Amplitude, Period and Phase Shift Calculator

This amplitude, period and phase shift calculator reads off the key features of a sinusoid written as y = A sin(Bx + C) + D. Enter the four coefficients A, B, C and D and the calculator returns the amplitude, the period, the phase shift and the vertical shift. The amplitude is how far the curve rises above and falls below its midline, the period is the length of one full cycle, the phase shift is how far the curve is moved left or right, and the vertical shift is the midline height. These appear in waves, sound, tides, alternating current and any periodic signal. The angle term Bx + C is taken in radians. The formulas, a worked example and the assumptions are set out below.

A (amplitude coeff)

B (inside x)

C (phase term)

D (vertical shift)

Period

Phase shift

The formula

For y = A sin(Bx + C) + D: amplitude = |A|; period = 2π / |B|; phase shift = −C / B (positive means a shift to the right); and vertical shift = D. The frequency is |B| / 2π cycles per unit.

Worked example

For y = 2 sin(3x + 1): amplitude = 2, period = 2π / 3, about 2.094395, phase shift = −1 / 3, about −0.333333 (a shift left), and vertical shift = 0. Enter 2, 3, 1, 0 to confirm.

Frequently asked questions

How do I find the period?

Period = 2 pi divided by the absolute value of B, the coefficient of x inside the sine.

What is the phase shift?

Phase shift = minus C over B. A positive value shifts the curve right, a negative value shifts it left.

What does D do?

D is the vertical shift, the height of the midline the wave oscillates around.

Who this calculator is for

This calculator is for students and engineers analysing waves, sound, AC signals and other sinusoids.

What this calculator assumes

The function is in the form y = A sin(Bx + C) + D.
The angle Bx + C is in radians.
Results are rounded for display.

Formula and sources

Related calculators

Trig Calculator: all six trig functions.
Sine Calculator: sine of an angle.
Right Angle Triangle Calculator: solve a right triangle.

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