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# Coterminal Angles

## Set of angles with the same terminal or end side.

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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 $60^\circ$ . 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.

### Guidance

Consider the angle $30^\circ$ , in standard position.

Now consider the angle $390^\circ$ . We can think of this angle as a full rotation $(360^\circ)$ , plus an additional 30 degrees.

Notice that $390^\circ$ looks the same as $30^\circ$ . 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 $360^\circ$ , we get the angle $750^\circ$ . Or, if we create the angle in the negative direction (clockwise), we get the angle $-330^\circ$ . 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 $30^\circ$ .

#### Example A

Is the following angle co-terminal with $45^\circ$ ?

$-45^\circ$

Solution: No, it is not co-terminal with $45^\circ$

#### Example B

Is the following angle co-terminal with $45^\circ$ ?

Solution: $405^\circ$ Yes, $405^\circ$ is co-terminal with $45^\circ$ .

#### Example C

Is the following angle co-terminal with $45^\circ$ ?

$-315^\circ$

Solution: Yes, $-315^\circ$ is co-terminal with $45^\circ$ .

### 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 $23^\circ$

2. Find a coterminal angle to $-90^\circ$

3. Find two coterminal angles to $70^\circ$ 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 $23^\circ$ but has a different value. In this case, $-337^\circ$ is a coterminal angle.

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

3. Rotating once around the circle gives a coterminal angle of $430^\circ$ . Rotating again around the circle gives a coterminal angle of $790^\circ$ .

### Concept Problem Solution

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

### Practice

1. Is $315^\circ$ co-terminal with $-45^\circ$ ?
2. Is $90^\circ$ co-terminal with $-90^\circ$ ?
3. Is $350^\circ$ co-terminal with $-370^\circ$ ?
4. Is $15^\circ$ co-terminal with $1095^\circ$ ?
5. Is $85^\circ$ co-terminal with $1880^\circ$ ?

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

1. Name the angle of the 8 on a standard clock two different ways.
2. Name the angle of the 11 on a standard clock two different ways.
3. Name the angle of the 4 on a standard clock two different ways.
4. Explain how to determine whether or not two angles are co-terminal.
5. How many rotations is $4680^\circ$ ?

### Vocabulary Language: English

Coterminal Angles

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.