Periodic Functions

This chapter covers notes on periodic functions. The notes assume assume the reader is familiar with right-angle trigonometry. Notes on this topic may be found here. From that chaper, we saw that any point ${(x,y) \in \reals^2}$ can be expressed as:

(x=\cos \theta, y=\sin \theta)

where ${\theta}$ is some fraction of ${2 \pi.}$ Substituting a value for ${\theta}$ in ${\cos \theta}$ gives us the number of steps we take along the ${x}$ -axis, and substituting a value for ${\theta}$ in ${\sin \theta}$ gives us the number of steps along the ${y}$ -axis:

Using the ratio identities, we get the following relations:

\cos(t) = x

\sin(t) = y

\tan(t) = \dfrac{y}{x},~x \neq 0

\sec(t) = \dfrac{1}{x},~x \neq 0

\csc(t) = \dfrac{1}{y},~y \neq 0

\cot(t) = \dfrac{x}{y},~y \neq 0

Because the trigonometric ratios map real numbers ( ${\theta}$ ) to real numbers, they are, in fact, functions. Collectively, they form a family of functions called the periodic functions:

f(x) = \cos(x)

f(x) = \sin(x)

f(x)= \tan(x)

f(x) = \sec(x)

f(x) = \csc(x)

f(x) = \cot(t)

We can now turn to examining each of these functions.

The Sine Function

The first function we examine is ${f(x) = \sin x.}$ First, let's see some sample values:

${\bm x}$	${0}$	${\dfrac{\pi}{6}}$	${\dfrac{\pi}{4}}$	${\dfrac{\pi}{3}}$	${\dfrac{\pi}{2}}$	${\dfrac{2 \pi}{3}}$	${\dfrac{3 \pi}{4}}$	${\dfrac{5 \pi}{6}}$	${\pi}$
${\bm{\sin(x)}}$	${0}$	${\dfrac{1}{2}}$	${\dfrac{\sqrt{2}}{2}}$	${\dfrac{\sqrt{3}}{2}}$	${1}$	${\dfrac{\sqrt{3}}{2}}$	${\dfrac{\sqrt{2}}{2}}$	${\dfrac{1}{2}}$	${0}$

Notice how the output values are cyclical. This isn't surprising — we're getting back points on a circle, as opposed to the straight real number line we're used to. And because we're getting back points on a circle, we see output values repeated at various points. This is even more apparent when we examine the graph of ${f(x) = \sin x:}$

Having seen the sine function, we can now provide a description.

sine function. The function ${f(x) = \sin(x),}$ called the sine function, is an odd function that maps the reals to members of the interval ${[-1,1].}$

\sin(x) : x \in \reals \to [-1,1]

Thus, the function ${f(x) = \sin (x)}$ has the domain of all real numbers, and the range of ${[-1,1].}$ There's another interesting property of the sine function. If we look closely at its graph, we see that it repeats after ${2 \pi.}$ This isn't surprising — from ${0}$ to ${2 \pi,}$ we've gone full circle. This "full circle" nature is where the name "periodic function" comes from. More formally, the notion of "going full circle" is called a cycle — the portion of the graph from one point to the next point where the graph commences repeating.

periodic function. Let ${f(x)}$ be a function, and ${P \in \reals}$ be a constant. If, for all values ${x}$ in the domain of ${f,}$

f(x + P) = f(x)

then ${f}$ is said to be a periodic function.

For the function ${f(x) = \sin(x),}$ ${P = 2 \pi.}$ We call the constant ${P}$ the periodic function's period — the horizontal length of one cycle. Returning to the graph of ${\sin x,}$ we see that it's symmetric about the origin. Accordingly, ${f(x) = \sin x}$ is an odd function.

The Cosine Function

Once again, let's take a look at some sample values of cosine:

${\bm x}$	${0}$	${\dfrac{\pi}{6}}$	${\dfrac{\pi}{4}}$	${\dfrac{\pi}{3}}$	${\dfrac{\pi}{2}}$	${\dfrac{2 \pi}{3}}$	${\dfrac{3 \pi}{4}}$	${\dfrac{5 \pi}{6}}$	${\pi}$
${\bm{\cos(x)}}$	${1}$	${\dfrac{\sqrt{3}}{2}}$	${\dfrac{\sqrt{2}}{2}}$	${\dfrac{1}{2}}$	${0}$	${-\dfrac{1}{2}}$	${-\dfrac{\sqrt{2}}{2}}$	${-\dfrac{\sqrt{3}}{2}}$	${-1}$

Once again, we see repeated values. Examining the graph of cosine:

Notice that this looks very similar to ${f(x) = \sin x,}$ only this time, the graph is symmetric about the ${y}$ -axis. As such, the cosine function is an even function, in contrast to its sibling sine. Juxtaposing the two plots:

cosine function. The function ${f(x) = \cos(x),}$ called the cosine function, is an even function that maps the reals to members of the interval ${[-1,1].}$

\cos(x) : x \in \reals \to [-1,1]

Sinusoids

Because the sine and cosine functions have the same period ( ${2 \pi}$ ) and range ( ${[-1,1]}$ ), their transformations are collectively called the sinusoids (or sinusoidal functions).

sinusoid. Let ${f}$ be a function of the form:

f(x) = A \sincos (B x - C) + D, \where{x \in \reals}

where ${A, B, C, D \in \reals}$ are constants, and ${\sincos}$ is either ${\sin}$ or ${\cos.}$

Each of the constants in a sinusoid results in some transformation of the base function ( ${y = \cos x}$ or ${y = \sin(x)}$ ).

Period

Given the function:

f(x) = A \sincos (B x - C) + D

the constant ${B}$ is called the period coefficient. It's related to the sinusoid's period ${P}$ by the equation:

P = \dfrac{2 \pi}{\abs{B}}

Changes to the period coefficient results in the following transformations:

${B \ltn 1}$	${B = 1}$	${B \gtn 1}$
horizontal stretching	base	horizontal compression

To illustrate, consider the following functions:

f(x) = \sin \ar{\dfrac{x}{2}}

f(x) = \sin(x)

f(x) = \sin(4x)

f(x) = \cos \ar{\dfrac{x}{2}}

f(x) = \cos(x)

f(x) = \cos(4x)

This behavior is intuitive. If we divide every input that goes in, we get to full circle much more slowly, resulting in a function with a "slower" or "wider" plot. In contrast, if we multiply every input that goes in, we get to full circle much faster, resulting in a "faster" or "tighter" plot.

Midline

Given the function:

f(x) = A \sincos (B x - C) + D

The constant ${A}$ communicates how much the function's graph stretches vertically. Moreover, ${\abs{A}}$ gives us ${f's}$ ampltiude.