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:

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

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:

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

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