Coterminal Angles

When an angle is drawn in standard position, its vertex sits at the origin and its initial side lies along the positive x-axis. The terminal side is the ray that sweeps around to show the angle's measure. Two angles are called coterminal angles when they are both drawn in standard position and their terminal sides land on exactly the same ray, even though the two angle measures are different.

For example, \(40^\circ\) and \(400^\circ\) are coterminal, because rotating one extra full turn of \(360^\circ\) brings the terminal side right back to the same position. The same idea works with negative rotation: \(40^\circ - 360^\circ = -320^\circ\) is also coterminal with \(40^\circ\).

Since one full trip around the circle is \(360^\circ\) (or \(2\pi\) radians), every angle coterminal with a given angle \(\theta\) can be written with the coterminal angle formula:

\(\theta + 360^\circ \cdot k\) in degrees, or \(\theta + 2\pi k\) in radians, where \(k\) is any integer, positive, negative, or zero.

Choosing \(k = 1\) adds one full turn, \(k = -1\) subtracts one full turn, \(k = 2\) adds two full turns, and so on. Because there are infinitely many integers, every angle has infinitely many coterminal angles.

A very common task is to find an angle between \(0^\circ\) and \(360^\circ\) that is coterminal with a given angle. Add or subtract multiples of \(360^\circ\) until the result falls in that range.

Example: Find a coterminal angle for \(750^\circ\) between \(0^\circ\) and \(360^\circ\).

\(750^\circ - 360^\circ = 390^\circ\), which is still too big, so subtract again: \(390^\circ - 360^\circ = 30^\circ\). The angle \(30^\circ\) is coterminal with \(750^\circ\) and lies between \(0^\circ\) and \(360^\circ\).

Example: Find a coterminal angle for \(-100^\circ\) between \(0^\circ\) and \(360^\circ\).

Since \(-100^\circ\) is negative, add \(360^\circ\): \(-100^\circ + 360^\circ = 260^\circ\). This lands in range on the first try, so \(260^\circ\) is the answer.

Sometimes you need both a positive and a negative coterminal angle for the same starting angle. Just add \(360^\circ\) for the positive one and subtract \(360^\circ\) for the negative one.

Example: Determine two coterminal angles for \(75^\circ\), one positive and one negative (other than \(75^\circ\) itself).

Positive: \(75^\circ + 360^\circ = 435^\circ\). Negative: \(75^\circ - 360^\circ = -285^\circ\). Both \(435^\circ\) and \(-285^\circ\) share the same terminal side as \(75^\circ\).

The same idea applies to radian measure, using \(2\pi\) as a full rotation instead of \(360^\circ\). For instance, \(\frac{\pi}{6}\) and \(\frac{\pi}{6} + 2\pi = \frac{13\pi}{6}\) are coterminal. If a problem mixes degrees and radians, it helps to convert between degrees and radians first so every angle is measured the same way before you add or subtract full rotations.

Because coterminal angles always end on the same ray, they always fall in the same quadrant and always have the same reference angle. That also means their sine, cosine, and tangent values are identical. Once you know which quadrant an angle lands in, the ASTC rule tells you the sign of each trig ratio, and that sign pattern stays the same for every angle coterminal with it.

To test whether two angles are coterminal, subtract one from the other. If the difference is a multiple of \(360^\circ\) (or \(2\pi\) in radians), the angles are coterminal. For example, \(-45^\circ\) and \(675^\circ\) differ by \(720^\circ = 2 \times 360^\circ\), so they are coterminal, while \(-45^\circ\) and \(300^\circ\) differ by \(345^\circ\), which is not a multiple of \(360^\circ\), so they are not.

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