Question

What is the period of $y = 3\cos 5x$ ?

Answer 1

Here we are going to find the period of trigonometric function using the definition of periodic function.
Definition used:
A function is said to be periodic if there exists a positive real number $T$ such that $f\left( {x + T} \right) = f(x)$ for all $x \in D$ where $D$ is the domain of the function$f(x)$ .
Now for trigonometric function, graphs of trigonometric function clearly show that periods of $\cos x$ is $2\pi$ . Here we shall mathematically determine periods of few of these trigonometric functions using definition of period.

Complete step-by-step solution:
For $\cos x$ to be periodic function,
$\cos \left( {x + T} \right) = \cos x$
$\Rightarrow x + T = 2n\pi \pm x$ , $n \in Z$
Therefore $x + T = 2n\pi + x$ or $x + T = 2n\pi - x$
First set of value is independent of $x$ hence,
$T = 2n\pi , n \in Z$
The least positive value of $T$ that is the period of the function is $T = 2\pi$ .
Now we have to find the periodic function $y = 3\cos 5x$ . We know that the period of the function $\cos x$ is $2\pi$.
If we have a function $f(x) = \cos (ax)$ where $a > 0$ is the coefficient of the $x$ term, then the period of the functions is $T = \dfrac{{2\pi }}{a}$ .
Here we have the periodic function is $f(x) = 3\cos 5x$ The formula for the period of the function is $T = \dfrac{{2\pi }}{a}$.
From given function, $a = 5$
Hence the period of the function $T = \dfrac{{2\pi }}{5}$

Note: The smallest positive $\alpha \in \mathbb{R}$ for a periodic function $f$ is defined as the fundamental period of $f$.
Ex:
The function $f = \sin x$ is periodic function with set of periods \left\\{ {2n\pi :n \in \mathbb{Z}} \right\\} fundamental period of $f = \sin x$ is $2\pi$

Periodicity is the domain based property.
A periodic function may or may not have a fundamental period.
Ex:
$f: R \to R$
$f(x) = c, \forall x \in \mathbb{R},\,c \in {\mathbb{R}}$
Then for any $\alpha \in \mathbb{R}\,\,\,\,\,\,f\left( {x + \alpha } \right) = f(x) = c, \forall x \in \mathbb{R}$
Set of periods is $\mathbb{R}$
But the fundamental period does not exist.
Sum or difference of periodic function may not be periodic. Sum of two non periodic functions may be a periodic function.