Q 108 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

The cosine function is an even function.

It is symmetrical about y-axis.

Step 1. Given Information

We have to check that the cosine 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 Cosine 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  $\cos \left(-\theta \right)=\cos \left(\theta \right)$.

So we can conclude that Cosine function is an even function. 

Step 3. To check about symmetric

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

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

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

So we can conclude that cosine function is symmetric about y-axis.