Amplitude, Period and Phase Shift Calculator

This amplitude, period and phase shift calculator reads the key features straight off a sinusoid written in the standard form y = A sin(Bx + C) + D, the equation used to describe waves, sound, tides, alternating current and any signal that repeats. You enter the four coefficients: A, the amplitude term; B, the coefficient of x inside the sine; C, the phase term; and D, the vertical shift. From these the calculator instantly returns the amplitude, the period, the phase shift and the vertical shift, updating as you type. The amplitude is how far the curve rises above and falls below its midline, worked out as the absolute value of A. The period is the length of one full cycle, found by dividing two pi by the absolute value of B. The phase shift shows how far the curve is moved left or right, calculated as minus C divided by B, with a positive result meaning a shift right. The vertical shift is simply D, the midline height the wave oscillates around. Note that the angle term Bx + C is always taken in radians, not degrees. This suits students checking homework and engineers wanting a quick, reliable read on a sinusoid's properties. A full worked example, the underlying formulas and the assumptions behind the results are set out below the calculator.

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A (amplitude coeff)

B (inside x)

C (phase term)

D (vertical shift)

Period

Phase shift

Vertical 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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