Coterminal Angles: Definition & Examples

Coterminal Angles: Definition & Examples
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  • 0:01 What Are Coterminal Angles?
  • 0:23 Angle Terminology Review
  • 1:32 Measuring Coterminal Angles
  • 3:18 Examples
  • 6:00 Lesson Summary
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Lesson Transcript
Instructor: Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

This lesson defines coterminal angles measured in degrees. You will learn how to calculate positive and negative coterminal angles with a few examples.

What Are Coterminal Angles?

Coterminal angles are two (or more) angles that have their initial and terminal sides in the same positions. However, the angle measures differ either because:

  • One angle is measured clockwise and the other is measured counterclockwise
  • The angles' terminal sides completed different complete rotations

Angle Terminology Review

Let's review the standard set-up for angles in trigonometry. We will begin with the angle created by the hands of a clock at 3:15, when both the hour hand and the minute hand are pointing to the 3. This forms an angle that measures 0 degrees. Often you see angles in trigonometry on a coordinate plane. That's why I have also drawn x- and y-axes on the clock. So, the sides of our angle are aligned with the positive x-axis.

Clock showing 3:15

For all angles in this lesson, we keep one side of the angle on the positive x-axis and the vertex at the origin. This is considered the standard position for an angle. The side along the positive x-axis is called the initial side. In every diagram in this lesson, the red side is the initial side. We rotate the other side of the angle, called the terminal side, to sweep out the angle measure. Throughout the lesson, the blue side will be the terminal side.

Also, remember the direction in which angles are measured. An angle with a positive measure has a terminal side that rotated counterclockwise. An angle with a negative measure has a terminal side that rotated clockwise.

Measuring Coterminal Angles

What angle measurements would result in the same initial and terminal side position as a 0 degree angle? Imagine if you spin the terminal side counterclockwise once all the way around the circle until both sides of the angle point toward the 3 again. The final result looks the same as the 0 degree angle. However, the blue side was rotated through 360 degrees. Therefore, 360 degrees is a coterminal angle to 0 degrees. The only difference in the measures is indicated by the rotation arrow.

Clock faces showing that a 0 degree angle and 360 degree angle appear the same

If the terminal side completes a full rotation clockwise until it points toward the 3 again, you have rotated through a -360 degree angle. Therefore, -360 degrees is also a coterminal angle to 0 degrees. Notice below that the final position of the sides of the angle looks the same for 0 degrees and -360 degrees (as well as the 360 degrees from earlier).

Clock faces showing 0 degrees and -360 degrees

The terminal side could rotate several full rotations and as long as the final position of the sides of the angle has them both pointing at the 3, you would have a coterminal angle to 0 degrees. For example, if you did 5 counterclockwise rotations, then you would have an angle that measures 1800 degrees because 5 x 360 = 1800.

It might be useful to know how many degrees are in multiples of full rotations:

  • 1 rotation = 360 degrees
  • 2 rotations = 720 degrees
  • 3 rotations = 1080 degrees

Examples

Now, let's go through a couple of common example problems related to coterminal angles.

For the first example, you have to find a positive and a negative angle that is coterminal to an angle measuring 60 degrees.

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