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 . 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 , in standard position.

Now consider the angle . We can think of this angle as a full rotation , plus an additional 30 degrees.

Notice that
looks the same as
. 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
, we get the angle
. Or, if we create the angle in the negative direction (clockwise), we get the angle
. 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
.

#### Example A

Is the following angle co-terminal with ?

**
Solution:
**
No, it is not co-terminal with

#### Example B

Is the following angle co-terminal with ?

**
Solution:
**
Yes,
is co-terminal with
.

#### Example C

Is the following angle co-terminal with ?

**
Solution:
**
Yes,
is co-terminal with
.

### 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

2. Find a coterminal angle to

3. Find two coterminal angles to 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 but has a different value. In this case, is a coterminal angle.

2. A coterminal angle would be an angle that is at the same terminal place as but has a different value. In this case, is a coterminal angle.

3. Rotating once around the circle gives a coterminal angle of . Rotating again around the circle gives a coterminal angle of .

### Concept Problem Solution

You can either think of as if you rotate all the way around the circle once and continue the rotation to where the spinner has stopped, or as if you rotate clockwise around the circle instead of counterclockwise to where the spinner has stopped.

### Practice

- Is co-terminal with ?
- Is co-terminal with ?
- Is co-terminal with ?
- Is co-terminal with ?
- Is co-terminal with ?

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 ?