π ⁄ 4 exactly.
Questions
What does a negative phase shift mean?
It means the curve has moved left along the x-axis compared with the unshifted sin(bx). For y = sin(2x + π ⁄ 2), the shift is −π ⁄ 4, roughly −45°, so every feature of the wave — its peak, its zero crossings — turns up earlier, at a smaller x, than the matching feature on sin(2x).
Why is the formula −c ⁄ b instead of c ⁄ b?
Because sin(bx + c) only becomes a clean sideways translation of sin(bx) after factoring out the coefficient: bx + c = b(x + c ⁄ b). Set that beside the standard form y = f(x − h) and h works out to −c ⁄ b, so the constant you type in flips sign on its way to becoming the reported shift.
How is phase shift different from the period?
Phase shift slides the wave sideways without changing its shape; the period, 2π ⁄ b, is the length of the shape's own repeat. Changing the coefficient alone rescales the period and, since it also sits underneath the constant term in the shift formula, rescales the shift computed from a fixed c at the same time.
Does the angle unit I pick change the shift I get?
No — whatever unit that constant is entered in, the shift comes back in that same unit, since −c ⁄ b is a plain ratio. Switching the selector on the c field or on Phase shift between degrees, radians, and turns only changes the display; the underlying wave and its physical shift stay identical.
What happens if b is zero?
The formula has no answer, because dividing by b = 0 is undefined — and the geometry agrees: with b = 0, y = sin(c) is no longer a wave in x at all, just a flat line at a constant height, so there is no period and no shift left to describe.
Where does phase shift matter outside a math class?
Anywhere two periodic signals get compared: AC circuits describe how voltage lags current, acoustics describe how one sound wave lags another, and signal processing tracks a filtered wave against the original — each using this same b(x + c ⁄ b) shift, with time standing in for x.