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:

$x\mapsto \mu +A\sin (mx+\phi ),A>0,m\ne 0$

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 $2\pi /m$.
• $\phi$ 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:

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

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 $\phi$ as an angle so that $\cos \phi =\alpha /A$ and $\sin \phi =\beta /A$. $\phi$ 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.
• $\alpha =A\cos \phi$.
• $\beta =A\sin \phi$.

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	$0+1\sin (1x+0)$: $\mu =0,A=1,m=1,\phi =0$	$0+1\sin (1x)+0\cos (1x)$ $\mu =0,m=1,\alpha =1,\beta =0$
cosine function	$0+1\sin (1x+\pi /2)$ $\mu =0,A=1,m=1,\phi =\pi /2$	$0+0\sin (1x)+1\cos (1x)$ $\mu =0,m=1,\alpha =0,\beta =1$.
sine-squared function ${\sin }^2$	$1/2+(1/2)\sin (2x-\pi /2)$ $\mu =1/2,A=1/2,m=2,\phi =-\pi /2$	$(1/2)+0\sin (2x)+(-1/2)\cos (2x)$ $\mu =1/2,m=2,\alpha =0,\beta =-1/2$
cosine-squared function ${\cos }^2$	$1/2+(1/2)\sin (2x+\pi /2)$ $\mu =1/2,A=1/2,m=2,\phi =\pi /2$	$(1/2)+0\sin (2x)+(1/2)\cos (2x)$ $\mu =1/2,m=2,\alpha =0,\beta =1/2$