Help, please. I don't really the concept of "angle" in periodic functions.

Question

I have trouble with it, more specifically with the trig. functions. I know that if you multiply the angle of a function by a constant the period of the function gets divided by it but what is exactly the angle? Is it the rate of change or something like that? Thanks a lot.

Answer 1

Let's take $\sin$ as an example. We know that $\sin(x)$ is periodic with period $2\pi$. That is, $\forall x\in\mathbb{R},$ $\sin(x+2\pi)=\sin(x).$ So what can we say about the function $f(x)=\sin(kx)?$ Define a new variable $x'=kx$. Then $f(x)=\sin(x')$. We know that $\sin(x'+2\pi)=\sin(x')$, or instead we can write $$\sin\left(k\left(x+\frac{2\pi}{k}\right)\right)=\sin(kx)$$ Thus $f$ is periodic with period $2\pi/k$.

Answer 2

Welcome to Math Stack Exchange! There are a couple of things in your question I would like to address to hopefully help you clarify the concepts you mentioned.

$\textbf{1.}$ Definition of angle

There are several ways to define what an angle is, but for our purposes, I think the best way to think about an angle is as follows:

An angle is a measure of how much you rotate around some point.

This intuition will come in handy later when we interpret "rotation" as "walking around a circle".

$\textbf{2.}$ Periodic functions

A periodic function is defined as follows

A function $f$ is said to be periodic if, for some nonzero constant $P$, it is the case that $$f(x +P) = f(x) $$ for all values of $x$ in the domain of the function.

The key thing to notice here is that this definition does not apply exclusively to trig functions. It just so happens that trig functions also satisfy this definition.

For an example of a periodic function that is not related to trigonometry, you can take the function $$ f(x) =x- \lfloor x\rfloor $$ where $\lfloor x\rfloor$ denotes the floor function. You can find a proof that the above function is indeed periodic in this answer.

$\textbf{3.}$ Trig functions

Now we come to the important part. To understand how angles relate to trig functions we first need to understand how these trig functions are defined.

Let's say that you draw a circle of radius $1$ centered at the origin $(0,0)$ on the $xy$ plane. Now, let's suppose that you're standing at the point $(1,0)$. The diagram representing this scenario looks like this:

Let's now suppose that you start walking around this circle in a counterclockwise direction and that after a while you stop at some other point on the circle.

We can say that the distance you walked around the circle is, in some sense, equivalent to how much you rotated around the circle. This measurement of how much you rotated is precisely what we'll call the angle you rotated (recalling our definition in section $\textbf{1}$).

We can visually see the previously described scenario in the following diagram:

where here the angle you rotated is denoted by the symbol $\color{orange}{\theta}$ and is visually seen as the $\color{orange}{\text{curved arrow}}$ pointing in the direction you walked.

In the above diagram, we also see that the yellow point you ended up on is at some $\color{purple}{\text{vertical}}$ and $\color{ForestGreen}{\text{horizontal}}$ distance from the origin (which is where the blue axes intersect).

These vertical and horizontal distances are determined by how much you rotated around the circle, or in other words, these distances are determined by your angle of rotation $\color{orange}{\theta}$.

As you may have guessed by now, these distances are the definitions of both the $\sin$ and $\cos$ functions:

• $\color{Purple}{\sin(}\color{orange}{\theta}\color{Purple}{)}$ is a function that gives you the $\color{Purple}{\text{vertical}}$ distance you are at after rotating some angle $\color{orange}{\theta}$ around the circle of radius $1$.
• $\color{ForestGreen}{\cos(}\color{orange}{\theta}\color{ForestGreen}{)}$ is a function that gives you the $\color{ForestGreen}{\text{horizontal}}$ distance you are at after rotating some angle $\color{orange}{\theta}$ around the circle of radius $1$.

$\textbf{4.}$ Why are trig functions periodic?

Let's now suppose that after you stopped walking at the point in the previous image you continue walking around the circle. What would happen if you walked all around the circle and you again ended up at the point you previously stopped at? Well, this scenario would look something like this:

Notice that your angle of rotation has changed and is now $\color{orange}{\theta} \color{red}{+ 2 \pi}$. This $\color{red}{2 \pi}$ we're adding just means that you rotated one additional full turn around the circle. Thus, we say that one full rotation around the circle is equivalent to an "angle" of $\color{red}{2 \pi}$.

Now, the key thing to notice is that you're angle of rotation is different now. This time you have clearly rotated more by completely walking around the circle, in contrast to the angle you rotated in section $\textbf{3}$.

Despite this, the position where you ended up on is exactly the same position as the one you had in section $\textbf{3}$. In particular, the horizontal and vertical distances of your position now are the same as the horizontal and vertical distances of your position before. Because of this, we conclude that $$ \color{purple}{\sin(}\color{orange}{\theta}\color{red}{+ 2 \pi}\color{purple}{)} = \color{purple}{\sin(}\color{orange}{\theta}\color{purple}{)} $$ $$ \color{ForestGreen}{\cos(}\color{orange}{\theta}\color{red}{+ 2 \pi}\color{ForestGreen}{)} = \color{ForestGreen}{\cos(}\color{orange}{\theta}\color{ForestGreen}{)} $$ where we see that both $\sin(\theta)$ and $\cos(\theta)$ satisfy the definition of periodic function established in section $\textbf{2}$, where the period in this case corresponds to $P = \color{red}{ 2 \pi}$.

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As for why the period of these functions changes when multiplying the input of $\sin$ and $\cos$ by some constant, K.defaoite's answer gives a really good explanation for this.

In summary, we know the following:

1. Angles are not related to periodic functions. Each one is its own unique concept.
2. In the case of the $\sin$ and $\cos$ functions, we choose to say the input of these functions is an angle because we define these functions in terms of an angle of rotation around a circle.
3. The reason $\sin$ and $\cos$ are periodic is because if you're at some point on a circle, you can walk one full rotation around the circle and you'll again end up at the same point.

I hope some of this helped make the concepts you mentioned a bit clearer. If you have any more questions don't be afraid to ask. Good day!