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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 \begin{align*}60^\circ\end{align*}60. Is it possible to represent the angle any other way?

Coterminal Angles

 Consider the angle \begin{align*}30^\circ\end{align*}30, in standard position.

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

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

 

Identifying Co-Terminal Angles 

For the following questions, determine if the angle is co-terminal with \begin{align*}45^\circ\end{align*}45

1. \begin{align*}-45^\circ\end{align*}45

No, it is not co-terminal with \begin{align*}45^\circ\end{align*}45

2. \begin{align*}405^\circ\end{align*}405

Yes, \begin{align*}405^\circ\end{align*}405 is co-terminal with \begin{align*}45^\circ\end{align*}45.

3. \begin{align*}-315^\circ\end{align*}315

Yes, \begin{align*}-315^\circ\end{align*}315 is co-terminal with \begin{align*}45^\circ\end{align*}45.

Examples

Example 1

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

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

Example 2

 Find a coterminal angle to \begin{align*}23^\circ\end{align*}23

 A coterminal angle would be an angle that is at the same terminal place as \begin{align*}23^\circ\end{align*} but has a different value. In this case, \begin{align*}-337^\circ\end{align*} is a coterminal angle.

Example 3

Find a coterminal angle to \begin{align*}-90^\circ\end{align*}

A coterminal angle would be an angle that is at the same terminal place as \begin{align*}-90^\circ\end{align*} but has a different value. In this case, \begin{align*}270^\circ\end{align*} is a coterminal angle.

Example 4

Find two coterminal angles to \begin{align*}70^\circ\end{align*} by rotating in the positive direction around the circle.

Rotating once around the circle gives a coterminal angle of \begin{align*}430^\circ\end{align*}. Rotating again around the circle gives a coterminal angle of \begin{align*}790^\circ\end{align*}.

Review

  1. Is \begin{align*}315^\circ\end{align*} co-terminal with \begin{align*}-45^\circ\end{align*}?
  2. Is \begin{align*}90^\circ\end{align*} co-terminal with \begin{align*}-90^\circ\end{align*}?
  3. Is \begin{align*}350^\circ\end{align*} co-terminal with \begin{align*}-370^\circ\end{align*}?
  4. Is \begin{align*}15^\circ\end{align*} co-terminal with \begin{align*}1095^\circ\end{align*}?
  5. Is \begin{align*}85^\circ\end{align*} co-terminal with \begin{align*}1880^\circ\end{align*}?

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 \begin{align*}4680^\circ\end{align*}?

Review (Answers)

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

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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.

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