# 1.16: Coterminal Angles

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

**Practice**Coterminal Angles

While playing a game with friends, you use a spinner that looks like this:

As you can see, the angle that the spinner makes with the horizontal is \begin{align*}60^\circ\end{align*}. Is it possible to represent the angle any other way?

At the completion of this Concept, you'll know more than one way to represent this angle.

### Watch This

James Sousa Example: Determine if Two Angles are Coterminal

### Guidance

Consider the angle \begin{align*}30^\circ\end{align*}, in standard position.

Now consider the angle \begin{align*}390^\circ\end{align*}. We can think of this angle as a full rotation \begin{align*}(360^\circ)\end{align*}, plus an additional 30 degrees.

Notice that \begin{align*}390^\circ\end{align*} looks the same as \begin{align*}30^\circ\end{align*}. Formally, we say that the angles share the same terminal side. Therefore we call the angles **co-terminal**. Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another \begin{align*}360^\circ\end{align*}, we get the angle \begin{align*}750^\circ\end{align*}. Or, if we create the angle in the negative direction (clockwise), we get the angle \begin{align*}-330^\circ\end{align*}. Because we can rotate in either direction, and we can rotate as many times as we want, we can continuously generate angles that are co-terminal with \begin{align*}30^\circ\end{align*}.

#### Example A

Is the following angle co-terminal with \begin{align*}45^\circ\end{align*}?

\begin{align*}-45^\circ\end{align*}

**Solution:** No, it is not co-terminal with \begin{align*}45^\circ\end{align*}

#### Example B

Is the following angle co-terminal with \begin{align*}45^\circ\end{align*}?

**Solution:** \begin{align*}405^\circ\end{align*} Yes, \begin{align*}405^\circ\end{align*} is co-terminal with \begin{align*}45^\circ\end{align*}.

#### Example C

Is the following angle co-terminal with \begin{align*}45^\circ\end{align*}?

\begin{align*}-315^\circ\end{align*}

**Solution:** Yes, \begin{align*}-315^\circ\end{align*} is co-terminal with \begin{align*}45^\circ\end{align*}.

### Vocabulary

**Coterminal Angles:** A set of ** coterminal angles** are angles with the same terminal side but expressed differently, such as a different number of complete rotations around the unit circle or angles being expressed as positive versus negative angle measurements.

### Guided Practice

1. Find a coterminal angle to \begin{align*}23^\circ\end{align*}

2. Find a coterminal angle to \begin{align*}-90^\circ\end{align*}

3. Find two coterminal angles to \begin{align*}70^\circ\end{align*} by rotating in the positive direction around the circle.

**Solutions:**

1. A coterminal angle would be an angle that is at the same terminal place as \begin{align*}23^\circ\end{align*} but has a different value. In this case, \begin{align*}-337^\circ\end{align*} is a coterminal angle.

2. A coterminal angle would be an angle that is at the same terminal place as \begin{align*}-90^\circ\end{align*} but has a different value. In this case, \begin{align*}270^\circ\end{align*} is a coterminal angle.

3. Rotating once around the circle gives a coterminal angle of \begin{align*}430^\circ\end{align*}. Rotating again around the circle gives a coterminal angle of \begin{align*}790^\circ\end{align*}.

### Concept Problem Solution

You can either think of \begin{align*}60^\circ\end{align*} as \begin{align*}420^\circ\end{align*} if you rotate all the way around the circle once and continue the rotation to where the spinner has stopped, or as \begin{align*}-300^\circ\end{align*} if you rotate clockwise around the circle instead of counterclockwise to where the spinner has stopped.

### Practice

- Is \begin{align*}315^\circ\end{align*} co-terminal with \begin{align*}-45^\circ\end{align*}?
- Is \begin{align*}90^\circ\end{align*} co-terminal with \begin{align*}-90^\circ\end{align*}?
- Is \begin{align*}350^\circ\end{align*} co-terminal with \begin{align*}-370^\circ\end{align*}?
- Is \begin{align*}15^\circ\end{align*} co-terminal with \begin{align*}1095^\circ\end{align*}?
- Is \begin{align*}85^\circ\end{align*} co-terminal with \begin{align*}1880^\circ\end{align*}?

For each diagram, name the angle in 3 ways. At least one way should use negative degrees.

- Name the angle of the 8 on a standard clock two different ways.
- Name the angle of the 11 on a standard clock two different ways.
- Name the angle of the 4 on a standard clock two different ways.
- Explain how to determine whether or not two angles are co-terminal.
- How many rotations is \begin{align*}4680^\circ\end{align*}?

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### Image Attributions

Here you'll learn how to identify coterminal angles.