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Trigonometry

Sine wave (amplitude, period and phase)

The general sine wave: A stretches it vertically, B sets how often it repeats, C slides it sideways and D lifts it.

y=Asin⁡(B(x−C))+D
PeriodP=360°|B|=2π|B|rad
RangeD−|A|≤y≤D+|A|

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−180°−90°90°180°270°360°450°540°−6−4−2246one period
  • Amplitude |A| = 3, period 360°/B = 180°
  • Phase shift C = 15° (a cycle starts there), midline y = 1
  • Range: −2 ≤ y ≤ 4

What the symbols mean

y
the height of the wave at x
A
the amplitude: half the distance from trough to peak (a negative A flips the wave)
B
the frequency factor: one cycle takes 360°/|B| (or 2π/|B| radians)
C
the phase shift: the wave is moved C to the right
D
the vertical shift: the midline is y = D
x
the input (an angle, or time)

When to use it

Use it to model anything periodic: tides, daylight hours over a year, sound and alternating current, a point on a turning wheel. Read off the amplitude, period, phase shift and midline to sketch the graph, or fit them to data (D is the average of the highest and lowest values, |A| half their difference). Solving for x gives two answers in each cycle, and more in every later cycle.

Worked example

For y = 3 sin(2(x − 15°)) + 1, find y when x = 60°.

  1. Start from the formula.

    y=A⁢sin⁡(B⁢(x−C))+D

  2. Substitute A = 3, B = 2, C = 15°, D = 1, x = 60° and work it out.

    y=3⁢sin⁡(2·(60°−15°))+1
    =3⁢sin⁡(2·45°)+1
    =3⁢sin⁡(90°)+1
    =3·1+1
    =3+1
    =4

  3. The wave has amplitude |A| = 3, period 180°, is shifted 15° horizontally, and swings between −2 and 4 about the midline y = 1.

Answer y = 4, the wave's maximum

Why it works

It is y=sin⁡x transformed: multiplying by A stretches it vertically, replacing x by Bx squeezes it horizontally by a factor of B, replacing x by x − C shifts it right by C, and adding D moves it up.

Common mistakes

  • Reading the period as B. The period is 360°/B: B = 2 means the wave repeats twice as often (every 180°).
  • Getting the phase shift's direction wrong: sin⁡(2(x−15°)) shifts right by 15°; sin⁡(2x−15°) shifts right by only 7.5°.
  • Forgetting that a negative A turns the wave upside down rather than changing its size.