পর্যায়কাল, কম্পাংক আর বিস্তার

We define a function $f$ to be periodic if there exists a $T$ such that

$$f(t+T) = f(t) \quad \quad (\text{for all t})$$

Which is to say that, the function repeats itself after a fixed time interval.

Before proceeding think of an example of such a function.

Okay, here is the most popular periodic function $\sin(x)$

(Graph borrowed from Draw Plots guide)

Given that $$f(t+T) = f(t) \quad \quad (\text{for all t})$$

we define the period of the function to be $T$.

We define frequency $\nu$ to be the number of times the function has completed a cycle in unit time.

Clearly,

$$\nu = \frac{1}{T}$$

Would it be possible to explain why this answer is correct? I'm confused (I'm commenting this as a learner, so unfortunately I don't know how to explain it).

A function is periodic if it repeats a set of outputs after a certain T. In mathematical terms this means that a function, $f(x)$, is periodic if it satisfies $f(x+T) = f(x)$.

There are 3 main parts of a periodic function,$f(x) = a*f(x+T)$ are period, frequency, and amplitude:

The period is how long the function takes to return to the same set of outputs. In the periodic function shown above, the period is $T$.

The frequency is how many times the function oscillates or goes through a period in a certain amount of time(usually 1 second). This is equal to $\frac{1}{T}$.

The amplitude is max amount the function deviates from the mid-line or average. This is always a positive number.. In the function shown above, the amplitude is $|a|$.

The most common periodic functions are sinusoids(functions that are composed of $sin$ or $cos$. Graph of $y=sin(x)(in blue) and$y=cos(x)$(in green):$

The period and frequency of both functions are $2*\pi$ and $\frac{1}{2*\pi}$. The amplitude are both $1$.

In a sinusoid function, $f(x)=a*sin(bx)$, the period is $\frac{2*\pi}{|b|}$. The frequency is $\frac{|b|}{2*\pi}$ and the amplitude is $a$.