[Math] 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

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 $T_g=\frac{2\pi}{k}$.