Periodic Functions

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repeating

In addition to gluing together pieces from several functions, we also have functions which just repeat one part over and over and over. They glue the same function next to itself over and over and over. These functions are said to be periodic.

Periodic Function

A function $f$ is said to be periodic if there exists a real number $p \ne 0$ such that

\[ f(x+p) = f(x) \]

for all values $x$ in the domain.

$p$ is called the period.

Sawtooth Functions

Let’s begin with a function whose graph looks like an iscoceles triangle.

Graph of $y = Tooth(t)$.

$\blacktriangleright $ desmos graph

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The tooth here is $1$ unit high and $2$ units wide and this Sawtooth function below just repeats this tooth over and over and over.

Graph of $y = SawTooth(t)$.

$\blacktriangleright $ desmos graph

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$SawTooth(0) = \answer {0}$
$SawTooth(1) = \answer {1}$
$SawTooth(2) = \answer {0}$
$SawTooth(57) = \answer {1}$
$SawTooth(102) = \answer {0}$

Different sawtooth functions are made by varying the height and width of the tooth. The single tooth that was repeated is called the wave of the function. The width of the wave is called the period. And, the height of the wave is called the amplitude.

Sawtooth functions are piecewise defined functions but there is really one piece that just repeats. We would like to express this idea in a formula. We define a formula for the initial piece and then state the period.

\[ SawTooth(x) = \begin{cases} x &\text {on $[0,1)$,} \\ -x+2 &\text {on $[1,2)$} \\ SawTooth(x) = SawTooth(x+2) & \text {otherwise} \end{cases} \]

$SawTooth(x) = SawTooth(x+2)$ tells us to keep jumping by $2$ until you get to a known value.
$2$ is the period of $SawTooth$.

Evaluation

Evaluate $SawTooth(9)$.

Using the equation $SawTooth(x) = SawTooth(x+2)$ several times, we get

\[ SawTooth(9) = SawTooth(7) = SawTooth(5) = SawTooth(3) = SawTooth(1) = -(1) + 2 = 1 \]

Evaluate. $SawTooth(-2) = \answer {0}$

Evaluate. $SawTooth\left ( \frac {5}{2} \right ) = \answer {\frac {1}{2}}$

Evaluate. $SawTooth(401) = \answer {1}$

This function gets its name because it looks like moving up and down on the teeth on a saw. Other periodic functions come from similarly moving on different objects.

You can make a periodic function out of any piece of any function, just use that piece as the wave. But we have two very special basic periodic functions in our set of Elementary Functions.

Traveling around the unit circle produces two important periodic functions.

The Unit Circle

The unit circle is the circle of radius $1$ centered at the origin in the Cartesian plane. A point on the unit circle has two coordinates.

The first coordinate gives the horizontal position of the point.
The second coordinate gives the vertical position of the point.

Both of these depend on an angle measurement, $\theta $. $\theta $ is the angle made between the positive horizontal axis and a radius formed by drawing a line from the origin (center of the circle) the point on the unit circle.

Sine and Cosine

As the angle changes, the corresponding point on the unit circle changes position and thus its coordinates change. In other words, given an angle, there is exactly one corresponding point in the unit circle at that angle.

Given an angle there is exactly one corresponding horizontal coordinate.
Given an angle there is exactly one corresponding vertical coordinate.

The horizontal and vertical coordinates of points on the unit circle are functions of the angle. These two functions are called cosine and sine - abbreviated as cos and sin.

Each point on the unit circle has coordinates described as

\[ (\cos (\theta ), \sin (\theta )) \]

As the angle turns around the unit circle, these coordinates repeat. Sine and cosine are periodic functions.

Cosine

Given an angle $\theta $, measured counterclockwise from the positive $x$-axis, draw a radius from the origin to the unit circle. This radius intersects the unit circle in exactly one point.

$\cos (\theta )$ is the value of the $x$-coordinate (horizontal) of that intersection point.

Sine

Given an angle $\theta $, measured counterclockwise from the positive $x$-axis, draw a radius from the origin to the unit circle. This radius intersects the unit circle in exactly one point.

$\sin (\theta )$ is the value of the $y$-coordinate (vertical) of that intersection point.

Etymology

Where do the trigonometric names come from? https://mathisonian.github.io/trig/etymology/

Angle Measurement

The domain of sine and cosine are real numbers interpreted as angle measurements measured couterclockwise from the positive horizontal axis. We have two units for measuring these angles.

degrees: A full circle is divided into $360$ degrees.
radians: A full circle is divided into $2\pi $ radians.

A half-circle rotation is $\answer {180}$ degrees.
A quarter-circle rotation is $\answer {90}$ degrees.
A eighth-circle rotation is $\answer {45}$ degrees.
A half-circle rotation s $\answer {\pi }$ radians.
A quarter-circle rotation is $\answer {\frac {\pi }{2}}$ radians.
A eighth-circle rotation is $\answer {\frac {\pi }{4}}$ radians.

Shorthand: Degrees has a little superscript circle as a shorthand abbreviations, like $90^\circ $. Radians doesn’t have a shorthand abbreviation. Therefore, if you see an angle measurement with no units, then the units are radians.

Cosine

With Degrees:
As the angle, $\theta $, rotates counter-clockwise from $0^\circ $, the value of cosine start at $1$, decreases to $0$ at $90^\circ $, keeps decreasing to $-1$ at $180^\circ $, increases to $0$ at $270^\circ $, keeps increasing to $1$ at $360^\circ $. These values continue to repeat with a period of $360^\circ $.
With Radians:
As the angle, $\theta $, rotates counter-clockwise from $0^\circ $, the values of cosine start at $1$, decrease to $0$ at $\frac {\pi }{2}$, keeps decreasing to $-1$ at $\pi $, increases to $0$ at $\frac {3\pi }{2}$, keeps increasing to $1$ at $2\pi $. These values continue to repeat with a period of $2\pi $.

Sine

With Degrees:
As the angle, $\theta $, rotates counter-clockwise the value of sine start at $0$, increases to $1$ at $90^\circ $, decreases to $0$ at $180^\circ $, keeps decreasing to $-1$ at $270^\circ $, increases to $0$ at $360^\circ $. These values continue to repeat with a period of $360^\circ $.
With Radians:
As the angle, $\theta $, rotates counter-clockwise the value of sine start at $0$, increases to $1$ at $\frac {\pi }{2}$, decreases to $0$ at $\pi $, keeps decreasing to $-1$ at $\frac {3\pi }{2}$, increases to $0$ at $2\pi $. These values continue to repeat with a period of $2\pi $.

Angles are measured counterclockwise from the positive horizontal axis. It is like a circular number line. Rotating counterclockwise is the positive direction. Rotating clockwise is the negative direction. After you rotate a full circle (positively or negatively), the values just keep repeating.

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The unit circle is a circle of radius $1$ centered at the origin.

Its coordinates satisfy the equation

\[ x^2 + y^ 2 = 1 \]

For instance the point $\left ( \frac {1}{\sqrt {2}}, \frac {1}{\sqrt {2}} \right )$ is on the unit circle since these coordinates satisfy the equation.

\[ \left ( \frac {1}{\sqrt {2}} \right )^2 + \left ( \frac {1}{\sqrt {2}} \right )^2 = \frac {1}{2} + \frac {1}{2} = 1 \]

Values

The radius from the origin at $45^{\circ }$ intersects the unit circle at $\left ( \frac {1}{\sqrt {2}}, \frac {1}{\sqrt {2}} \right )$.

Therefore,

\[ \cos (45^{\circ }) = \cos \left ( \frac {\pi }{4} \right ) = \frac {1}{\sqrt {2}} \]

$765^{\circ }$ = $45^{\circ } + 360^{\circ } + 360^{\circ }$

\[ \cos (765^{\circ }) = \cos (45^{\circ }) = \frac {1}{\sqrt {2}} \]

Values

The radius from the origin at $135^{\circ }$ intersects the unit circle at $\left ( -\frac {1}{\sqrt {2}}, \frac {1}{\sqrt {2}} \right )$.

Therefore,

\[ \cos (135^{\circ }) = \cos \left ( \frac {3\pi }{4} \right ) = -\frac {1}{\sqrt {2}} \]

$-225^{\circ }$ = $135^{\circ } - 360^{\circ }$

\[ \cos (-225^{\circ }) = \cos (135^{\circ }) = -\frac {1}{\sqrt {2}} \]

It turns out that there are many connections between sine and cosine. Many. One of these connections concerns rate of change, which is a very important thread in this course. The easiest way to describe these connections is with radians. Hence, Calculus only uses radians. Therefore, we will attempt to speak in radians as much as possible.

Sine and Cosine are periodic functions with periods of $2\pi $ (radians). Therefore, we only need examine the interval $[0, 2\pi )$ and then repeat our findings every $2\pi $.

We call the interval $[0, 2\pi )$, the principal interval for sine and cosine.

From the graphs we can see that Sine and Cosine have maximum and minimum values.

Extreme Values

The sine function has a maximum value of $\answer {1}$. Inside the interval $[0, 2\pi )$, this maximum value of sine occurs at $\answer {\frac {\pi }{2}}$.
The sine function has a minimum value of $\answer {-1}$. Inside the interval $[0, 2\pi )$, this minimum value of since occurs at $\answer {\frac {3\pi }{2}}$.
The sine function has two zeros inside the interval $[0, 2\pi )$. One of these zeros is $0$. The other zero is $\answer {\pi }$.

The cosine function has a maximum value of $\answer {1}$. Inside the interval $[0, 2\pi )$, this maximum value of cosine occurs at $\answer {0}$.
The cosine function has a minimum value of $\answer {-1}$. Inside the interval $[0, 2\pi )$, this minimum value of cosine occurs at $\answer {\pi }$.
The cosine function has two zeros inside the interval $[0, 2\pi )$. One of these zeros is $\frac {\pi }{2}$. The other zero is $\answer {\frac {3\pi }{2}}$.

Easy Angles

Sine and cosine are transcendental functions. They transcend Algebra. They are beyond our usual algebraic tools. That makes equations difficult when they involve sine and cosine.

This is true unless you work with angles that just happen to have nice values for sine and cosine. We have several easy angles: $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$.

$\sin (30^{\circ }) = \frac {1}{2}$
$\cos (30^{\circ }) = \frac {\sqrt {3}}{2}$

And, since $30^{\circ }$ and $60^{\circ }$ make up a right triangle, we have

$\sin (60^{\circ }) = \frac {\sqrt {3}}{2}$
$\cos (60^{\circ }) = \frac {1}{2}$

$45^{\circ }$ cuts the quadrant in half making sine and cosine equal.

$\sin (45^{\circ }) = \frac {1}{\sqrt {2}}$
$\cos (45^{\circ }) = \frac {1}{\sqrt {2}}$

Add these to $0^{\circ }$, $90^{\circ }$, $180^{\circ }$, and $270^{\circ }$ and we can walk around the unit circle.

Memorize this Circle

\[ \sin \left ( \frac {7\pi }{6} \right ) = \answer {-\frac {1}{2}} \]

\[ \cos \left ( \frac {5\pi }{3} \right ) = \answer {\frac {1}{2}} \]

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more examples can be found by following this link
More Examples of Piecewise-Defined Functions

2026-01-27 21:32:52