What Are Coterminal Angles?

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.

\(\text{Coterminal Angles for $\theta$:} \ \ \ \theta \pm 360 \text{ degrees} \ \ \ \theta \pm 2\pi \text{ radians}\)

In trigonometry, you draw an angle in standard position by scribing a line from the origin of a coordinate plane to a termination point. The angle is measured between the positive x-axis and the line you scribed, known as the terminal side of the angle. The angle is positive if you measure the counterclockwise distance to the line and negative if you move clockwise from the initial side. In this scenario, the origin also denotes the vertex of the angle.

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. In this manner, we can solve for any number of positive coterminal angles or negative coterminal angles.

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.

Angles are incredibly useful across many fields of math, from high school algebra and precalculus to imaginary numbers and wave functions; angle measures are found everywhere.

When looking at the coterminal angles of a given angle, we can always say that the value of sine, cosine, tangent, and the other trigonometric functions are equivalent. This can be demonstrated geometrically if we look at the unit circle where the values of these trig functions are defined by their location around the complete circle. Since coterminal angles will always terminate at the same location, they will always share the same values.

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:

\(3 \times 2\pi – 15 = 18.84 – 15 = 3.84 \text{ radians}\)

Because 2π radians = 360 degrees, 1 radian = 57.32 degrees. Therefore, 3.84 radians is:

\(3.84 \text{ radians} \times 57.32 \frac{\text{degrees}}{\text{radian}} = 220.13 \text{ degrees}\)

There are other useful angle relationships that describe an angle ‌θ.‌ A reference angle describes an acute angle that finds the smallest angle compared to the x-axis. Complementary angles and supplementary angles are pairs of angles that add together to 90 degrees and 180 degrees respectively.