Coterminal Angles: Definition, Formula, and Examples

What Are Coterminal Angles?

To define coterminal angles cleanly, picture an angle in standard position: its vertex sits at the origin of a coordinate plane, and its initial side lies along the positive $x$-axis. The angle opens to its terminal side — the ray you end on after rotating.

Coterminal angles are two or more angles in standard position that share the same terminal side. The initial side and the vertex are the same; only the amount of rotation differs, and it differs by a whole number of full turns. Because a full turn is $360°$ (or $2\pi$ radians), coterminal angles always differ by a multiple of $360°$ (or $2\pi$).

A direct consequence, and a reader question that comes up constantly — do coterminal angles have the same trig values? Yes. Because $30°$, $390°$, and $-330°$ all end on the same ray, their sine, cosine, and tangent are identical: $\sin 30° = \sin 390° = \sin(-330°) = 0.5$. The terminal side is what the trig functions "see," so coterminal angles are interchangeable inside $\sin$, $\cos$, and $\tan$.

The Coterminal Angles Formula

Finding a coterminal angle is one operation: add or remove full turns. The formula has two forms, one per unit.

In degrees, the coterminal angles of an angle $\theta$ are:

$$\theta + 360°n, \quad n = \pm 1, \pm 2, \pm 3, \dots$$

In radians, they are:

$$\theta + 2\pi n, \quad n = \pm 1, \pm 2, \pm 3, \dots$$

Here $n$ is any nonzero integer — it counts how many full turns you add ($n > 0$) or remove ($n < 0$). Each variable earns its place: $\theta$ is your starting angle, $360°$ (or $2\pi$) is one complete revolution, and $n$ is the number of revolutions. Since $n$ can be any integer, every angle has infinitely many coterminal angles — you can keep adding turns forever.

The two forms are the same idea in two languages, because $360° = 2\pi$ radians. If you need to move between them, that conversion is its own small skill — see the radians-to-degrees conversion for the bridge. Throughout this article we show both units side by side so neither form feels foreign.

Positive and Negative Coterminal Angles

The sign of $n$ chooses the direction of the extra turns.

• A positive coterminal angle comes from adding $360°$ (or $2\pi$) one or more times — rotating further counter-clockwise.

• A negative coterminal angle comes from subtracting $360°$ (or $2\pi$) one or more times — rotating clockwise.

For an angle of $45°$ ($\frac{\pi}{4}$ radians):

A useful habit: to find the smallest positive coterminal angle of any angle, keep adding or subtracting $360°$ (or $2\pi$) until the result lands between $0°$ and $360°$ (or $0$ and $2\pi$). That single representative is the one most problems want.

Coterminal Angles and Reference Angles

Coterminal angles are easy to confuse with reference angles, but they answer different questions. Coterminal angles ask which angles end on the same ray; the reference angle asks how far that ray sits from the nearest part of the $x$-axis, always as an acute angle between $0°$ and $90°$. You often use them together: reduce a large angle to its smallest positive coterminal partner first, then read off the reference angle. For the full treatment of that second step, see the reference angle article.

Examples of Coterminal Angles

With standard position, the formula, and the sign rule in place, here is the topic in action. The problems move from a single positive turn up to a mixed-unit verification.

Example 1

Find a positive coterminal angle of $50°$.

Add one full turn: $50° + 360° = 410°$. So $410°$ is coterminal with $50°$.

Example 2

Find a negative coterminal angle of $100°$ by adding $360°$.

Wrong attempt. A student reasons: "negative coterminal means I make the angle negative, so I just write $-100°$." That swaps the sign of the angle for the direction of rotation.

Why it breaks. The angle $-100°$ does not share a terminal side with $100°$. Rotating $100°$ counter-clockwise lands in the second quadrant; rotating $-100°$ (clockwise) lands in the third quadrant. Different terminal sides, so they are not coterminal at all.

Correct. A negative coterminal angle is reached by subtracting a full turn from the original: $100° - 360° = -260°$. Check: $-260°$ and $100°$ differ by exactly $360°$, so they share a terminal side. A coterminal angle keeps the terminal side fixed and changes only the number of full turns.

Example 3

Find the smallest positive coterminal angle of $-200°$.

It is negative, so add a full turn: $-200° + 360° = 160°$. Since $160°$ is already between $0°$ and $360°$, the smallest positive coterminal angle is $160°$.

Example 4

Find a positive and a negative coterminal angle of $\dfrac{\pi}{3}$ radians.

Add $2\pi$: $\dfrac{\pi}{3} + 2\pi = \dfrac{\pi}{3} + \dfrac{6\pi}{3} = \dfrac{7\pi}{3}$ (positive). Subtract $2\pi$: $\dfrac{\pi}{3} - 2\pi = \dfrac{\pi}{3} - \dfrac{6\pi}{3} = -\dfrac{5\pi}{3}$ (negative).

Example 5

Find the smallest positive coterminal angle of $480°$.

It is larger than $360°$, so subtract a full turn: $480° - 360° = 120°$. Since $0° < 120° < 360°$, the smallest positive coterminal angle is $120°$.

Example 6

Are $765°$ and $45°$ coterminal? Verify in degrees, then confirm with radians.

Subtract: $765° - 45° = 720° = 2 \times 360°$. The difference is a whole multiple of $360°$, so yes, they are coterminal. In radians, $765° = \dfrac{17\pi}{4}$ and $45° = \dfrac{\pi}{4}$; their difference is $\dfrac{16\pi}{4} = 4\pi = 2 \times 2\pi$ — again a whole number of full turns. Coterminal, confirmed in both units.

Why Coterminal Angles Matter Beyond the Classroom

Coterminal angles are how rotation gets bookkept — anywhere a thing turns past a full circle, the idea is doing quiet work.

• Circular motion and engineering. A spinning wheel, a turbine blade, or a motor shaft passes through $360°$ over and over; describing its position needs the coterminal idea, since $370°$ and $10°$ are physically the same orientation.

• Navigation and bearings. A heading of $400°$ is meaningless on a compass until you reduce it to its coterminal partner $40°$ — the actual direction.

• Trigonometry and waves. Because $\sin$ and $\cos$ repeat every $360°$ (or $2\pi$), coterminal angles explain why these functions are periodic — the same terminal side gives the same height on the unit circle, turn after turn.

• Animation and graphics. Rotating an object by $725°$ in a game engine is stored and rendered as its coterminal $5°$, because the screen only cares where it ends up.

For a Grade 10 or Grade 11 student, coterminal angles are the moment angle stops meaning "a corner" and starts meaning "an amount of turn" — the shift that makes trigonometry's repeating functions finally make sense.

Where Students Trip Up on Coterminal Angles

Mistake 1: Confusing a negative angle with a negative coterminal angle

Where it slips in: A problem asks for a negative coterminal angle, and the student simply negates the original angle.

Don't do this: Write $-60°$ as the negative coterminal angle of $60°$.

The correct way: A negative coterminal angle is found by subtracting a full turn: $60° - 360° = -300°$. The angle $-60°$ ends in a different quadrant entirely.Mistake 2: Mixing units — adding $360$ to an angle in radians

Where it slips in: The angle is given in radians, but the student adds $360$ instead of $2\pi$.

Don't do this: Compute $\dfrac{\pi}{6} + 360$, mixing a radian measure with a degree-sized number.

The correct way: Match the unit. In radians, one full turn is $2\pi$, so add $2\pi$: $\dfrac{\pi}{6} + 2\pi = \dfrac{13\pi}{6}$. The rusher who reaches for $360$ on autopilot gets a meaningless mixed-unit answer.

Mistake 3: Forgetting there are infinitely many

Where it slips in: A student finds one coterminal angle and assumes it is "the" answer, or thinks only one positive and one negative exist.

Don't do this: Treat $410°$ as the only coterminal angle of $50°$.

The correct way: Every angle has infinitely many coterminal partners — $50°, 410°, 770°, -310°, \dots$ — one for each integer $n$ in $\theta + 360°n$. When a problem wants a single answer, it usually means the smallest positive one.

Key Takeaways

• Coterminal angles are angles in standard position that share the same terminal side, differing by whole turns.

• Find them with $\theta + 360°n$ in degrees or $\theta + 2\pi n$ in radians, where $n$ is any nonzero integer.

• Adding turns gives positive coterminal angles; subtracting turns gives negative ones.

• Every angle has infinitely many coterminal partners; the smallest positive one (between $0°$ and $360°$) is usually what a problem wants.

• Coterminal angles share all trig values, which is why sine and cosine are periodic.

Practice These Problems to Solidify Your Understanding

1. Find the smallest positive coterminal angle of $-120°$.

2. Find a positive and a negative coterminal angle of $\dfrac{2\pi}{5}$ radians.

3. Are $-150°$ and $570°$ coterminal? Justify your answer.

Answer to Question 1: $240°$. Answer to Question 2: positive $\dfrac{12\pi}{5}$, negative $-\dfrac{8\pi}{5}$. Answer to Question 3: $570° - (-150°) = 720° = 2 \times 360°$, a whole number of turns, so yes — they are coterminal. If you answered $-150°$ for Question 1, revisit Mistake 1.

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