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*}

### Coterminal Angles

Consider the angle \begin{align*}30^\circ\end{align*}

Now consider the angle \begin{align*}390^\circ\end{align*}

Notice that \begin{align*}390^\circ\end{align*}**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*}

#### Identifying Co-Terminal Angles

For the following questions, determine if the angle is co-terminal with \begin{align*}45^\circ\end{align*}

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

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

2. \begin{align*}405^\circ\end{align*}

Yes, \begin{align*}405^\circ\end{align*}

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

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

### Examples

#### Example 1

Earlier, you were asked if it is possible to represent the angle any other way.

You can either think of \begin{align*}60^\circ\end{align*}

#### Example 2

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

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.

#### Example 3

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

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.

#### Example 4

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

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*}.

### Review

- 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*}?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.16.