# 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:

\[\begin{align} \cos(-\theta) &= \cos(\theta) \\ \sin(-\theta) &= -\sin(\theta). \end{align}\]

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

\[\begin{align} \tan(-\theta) &= -\tan(\theta)\\ \cot(-\theta) &= -\cot(\theta)\\ \csc(-\theta) &= -\csc(\theta)\\ \sec(-\theta) &= \sec(\theta). \end{align}\]

We have

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

## 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\[\begin{align} \tan (\theta + \pi) & = \frac{\sin(\theta + \pi ) }{\cos(\theta + \pi) } \\ &= \frac{-\sin(\theta)}{-\cos(\theta)}\\ &= \frac{\sin(\theta)}{\cos(\theta)}\\ &= \tan (\theta ). \end{align}\]

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.

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