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Identify Coterminal Angles

An angle in standard position on the coordinate plane has its vertex at the origin and its initial (beginning) side along the positive x-axis. An angle’s terminal (ending) side rotates counter-clockwise around the origin. Two angles are coterminal if they have the same terminal side.

Coterminal angles can also involve multiple revolutions (round and round the origin) or have negative values (rotating clockwise). There are an infinite number of ways to give an angle measure for a particular terminal ray.

Sometimes, using a negative angle rather than a positive angle is more convenient, or the answer to an application may involve more than one revolution (spinning around and around). Angles in these applications can have terminal sides that involve one or more full revolutions around the origin or terminal sides that go clockwise instead of counterclockwise — or both of these situations can happen.

Coterminal angles with more than one revolution

An angle measuring 70 degrees is coterminal with an angle measuring 430 degrees (see the following figure). The angle measuring 430 degrees is actually 360 + 70 (one full revolution plus the original 70). These two angles are also coterminal with an angle of 790 degrees (360 + 360 + 70 = 790). This pattern could go on and on, with the addition of another 360 degrees each time.

Three coterminal angles.
Three coterminal angles.

Negative coterminal angles

An angle of 70 degrees is coterminal with an angle of –290 degrees. Two rotations in the negative (clockwise) direction give you an angle of –650 degrees by subtracting another 360 degrees: –290 – 360 = –650.

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