Coterminal Angles Calculator
Find every angle that shares the same terminal side as your angle — with the general formula, a full list of coterminal angles, a rotation diagram, and complete step-by-step reduction to the principal angle.
Background
Two angles are coterminal when they start on the same initial side and end on the same terminal side, differing only by a whole number of full rotations. Because one full turn is 360° (or 2π radians), every angle has infinitely many coterminal angles — found by repeatedly adding or subtracting 360°. Trig functions treat coterminal angles identically: sin(750°) = sin(30°), because 750° and 30° are coterminal.
How to use this calculator
- Enter your angle: type any value — positive, negative, or larger than 360° — in degrees or radians. In radian mode you can type a decimal (1.047) or a pi-expression (pi/6, -3pi/4, 13pi/3).
- Choose how many rotations to list: ±3 shows coterminal angles for n = −3 through +3.
- Calculate: get the general formula, a full table of coterminal angles, the principal positive and negative angles, a rotation diagram, and the exact steps used to reduce your angle.
How coterminal angles work
The core idea. Angles are measured by rotation from the positive x-axis. Once you've rotated 360° (one full turn), you're back where you started — so adding or subtracting any whole number of full turns lands on the exact same terminal ray.
The general formula. θ + 360°n (degrees) or θ + 2πn (radians), where n is any integer. n = 1 spins one extra turn forward; n = −1 unwinds one turn backward.
The principal angle. Among all the coterminal angles, exactly one falls in [0°, 360°) — this is usually the most useful form, since it's the one you'll find directly on the unit circle.
Finding it by hand. If your angle is 360° or more, keep subtracting 360° until it drops below 360°. If it's negative, keep adding 360° until it's non-negative. Whatever you land on is the principal angle.
Formulas Used
General coterminal angle (degrees): θ + 360°n, n ∈ ℤ
General coterminal angle (radians): θ + 2πn, n ∈ ℤ
Principal positive angle: θ mod 360° (or θ mod 2π), adjusted to stay non-negative
Principal negative angle: (θ mod 360°) − 360°
Fully Worked Example Problems
Example 1 — Angle greater than 360°
Find the principal coterminal angle for 750°.
Step 1: 750° is greater than 360°, so subtract 360°: 750° − 360° = 390°
Step 2: 390° is still greater than 360°, subtract again: 390° − 360° = 30°
Result: 30° is the principal coterminal angle. So 750° and 30° share the same terminal side, and sin(750°) = sin(30°) = 0.5.
Example 2 — Negative angle
Find the principal coterminal angle for −450°.
Step 1: −450° is negative, so add 360°: −450° + 360° = −90°
Step 2: −90° is still negative, add 360° again: −90° + 360° = 270°
Result: 270° is the principal coterminal angle, sitting on the negative y-axis (the border of Quadrants III and IV).
Example 3 — Working in radians
Find the principal coterminal angle for 13π/3.
Step 1: One full turn is 2π = 6π/3. Since 13π/3 > 6π/3, subtract: 13π/3 − 6π/3 = 7π/3
Step 2: 7π/3 is still greater than 2π (6π/3), subtract again: 7π/3 − 6π/3 = π/3
Result: π/3 is the principal coterminal angle. General form: 13π/3 = π/3 + 2π(2).
Example 4 — Listing multiple coterminal angles
List three positive and three negative coterminal angles for 50°.
Using θ + 360°n:
n = 1: 410° n = 2: 770° n = 3: 1130°
n = −1: −310° n = −2: −670° n = −3: −1030°
Key insight: every one of these six angles — and infinitely many more — has the exact same sine, cosine, and tangent as 50°.
Frequently Asked Questions
What's the difference between coterminal and equal angles?
Equal angles have the same measure. Coterminal angles can have completely different measures (like 30° and 750°) but still point in the same direction, because they differ by a whole number of full rotations.
Can a coterminal angle be negative?
Yes. Negative angles rotate clockwise instead of counterclockwise. Every positive angle has a negative coterminal angle, found with θ − 360° (or the principal angle minus 360°).
How many coterminal angles does an angle have?
Infinitely many. For any integer n, θ + 360°n is coterminal with θ — there's no limit to how many extra rotations you can add or subtract.
Why do coterminal angles matter for trig functions?
Sine, cosine, and tangent depend only on where the terminal side points — not on how many times you spun to get there. So coterminal angles always produce identical trig function values.
What if my angle is already between 0° and 360°?
Then it's already its own principal angle — no reduction needed. You can still find other coterminal angles by adding or subtracting 360° as many times as you like.
Does this work the same way in radians?
Yes — just use 2π instead of 360° as the size of one full rotation. Everything else about the concept and the reduction process is identical.
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