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2.3.4: Coterminal Angles

  • Page ID
    14353
  • Set of angles with the same terminal or end side.

    While playing a game with friends, you use a spinner that looks like this:

    f-d_f7240c6d67dc4c11a91ef52a9bd37ee4fffeff6e8b3f1b75cad23ab6+IMAGE_TINY+IMAGE_TINY.jpg
    Figure \(\PageIndex{1}\)

    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?

    Coterminal Angles

     Consider the angle \(30^{\circ}\), in standard position.

    f-d_bf84fee1daaa9b351764625d762ccde992c539f0991bc0c27057a764+IMAGE_TINY+IMAGE_TINY.jpg
    Figure \(\PageIndex{2}\)

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

    f-d_1d389b6f5ea40c5442bde599b63f42ab1653590f787828e2395c5057+IMAGE_TINY+IMAGE_TINY.jpg
    Figure \(\PageIndex{3}\)

    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 \(30^{\circ}\).

    Identifying Co-Terminal Angles 

    For the following questions, determine if the angle is co-terminal with \(45^{\circ}\). 

    1. \(−45^{\circ}\)

    No, it is not co-terminal with \(45^{\circ}\)

    2. \(405^{\circ}\)

    Yes, \(405^{\circ}\) is co-terminal with \(45^{\circ}\).

    3. \(−315^{\circ}\)

    Yes, \(−315^{\circ}\) is co-terminal with \(45^{\circ}\).

    Example \(\PageIndex{1}\)

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

    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. 

    Example \(\PageIndex{2}\)

     Find a coterminal angle to \(23^{\circ}\)

    Solution

     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.

    Example \(\PageIndex{3}\)

    Find a coterminal angle to \(−90^{\circ}\)

    Solution

    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.

    Example \(\PageIndex{4}\)

    Find two coterminal angles to \(70^{\circ}\) by rotating in the positive direction around the circle.

    Solution

    Rotating once around the circle gives a coterminal angle of \(430^{\circ}\). Rotating again around the circle gives a coterminal angle of \(790^{\circ}\).

    Review

    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. f-d_73ad3626b6e056a4f29657681a70a6979c62626cff32ce11e312cf7c+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{4}\)
    2. f-d_d0ae6b7496b390c1bd62fd090a6dd3d680dd33dd1a948de9f9be4fab+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{5}\)

    3. f-d_d893036b46a5fee55f3dcec674b486cba8c551546730cf630213000a+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{6}\)
    4. f-d_a3ec27a8e07497ae5cd8d6128fb6ecd61f0e26d4197a4c7937153ba5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    5. f-d_17ad46e119a60425bf5acd3ffe5912f278a391c4347307b05167273e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    6. Name the angle of the 8 on a standard clock two different ways.
    7. Name the angle of the 11 on a standard clock two different ways.
    8. Name the angle of the 4 on a standard clock two different ways.
    9. Explain how to determine whether or not two angles are co-terminal.
    10. How many rotations is \(4680^{\circ}\)?

    Review (Answers)

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

    Vocabulary

    Term Definition
    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.

    Additional Resources

    Interactive Element

     

    Video: Example: Determine if Two Angles are Coterminal

    Practice: Coterminal Angles

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