What Are Coterminal Angles?

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The word "coterminal" is slightly confusing, but all it's meant to denote is angles that terminate at the same point. If you're confused, you won't be when you realize that, to find an angle coterminal to a given angle which has its origin on the 0-point of an x-y axis, you simply add or subtract multiples of 360 degrees. If you're measuring angles in radians, you get coterminal angles by adding or subtracting multiples of 2π.

There Are an Infinite Number of Coterminal Angles

In trigonometry, you draw an angle in standard position by scribing a line from the origin of a set of coordinate axes to a termination point. The angle is measured between the x-axis and the line you scribed. The angle is positive if you measure the counterclockwise distance to the line and negative if you move clockwise.

A line parallel with the x-axis and extending in the positive direction has an angle of 0 degrees, but you can also denote that angle as 360 degrees. Consequently, 0 degrees and 360 degrees are coterminal angles. It's also possible to measure that same angle in the negative direction, which makes it -360 degrees. This is another angle coterminal with 0 degrees.

There is nothing to stop you from making two complete rotations in either the counterclockwise or clockwise direction to form angles of 720 and -720 degrees, which are also coterminal angles. In fact, you can make as many rotations as you want in either direction, which means that a 0-degree angle has an infinite number of coterminal angles. This is true for any angle.

Degrees or Radians

If you have a given angle, say 35 degrees, you can find the angles coterminal with it by adding or subtracting multiples of 360 degrees. This is because the degree is defined in such a way that a circle contains 360 of them.

A radian is defined as the angle formed by a line that scribes an arc length on the circumference of a circle equal to the radius of the circle. If the line scribes out the entire circumference of the circle, the angle it forms, in radians, is 2π. Consequently, if you measure an angle in radians, all you have to do to find angles coterminal to it is to add or subtract multiples of 2π.

Examples

1. Find two angles coterminal with 35 degrees.

Add 360 degrees to get ​395 degrees​ and subtract 360 degrees to get ​-325 degrees​. Equivalently, you could add 360 degrees to get 395 degrees and add 720 degrees to get ​755 degrees.​ You could also subtract 360 degrees to get -325 degrees and subtract 720 degrees to get ​-685 degrees​.

2. Find the smallest positive angle, in degrees, coterminal with -15 radians.

Add multiples of 2π until you get a positive angle. Since 2π = 6.28, we need to multiply by 3 to end up with a positive angle:

3(2\pi)+(-15)=18.84-15 = 3.84\text{ radians}

Because 2π radians = 360 degrees, 1 radian = 57.32 degrees.

Therefore, 3.84 radians is:

3.84\times 57.32 = 220.13\text{ degrees}

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