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How to Rename Coterminal Angles

Any angle can have many, many descriptions in terms of angle measures, because an angle measure is equivalent to the measures of its coterminal angles. The most frequently used positive angle measures are those that measure between 0 and 360 degrees. Rules for coterminal angles involve adding or subtracting rotations (or multiples of 360 degrees).

The first equation that follows shows what happens when you add a full rotation over and over. The second shows what happens when you subtract a full rotation many times. The results are all coterminal angles.

image0.png

So an angle measuring 100 degrees is coterminal with the following:

image1.png

Here’s an example: Suppose you want to find related measures, between 0 and 360 degrees, for angles of 800 degrees and –1040 degrees by using coterminal angles.

  1. Subtract 360 degrees from 800 until the result is less than 360.

    image2.png

    An angle measuring 800 degrees is coterminal with an angle of 80 degrees.

  2. Add 360 degrees to –1040 until the result is positive.

    image3.png

    An angle measuring –1040 degrees is coterminal with an angle of 40 degrees.

How about another example? Give new measures for angles of 280 degrees and –300 degrees by using coterminal angle measures that lie between –180 degrees and 180 degrees.

  1. Subtract 360 from 280.

    image4.png

    An angle measuring 280 degrees is coterminal with an angle of –80 degrees.

  2. Add 360 to –300.

    image5.png

    An angle measuring –300 degrees is coterminal with an angle of 60 degrees.

The angle measure that you want depends on what form you want the answer to be in. Sometimes, you only want a positive angle measure. And if you’re a pilot, you certainly don’t want to be given an angle or heading of 2000 degrees. You’d be spinning around until you were dizzy!

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