3.1.5: Even and Odd Identities

Functions symmetric with respect to the y-axis or about the origin.

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:

$\cos \left(\dfrac{\pi}{18}\right)$

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

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:

$f(x)=f(−x)$

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:

$−f(x)=f(−x)$

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, $y=x^2$ is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the $y$−axis. $y=x^3$ is considered an odd function for the opposite reason. The ends of a cubic function point in opposite directions and therefore the parabola is not symmetric about the $y$−axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd).

$\dfrac{\text { Even Function }}{y=(-x)^{2}=x^{2}} \quad \dfrac{\text { Odd Function }}{y=(-x)^{3}=-x^{3}}$

Let’s consider sine. Start with $\sin(−x)$. Will it equal $\sin x$ or $−\sin x$? Plug in a couple of values to see.

$\begin{aligned} \sin(−30^{\circ} )&=\sin 330^{\circ} =−\dfrac{1}{2}=−\sin 30^{\circ} \\ \sin(−135^{\circ} ) &=\sin 225^{\circ} =−\dfrac{\sqrt{2}}{2}=−\sin 135^{\circ}\end{aligned}$

From this we see that sine is odd. Therefore, $\sin(−x)=−\sin x$, for any value of $x$. For cosine, we will plug in a couple of values to determine if it’s even or odd.

$\begin{aligned} \cos(−30^{\circ} )&=\cos 330^{\circ} =\dfrac{\sqrt{3}}{2}=\cos 30^{\circ} \\ \cos(−135^{\circ} ) &=\cos 225^{\circ} =−\dfrac{\sqrt{2}}{2}=\cos 135^{\circ}\end{aligned}$

This tells us that the cosine is even. Therefore, $\cos(−x)= \cos x$, for any value of $x$. The other four trigonometric functions are as follows:

$\begin{aligned} \tan(−x)&=−\tan x \\ \csc(−x)&=−\csc x \\ \sec(−x)&=\sec x \\ \cot(−x)&=−\cot x \end{aligned}$

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

Finding Even and Odd Identities

1. Find $\sin x$

If $\cos(−x)=\dfrac{3}{4}$ and $\tan(−x)=−\dfrac{\sqrt{7}}{3}$, find $\sin x$.

We know that sine is odd. Cosine is even, so $\cos x=\dfrac{3}{4}$. Tangent is odd, so $\tan x=\dfrac{\sqrt{7}}{3}$. Therefore, sine is positive and $\sin x=\dfrac{\sqrt{7}}{4}$.

2. Find sin(-x)

If $\sin(x)=.25$, find $\sin(−x)$

Since sine is an odd function, $\sin(−\theta )=−\sin(\theta )$.

Therefore, $\sin(−x)=−\sin(x)=−.25$

3. Find cos(-x)

If $\cos(x)=.75$, find $\cos(−x)$

Since cosine is an even function, $\cos(x)=\cos(−x)$.

Therefore, $\cos(−x)=.75$

Example $\PageIndex{1}$

Earlier, you were asked to compute $\cos\left(\dfrac{\pi }{18}\right)$.

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 $\cos\left(\dfrac{\pi}{18}\right)=.9848$, you automatically know that $\cos\left(-\dfrac{\pi}{18}\right)=\cos\left(\dfrac{17\pi}{18}\right)=.9848$.

Example $\PageIndex{2}$

What two angles have a value for cosine of $\dfrac{\sqrt{3}}{2}$?

Solution

On the unit circle, the angles $30^{\circ}$ and $330^{\circ}$ both have $\dfrac{\sqrt{3}}{2}$ as their value for cosine. $330^{\circ}$ can be rewritten as $−30^{\circ}$

Example $\PageIndex{3}$

If $\cos\theta =\dfrac{\sqrt{3}}{2}$, find $sec(−\theta )$

Solution

There are 2 ways to think about this problem. Since $\cos\theta =\cos−\theta$, you could say $\sec(−\theta )=\dfrac{1}{\cos(−\theta )}=\dfrac{1}{cos(\theta )}$ Or you could leave the cosine function the way it is and say that $\sec(−\theta )=\sec(\theta )=\dfrac{1}{\cos\theta}$. But either way, the answer is $\dfrac{2}{\sqrt{3}}$

Example $\PageIndex{4}$

If $cot\theta =−\sqrt{3}$ find $cot−\theta$

Solution

Since $\cot(−\theta )=−cot(\theta )$, if $\cot \theta =−\sqrt{3}$ then $−\cot(−\theta )=−\sqrt{3}$. Therefore, $\cot(−\theta )=\sqrt{3}$.

Review

Identify whether each function is even or odd.

1. $y=\sin(x)$
2. $y=\cos(x)$
3. $y=\cot(x)$
4. $y=x^4$
5. $y=x$
6. If $\sin(x)=.3$, what is $\sin(−x)$?
7. If $\cos(x)=.5$, what is $\cos(−x)$?
8. If $\tan(x)=.1$, what is $\tan(−x)$?
9. If $\cot(x)=.3$, what is $\cot(−x)$?
10. If $\csc(x)=.3$, what is $\csc(−x)$?
11. If $\sec(x)=2$, what is $\sec(−x)$?
12. If $\sin(x)=−.2$, what is $\sin(−x)$?
13. If $\cos(x)=−.25$, what is $\sec(−x)$?
14. If $\csc(x)=4$, what is $\sin(−x)$?
15. If $\tan(x)=−.2$, what is $\cot(−x)$?
16. If $\sin(x)=−.5$ and $\cos(x)=−\dfrac{\sqrt{3}}{2}$, what is $\cot(−x)$?
17. If $\cos(x)=−.5$ and $\sin(x)=\dfrac{\sqrt{3}}{2}$, what is $\tan(−x)$?
18. If $\cos(x)=−\dfrac{\sqrt{2}}{2}$ and $\tan(x)=−1$, what is $\sin(−x)$?

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

Vocabulary

Term	Definition
Even Function	An even function is a function with a graph that is symmetric with respect to the y-axis and has the property that $f(−x)=f(x)$.
Odd Function	An odd function is a function with the property that $f(−x)=−f(x)$. Odd functions have rotational symmetry about the origin.

Additional Resources

Video: Even and Odd Trigonometric Identities

Practice: Even and Odd Identities