Coterminal Angles: Definition, Formula, and Examples
Coterminal Angles: Definition, Formula, and Examples
Coterminal angles are angles in standard position that share the same terminal side — they land in the same place after differing by a whole number of full turns. You find them by adding or subtracting 360° (or 2π radians). This article covers the definition, the formula in degrees and radians, positive and negative coterminal angles, the unit-circle picture, and worked 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 2π radians), coterminal angles always differ by a multiple of 360° (or 2π).
A direct consequence — 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: sin30°=sin390°=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 θ are:
θ + 360°n, n = ±1, ±2, ±3,…
In radians, they are:
θ + 2πn, n = ±1, ±2, ±3,…
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: θ is your starting angle, 360° (or 2π) 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.
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 2π) one or more times — rotating further counter-clockwise.
- A negative coterminal angle comes from subtracting 360° (or 2π) one or more times — rotating clockwise.
For an angle of 45° (π/4 radians):
| Operation | Degrees | Radians |
|---|---|---|
| Given angle | 45° | π/4 |
| + one turn | 45°+360°=405° | π/4+2π=9π/4 |
| − one turn | 45°−360°=−315° | π/4−2π=−7π/4 |
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.
Examples of Coterminal Angles
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°.
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.
Example 3
Find the smallest positive coterminal angle of −200°.
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 π/3 radians.
Add 2π: π/3+2π=7π/3 (positive). Subtract 2π: π/3−2π=−5π/3 (negative).
Example 5
Find the smallest positive coterminal angle of 480°.
Subtract one full turn: 480°−360°=120°. The smallest positive coterminal angle is 120°.
Example 6
Are 765° and 45° coterminal?
Subtract: 765°−45°=720°=2×360°. The difference is a whole multiple of 360°, so yes, they are coterminal.
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 or turbine blade passes through 360° over and over; describing its position needs the coterminal idea.
- 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 2π), coterminal angles explain why these functions are periodic.
- Animation and graphics. Rotating an object by 725° in a game engine is stored and rendered as its coterminal 5°.
Where Students Trip Up on Coterminal Angles
Mistake 1
Confusing a negative angle with a negative coterminal angle.
The correct way: A negative coterminal angle is found by subtracting a full turn.
Mistake 2
Mixing units — adding 360 to an angle in radians.
The correct way: In radians, one full turn is 2π, so add 2π.
Mistake 3
Forgetting there are infinitely many.
Key Takeaways
- Coterminal angles are angles in standard position that share the same terminal side, differing by whole turns.
- Find them with θ + 360°n in degrees or θ + 2πn 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 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
- Find the smallest positive coterminal angle of −120°.
- Find a positive and a negative coterminal angle of 2π/5 radians.
- Are −150° and 570° coterminal? Justify your answer.