§1What are coterminal angles?
Two angles in standard position are coterminal when they share the same terminal side. Starting from the positive x-axis, you can reach the same side by turning less, by turning one or more extra full turns, or by turning the other way. All of those angles are coterminal.
A full turn is 360° (or 2π radians), so the coterminal angles of θ are θ + 360°·k, where k is any integer: positive k adds turns, negative k removes them. Every angle has infinitely many coterminal angles.
θ = 30°. 30° + 360° = 390° · 30° + 720° = 750° · 30° − 360° = −330°
§2The coterminal angle between 0° and 360°
Most problems want the single coterminal angle in [0°, 360°). Divide by 360° to see how many full turns are inside the angle, then remove them.
1000°. 1000 / 360 ≈ 2.78 → remove 2 turns → 1000° − 720° = 280°
−765°. −765 / 360 ≈ −2.13 → add 3 turns → −765° + 1080° = 315°
For a negative angle you always add one more turn than the whole part of the division, because you need to cross back over 0°. The calculator shows how many turns it added or removed.
§3Coterminal angles in radians
In radians a full turn is 2π, so coterminal angles are θ + 2π·k. Write 2π with the same denominator as the angle before adding or subtracting.
17π/6. 2π = 12π/6 → 17π/6 − 12π/6 = 5π/6
−π/3. 2π = 6π/3 → −π/3 + 6π/3 = 5π/3
If you enter a multiple of π, the calculator keeps the whole answer in exact radians. With a decimal like 1.5 rad it converts to degrees first.
§4Are two angles coterminal?
Subtract one angle from the other. If the difference is a whole multiple of 360° (or of 2π), they are coterminal; otherwise they are not. Type a second angle in the “compare with” box to check.
100° and −260°. 100° − (−260°) = 360° = 1 turn → coterminal
30° and 300°. 300° − 30° = 270°, not a whole turn → not coterminal
§5Why coterminal angles matter
Coterminal angles end at the same point on the unit circle, so all six trig functions give the same values for them. That is why sin 390° = sin 30° = 1/2 and cos(−300°) = cos 60° = 1/2. Once the angle is in one turn, the reference angle calculator takes you the rest of the way to an exact value, and the interactive unit circle shows where the angle ends.
The same idea explains why sine and cosine repeat every 360°: their period is 360°, or 2π.
§6Common mistakes
- Adding 180° instead of 360°. Half a turn ends on the opposite side, not the same one.
- Stopping one turn short with negative angles: −765° + 720° = −45°, which is coterminal but still negative.
- Adding 2π to an angle given in degrees, or 360 to one given in radians.
- Thinking an angle and its negative are coterminal. 60° and −60° end on different sides; they are reflections across the x-axis.
§7FAQ
How many coterminal angles does an angle have?
Infinitely many: one for every integer k in θ + 360°·k. The calculator lists the three closest positive and negative ones.
Are 0° and 360° coterminal?
Yes. They differ by exactly one full turn and both end on the positive x-axis.
What is the smallest positive coterminal angle?
It is the coterminal angle in (0°, 360°]. For most angles it is the same as the one in [0°, 360°); for multiples of 360° it is 360° itself.
Can a coterminal angle be negative?
Yes. Negative angles turn clockwise. −330° is coterminal with 30° because −330° + 360° = 30°.