Q 110 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

The cotangent function is an odd function.

It is symmetrical about origin.

Step 1. Given Information

We have to check that the cotangent 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 Cotangent 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 :- 

$cot\left(-\theta \right)=-cot\theta$.

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

Step 3. To check about symmetric

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

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

From the graph we can see that the graph is neither symmetric about x-axis not y-axis, but it is symmetric about origin.

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