What Are Coterminal Angles?

# What Are Coterminal Angles?

Coterminal angles are angles in standard position that share the same terminal side. To find a coterminal angle, simply add or subtract 360 degrees from any angle; the result is a new angle that is coterminal to the first angle. The same can be done with radians instead of degrees; instead of adding or subtracting 360 degrees, add or subtract 2π.

Any multiple of 360 degrees can be added or subtracted from an angle, and the answer is always a coterminal angle. A few examples on how to find coterminal angles are 45 + 360 = 405, 120 - 360 = -240, -77 + (2)360 = 643 and 2π/3 + 2π = 8π/3. Forty-five degrees and 405 degrees are coterminal to each other; however, they are not coterminal to any of the other examples listed.

The properties of coterminal angles can be used to find the reference angle of an angle that is larger than 360 degrees. For example, if the reference angle of 765 degrees must be solved for, it is known that 765/360 = 2.125. Since 360 only goes into 765 two times, that means in order to find the coterminal angle, the equation is 765 - (2)360 = 45. This means that 45 degrees is the coterminal angle of 765 degrees, and it is also in the first quadrant and is therefore also the reference angle.

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