Q 126 - Precalculus Enhanced with Graphing Utilities

Question

Show that the period of $f\left(\theta \right)=csc\theta$ is $2\pi$.

Step-by-Step Solution

Answer

From the following graph of cosecant function $f\left(\theta \right)=csc\theta$ it is clear that the period of cosecant function is $2\pi$.

Step 1. Given Information

We have to prove that the period of cosecant function is $2\pi$.

We will draw the graph of this function. After that from the graph we will examine the period of the function.

Step 2. To find period of cosecant function

The given function is :- 

$f\left(\theta \right)=csc\theta$

By using graphing utilities, we have the following graph of this function :-

We can see that curve of the graph repeated itself.

We know that the period of a function is where from the graph of function repeated itself.

We can see that the graph from $\frac{\pi }{2}$to $\frac{5\pi }{2}$ is same as $\frac{5\pi }{2}$ to $\frac{9\pi }{2}$ and $\frac{-3\pi }{2}$ to $\frac{\pi }{2}$.

So we can conclude that the period of this function is :-

$\frac{5\pi }{2}-\frac{\pi }{2}=2\pi$.