Question

Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?

Short answer

The cosecant function is an oddfunction.

It is symmetrical about origin.

Step-by-step solution

Step 1. Given Information 

We have to check that the cosecant function is an even function, odd function or neither.

Also we have to check that it is symmetric or not.

If yes, then we also to find that it is symmetric about what.

Step 2. To check that Cosecant function is even, odd or neither. 

We know that a function is even if $f(-x)=f\left(x\right)$and is odd if $f(-x)=-f\left(x\right)$.

We also know that :-

localid="1646574467907" $\cos ec(-\theta )=-\cos ec\theta$.

So we can conclude that Cosecant function is an odd function.

Step 3. To check about symmetric 

To check the symmetry of the cosecant function we will use graphing utility.

By using graphing utility, we can graph the cosecant function as following :-

From the graph we can see that the graph is neither symmetric about x-axis nor y-axis, but it is symmetric about origin.

So we can conclude that cosecant function is symmetric about origin.