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Sinusoidal function

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Definition

As a linear transform of the sine function

The term sinusoidal function refers to a function of the form f \circ \sin \circ g where f and g are linear functions and \sin is the sine function. Specifically, it is a function of the form:

x \mapsto \mu + A\sin(mx + \varphi), \qquad A > 0, m \ne 0

Here:

  • \mu is the mean value about which the function is oscillating, i.e., the graph of a function looks like a scaled sine function about the horizontal line y = \mu.
  • A is the amplitude of oscillations, i.e., the function oscillates between a minimum value of \mu - A and a maximum value of \mu + A.
  • m is the angular frequency parameter and controls the period of oscillations, which is given by 2\pi/m.
  • \varphi is a phase parameter that roughly describes the head start of the function relative to a function that starts at its mean value at x = 0.

As a linear combination of sine and cosine functions

The term sinusoidal function can be used for a function of the form:

x \mapsto \mu + \alpha \sin(mx) + \beta \cos(mx), \qquad m \ne 0, \alpha^2 + \beta^2 > 0

Conversion between the two versions

Here's how we convert the linear combination version to the linear transform version:

  • \mu,m remain the same.
  • Set A = \sqrt{\alpha^2 + \beta^2}.
  • Set \varphi as an angle so that \cos \varphi = \alpha/A and \sin \varphi = \beta/A. \varphi is uniquely determined up to additive multiples of 2\pi.

Here's how we convert the linear transform version to the linear combination version:

  • \mu, m remain the same.
  • \alpha = A \cos \varphi.
  • \beta = A \sin \varphi.

Examples

Function How it's a sinusoidal function in the linear transform sense How it's a sinusoidal function in the linear combination sense
sine function 0 + 1\sin(1x + 0):
\mu = 0, A = 1, m = 1, \varphi = 0
0 + 1\sin(1x) + 0\cos(1x)
\mu = 0, m = 1, \alpha = 1, \beta = 0
cosine function 0 + 1\sin(1x + \pi/2)
\mu = 0, A = 1, m = 1, \varphi = \pi/2
0 + 0\sin(1x) + 1\cos(1x)
\mu = 0, m = 1, \alpha = 0, \beta = 1.
sine-squared function \sin^2 1/2 + (1/2)\sin(2x - \pi/2)
\mu = 1/2, A = 1/2, m = 2, \varphi = -\pi/2
(1/2) + 0\sin(2x) + (-1/2)\cos(2x)
\mu = 1/2, m = 2, \alpha = 0, \beta = -1/2
cosine-squared function \cos^2 1/2 + (1/2)\sin(2x + \pi/2)
\mu = 1/2, A = 1/2, m = 2, \varphi = \pi/2
(1/2) + 0\sin(2x) + (1/2)\cos(2x)
\mu = 1/2, m = 2, \alpha = 0, \beta = 1/2