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):

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

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 Desmos.com 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).

Where:

- 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 ω
*t*at the maximum point on the graph) - ω = angular frequency.

## References

AzimaDLI, (2009). The Concept of Phase from: http://azimadli.com/vibman/theconceptofphase.htm

MIT. Sinusoidal Functions. Retrieved December 22, 2019 from: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/sinusoidal-functions/MIT18_03SCF11_s7_1text.pdf

U. of Washington. Chapter 17. Sinusoidal Functions. Retrieved December 22, 2019 from: https://sites.math.washington.edu/~m124/source/supps/sinusoidal.pdf