How to determine the periods of a periodic function?

Question

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here.

Given a periodic function $f(x)=sin(x)$,

Why is the period of above function becomes $2\pi$?

for function $sin(nx)$ , I am being told to use this formula $2\pi/n$ to get the period. But I am not able to fully understand the concept behind it.

I am using sin function for the sake of simplicity. My question is how do I quickly work out what are the periods of a certain periodic function(preferably without drawing out the graph)?

for example, $cos2t$ has periods $\pi,2\pi,3\pi,...$ (I have no idea why)

Also, there are many different types of periodic waves apart from sine and cosine. So, what would be the best way to get their periods if the only thing I am being given is the formula of the function.

For your info, I am supposed to find the smallest period so that I can work out the Fourier series of that function.

Answer 1

A periodic function is a function that repeats its values in regular intervals or periods. We say that a function has a period of $L$ if $$f(x) = f(x + L)$$ for all $x$ in the domain of $f(x)$.

For example, the function $\sin(x)$ has period $2 \pi$ since $\sin(x) = \sin(x + 2 \pi)$ (as you can easily verify from the graph). The function returns to the same value that it starts with after every $2 \pi$.

NOTE: Wikipedia has a great explanation for this.

Answer 2

First, let's discuss what the definition of a period is for a periodic function. A function $f$ is periodic with period $T$ means $f(t+T) = f(t)$ for all $t$.

The period of $\sin$ is $2\pi$ by definition. (You might ask why $\sin$ is defined this way, but that question may be outside the scope of this thread.) This means that $\sin(t+2\pi)=\sin(t)$ for all $t$.

Now that we know that the period of $\sin$ is, what is the period of $\sin(kt)$? Let us define a function $g$ as $g(t)=\sin(kt)$. We are asking, what is the period of $g$. That is, what value $T_g$ satisfies $g(t+T_g)=g(t)$ for all $t$.

We know $\sin(kt+2\pi)=\sin(kt)$ for all $t$. So what value of $T_g$ satisfies $k(t+T_g)=kt+2\pi$? Solving for $T_g$, we see that $T_g=\frac{2\pi}{k}$.

Answer 3

Because the sine function is defined to be the ratio of the $y$ value to the radius in a circle of radius $r$ the values will repeat every $2\pi$. Thus the period of $\sin{x}$ is $2\pi$. From there it reduces to recognizing, for any function $g(x)$, $g(nx)$ will compress the graph in the $x$ direction by a factor of $1/n$.

Answer 4

You just need to solve the equation $$f(x) = f(x + T)$$ for $T$ using all available means suitable for solving the equation you get.