# Symmetry in Trigonometric Graphs

The trigonometric functions cosine, sine, and tangent satisfy several properties of **symmetry** that are useful for understanding and evaluating these functions.

## Symmetry

The cosine and sine functions satisfy the following properties of symmetry:

$\begin{aligned} \cos(-\theta) &= \cos(\theta) \\ \sin(-\theta) &= -\sin(\theta). \end{aligned}$

From the definition of cosine and sine in the unit circle,

$x= \cos \theta \quad \text{ and } \quad y= \sin \theta.$

We can see that for both $\theta$ and $-\theta$, $x$ remains the same. Thus, $\cos \theta=\cos (-\theta)$.

Similarly, we can see that the $y$ in two cases are additive inverse of each other. Thus, $\sin (-\theta)=-\sin\theta.\ _\square$

Now that we have the above identities, we can prove several other identities, as shown in the following example.

Prove the identities

$\begin{aligned} \tan(-\theta) &= -\tan(\theta)\\ \cot(-\theta) &= -\cot(\theta)\\ \csc(-\theta) &= -\csc(\theta)\\ \sec(-\theta) &= \sec(\theta). \end{aligned}$

We have

$\begin{aligned} \tan(-\theta) &=\frac{\sin(-\theta)}{\cos(-\theta)}=\frac{-\sin(\theta)}{\cos(\theta)}=-\tan(\theta)\\ \cot(-\theta) &=\frac{1}{\tan(-\theta)}=\frac{1}{-\tan(\theta)}=-\cot(\theta)\\ \csc(-\theta) &=\frac{1}{\sin(-\theta)}=\frac{1}{-\sin(\theta)}=-\csc(\theta)\\ \sec(-\theta) &=\frac{1}{\cos(-\theta)}=\frac{1}{\cos(\theta)}=\sec(\theta).\ _\square \end{aligned}$

## Even and Odd Functions

Using the properties of symmetry above, we can show that sine and cosine are special types of functions.

A function $f(x)$ is an

even functionif and only if for all real values of $x$, $f(-x)=f(x)$. In other words, the graph is symmetric about $y$-axis.

A function $f(x)$ is an

odd functionif and only if for all real values of $x$, $f(-x)=-f(x)$. In other words, the graph is symmetric about origin.

Also, $f(-0)=-f(0)\implies f(0)=0$. That is, an odd function must pass through the origin.

From this definition, the cosine function is an **even function** and the sine function is an **odd function**.

## Symmetry in Angles

What symmetry is there between the angles $\theta$ and $(\theta + \pi)?$ If we plug in a few values for $\theta$, how do the basic trigonometric functions change?

By the properties of symmetry, we can write $\sin (\theta + \pi)$ in terms of $\sin(\theta)$ as follows:

$\sin (\theta+\pi)=-\sin(\theta).$

Similarly, we can write $\cos (\theta + \pi)$ in terms of $\cos(\theta)$ as follows:

$\cos (\theta+\pi)=-\cos(\theta).$

## Examples

Determine whether the function

$f(x) = \tan^2(x) + \cos(x)$

is an odd function, an even function, or neither.

The function satisfies

$f(-x) = \tan^2(-x) + \cos(-x) = \tan^2(x) + \cos(x) = f(x)$

since $\cos(x)$ is an even function. Therefore, $f(x)$ is an even function. $_\square$

Find a relationship between $\tan (\theta + \pi)$ and $\tan (\theta ).$

Solution 1:We have$\begin{aligned} \tan (\theta + \pi) & = \frac{\sin(\theta + \pi ) }{\cos(\theta + \pi) } \\ &= \frac{-\sin(\theta)}{-\cos(\theta)}\\ &= \frac{\sin(\theta)}{\cos(\theta)}\\ &= \tan (\theta ). \end{aligned}$

Therefore, $\tan (\theta + \pi) = \tan (\theta ).$

Solution 2:We have$\tan(x + \pi) = \frac{\tan(x) + \tan(\pi)}{ 1 - \tan(x) \tan(\pi)} = \frac{\tan(x) + 0}{ 1 - \tan(x) \cdot 0} = \tan(x).$

This shows the period of the tangent function is at most $\pi$. $_\square$

Since the cosine function satisfies $\cos(-\theta) = \cos(\theta)$, the graph of the function $\cos(x)$ is symmetric about the $y$-axis. What is the symmetry satisfied by the graph of $\sin(x)?$

Since the function $\sin(x)$ satisfies $\sin(-x) = \sin(x)$, the graph of $\sin(x)$ is symmetric about the origin. $_\square$

In general, for any even function $f(x)$, the the graph of $f(x)$ is symmetric about the $y$-axis; for any odd function $g(x)$, the graph of $g(x)$ is symmetric about the origin.

See Sine and Cosine graphs for more properties of the sine and cosine graphs.

**Cite as:**Symmetry in Trigonometric Graphs.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/symmetry-in-trigonometric-graphs/