Calculus How To

Sinusoidal Function: Definition, Formula, Examples

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A sinusoidal function (also called a sinusoidal oscillation or sinusoidal signal) is a generalized sine function. In other words, there are many sinusoidal functions; The sine is just one of them.

Formula for a Sinusoidal Function

A sinusoidal function can be written in terms of the sine (U. Washington):
definition of sinusoidal function

Where A, B, C, D are fixed constants and A & B are positive.

sinusoidal graph

Sinusoidal graph (blue) with constants A = 2, B = 3, C = 4, D = 5 and sin x (red).

One of the main differences in the graphs of the sine and sinusoidal functions is that you can change the amplitude, period, and other features of the sinusoidal graph by tweaking the constants. For example:

  • “A” is the amplitude.
  • “B” is the period, so you can elongate or shorten the period by changing that constant.
  • “C” is the phase shift (the horizontal shift).
  • “D” is the vertical shift. The line y = D – A is where the graph is at a minimum, and y = D + A is where the graph is at a maximum.

Tip: Access the sin vs sinusoidal graph I created on and play around with the different constants to see what each does to the graph.

Alternatively, a sinusoidal function can be written in terms of the cosine (MIT, n.d.):

f (t) = A cos(ω t – Φ).

  • A = amplitude (maximum displacement or distance)
  • Φ = phase lag (commonly defined as the delay of the waveform relative to another, but here it’s the value of ωt at the maximum point on the graph)
  • ω = angular frequency.


AzimaDLI, (2009). The Concept of Phase.
MIT. Sinusoidal Functions. Retrieved December 22, 2019 from:
U. of Washington. Chapter 17. Sinusoidal Functions. Retrieved December 22, 2019 from:

Stephanie Glen. "Sinusoidal Function: Definition, Formula, Examples" From Calculus for the rest of us!

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