# 3.1: Even and Odd Identities

**At Grade**Created by: CK-12

**Practice**Even and Odd Identities

You and your friend are in math class together. You enjoy talking a lot outside of class about all of the interesting topics you cover in class. Lately you've been covering trig functions and the unit circle. As it turns out, trig functions of certain angles are pretty easy to remember. However, you and your friend are wishing there was an easy way to ‘‘shortcut’’ calculations so that if you knew a trig function for an angle you could relate it to the trig function for another angle; in effect giving you more reward for knowing the first trig function.

You're examining some notes and starting writing down trig functions at random. You eventually write down:

Is there any way that if you knew how to compute this, you'd automatically know the answer for a different angle?

As it turns out, there is. Read on, and by the time you've finished this Concept, you'll know what other angle's value of cosine you already know, just by knowing the answer above.

### Watch This

James Sousa: Even and Odd Trigonometric Identities

### Guidance

An even function is a function where the value of the function acting on an argument is the same as the value of the function when acting on the negative of the argument. Or, in short:

So, for example, if f(x) is some function that is even, then f(2) has the same answer as f(-2). f(5) has the same answer as f(-5), and so on.

In contrast, an odd function is a function where the negative of the function's answer is the same as the function acting on the negative argument. In math terms, this is:

If a function were negative, then f(-2) = -f(2), f(-5) = -f(5), and so on.

Functions are even or odd depending on how the end behavior of the graphical representation looks. For example,

Let’s consider sine. Start with

From this we see that sine is **odd**. Therefore,

This tells us that the cosine is **even**. Therefore,

Notice that cosecant is odd like sine and secant is even like cosine.

#### Example A

If

**Solution:** We know that sine is odd. Cosine is even, so

#### Example B

If

**Solution:** Since sine is an odd function,

Therefore,

#### Example C

If

**Solution:**

Since cosine is an even function,

Therefore,

### Guided Practice

1. What two angles have a value for cosine of

2. If

3. If

**Solutions:**

1. On the unit circle, the angles

2. There are 2 ways to think about this problem. Since

3. Since

### Concept Problem Solution

Since you now know that cosine is an even function, you get to know the cosine of the negative of an angle automatically if you know the cosine of the positive of the angle.

Therefore, since \begin{align*}\cos \left( \frac{\pi}{18} \right) = .9848\end{align*}, you automatically know that \begin{align*}\cos \left( -\frac{\pi}{18} \right) = \cos \left( \frac{17\pi}{18} \right)= .9848\end{align*}.

### Explore More

Identify whether each function is even or odd.

- \begin{align*}y=\sin(x)\end{align*}
- \begin{align*}y=\cos(x)\end{align*}
- \begin{align*}y=\cot(x)\end{align*}
- \begin{align*}y=x^4\end{align*}
- \begin{align*}y=x\end{align*}
- If \begin{align*}\sin(x)=.3\end{align*}, what is \begin{align*}\sin(-x)\end{align*}?
- If \begin{align*}\cos(x)=.5\end{align*}, what is \begin{align*}\cos(-x)\end{align*}?
- If \begin{align*}\tan(x)=.1\end{align*}, what is \begin{align*}\tan(-x)\end{align*}?
- If \begin{align*}\cot(x)=.3\end{align*}, what is \begin{align*}\cot(-x)\end{align*}?
- If \begin{align*}\csc(x)=.3\end{align*}, what is \begin{align*}\csc(-x)\end{align*}?
- If \begin{align*}\sec(x)=2\end{align*}, what is \begin{align*}\sec(-x)\end{align*}?
- If \begin{align*}\sin(x)=-.2\end{align*}, what is \begin{align*}\sin(-x)\end{align*}?
- If \begin{align*}\cos(x)=-.25\end{align*}, what is \begin{align*}\sec(-x)\end{align*}?
- If \begin{align*}\csc(x)=4\end{align*}, what is \begin{align*}\sin(-x)\end{align*}?
- If \begin{align*}\tan(x)=-.2\end{align*}, what is \begin{align*}\cot(-x)\end{align*}?
- If \begin{align*}\sin(x)=-.5\end{align*} and \begin{align*}\cos(x)=-\frac{\sqrt{3}}{2}\end{align*}, what is \begin{align*}\cot(-x)\end{align*}?
- If \begin{align*}\cos(x)=-.5\end{align*} and \begin{align*}\sin(x)=\frac{\sqrt{3}}{2}\end{align*}, what is \begin{align*}\tan(-x)\end{align*}?
- If \begin{align*}\cos(x)=-\frac{\sqrt{2}}{2}\end{align*} and \begin{align*}\tan(x)=-1\end{align*}, what is \begin{align*}\sin(-x)\end{align*}?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.1.

Even Function

An even function is a function with a graph that is symmetric with respect to the -axis and has the property that .Odd Function

An odd function is a function with the property that . Odd functions have rotational symmetry about the origin.### Image Attributions

Here you'll learn what even and odd functions are and how you can use them in solving for values of trig equations.

## Concept Nodes:

Even Function

An even function is a function with a graph that is symmetric with respect to the -axis and has the property that .Odd Function

An odd function is a function with the property that . Odd functions have rotational symmetry about the origin.**Save or share your relevant files like activites, homework and worksheet.**

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