Question

How do you find the period of $\tan \left( {2\pi x} \right)?$

Answer 1

Hint: This problem deals with finding the period of the given function. The period of a periodic function is the interval between two matching points on the graph. In other words, it is the distance along the x-axis that the function has to travel before it starts to repeat its pattern. The basic sine and cosine functions have a period of $2\pi $, while tangent has a period of $\pi $.

Complete step-by-step answer:
Given an expression which is a trigonometric function.
The given trigonometric functional expression is $\tan \left( {2\pi x} \right)$
We know that the period of tangent trigonometric function is $\pi $.
To find the period of any function, if the function is in the standard form of an equation is given by:
The standard form of an equation is given by:
$ \Rightarrow y = f(x)$
Here $f(x) = A\tan \left( {Bx - C} \right) + D$, hence substituting the function in the above expression.
$ \Rightarrow y = A\tan \left( {Bx - C} \right) + D$
Here the period, $P$ of the tangent trigonometric standard function is given by:
$ \Rightarrow P = \dfrac{\pi }{B}$
Now finding the period of the given tangent function.
Consider the given function as given below:
$ \Rightarrow y = \tan \left( {2\pi x} \right)$
Here the period of the tangent function is given by:
Here by comparing with the standard form, $B = 2\pi $.
$ \Rightarrow P = \dfrac{\pi }{{2\pi }}$
$\therefore P = \dfrac{1}{2}$

The period of the function $\tan \left( {2\pi x} \right)$ is $\pi $.

Note:
Please note that the fundamental period of a function is the period of the function which are of the form, $f\left( {x + k} \right) = f\left( x \right)$ and $f\left( x \right) = f\left( {x + k} \right)$, then $k$ is called the period of the function and the function $f$ is called a periodic function. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. Any part of the graph that shows this pattern over one period is called a cycle.