Question

What is the period of the function $y = \cos 4x$ ?

Answer 1

Here we are asked to find the period of the given function. The period of the function can be found from the coefficients of the function. The coefficients of the function tell us what the amplitude and period of the function are. We know that the period of the function cosine is $2\pi$ then the period of the function can be found by dividing $2\pi$ by its coefficient.

Complete step-by-step answer:
Given the function is $y = \cos 4x$ .
Find the period of the function.
First of all period of the function means,
A function with a period $P$ will repeat on intervals of the length $P$, these intervals are sometimes referred to as periods of the function.
And the period of a function is used to mean its fundamental period.
So that,
In this sum, we find the period of $\cos 4x$. It means we find the length of the interval in this function.
Every function to find the period has a separate function format.
$\cos x$ the function also has a unique format.
The format of the function is $y = a\cos (bx - c) + d$.
Here,
$a,\,b,\,c\,$and $d$ are some constant values.
In question, they given the format of the function to the given function format.

$$y = \cos (bx - c) + d \\\ y = \cos 4x \\\$$

Here,

$$a = 1, \\\ b = 4, \\\ c = 0, \\\ d = 0. \\\$$

The general formal to find the period of the function have formulae.
The formulae $\dfrac{{2\pi }}{{\left| b \right|}}$ are.
here $b$ is the coefficient on the $x$ term.
Now we apply our function values in this formulae,
$= \dfrac{{2\pi }}{{\left| 4 \right|}}$
$= \dfrac{{2\pi }}{4}$
Divide the values,
$= \dfrac{\pi }{2}$.
So that $\dfrac{\pi }{2}$ is the period of the function $y = \cos 4x$.

Note: The period of all the functions is not the same, they vary from function to function. Here we have seen that the period of the function cosine as $2\pi$ likewise the period of the function tangent is $\pi$ so to find the period of the function with $\tan$ will be found by the following way: $\dfrac{\pi }{B}$ where $B$ is the coefficient of that function.