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Coterminal Angles

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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.

Watch This

James Sousa Example: Determine if Two Angles are Coterminal

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 ?

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