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1.16: Coterminal Angles

Difficulty Level: At Grade Created by: CK-12
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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. Is it possible to represent the angle any other way?

Coterminal Angles

 Consider the angle 30, in standard position.

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

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

Identifying Co-Terminal Angles 

For the following questions, determine if the angle is co-terminal with 45

1. 45

No, it is not co-terminal with 45

2. 405

Yes, 405 is co-terminal with 45.

3. 315

Yes, 315 is co-terminal with 45.

Examples

Example 1

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

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

Example 2

 Find a coterminal angle to 23

 A coterminal angle would be an angle that is at the same terminal place as 23 but has a different value. In this case, 337 is a coterminal angle.

Example 3

Find a coterminal angle to 90

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

Example 4

Find two coterminal angles to 70 by rotating in the positive direction around the circle.

Rotating once around the circle gives a coterminal angle of 430. Rotating again around the circle gives a coterminal angle of 790.

Review

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

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?

Review (Answers)

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

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Vocabulary

TermDefinition
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.

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Difficulty Level:
At Grade
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Date Created:
Sep 26, 2012
Last Modified:
Aug 11, 2016
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