# Sinusoidal function

## 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 $\cos^2$ $1/2 + (1/2)\sin(2x + \pi/2)$
$\mu = 1/2, A = 1/2, m = 2, \varphi = \pi/2$
$(1/2) + 0\sin(2x) + (1/2)\cos(2x)$
$\mu = 1/2, m = 2, \alpha = 0, \beta = 1/2$