Coterminal Angle Calculator

An angle in standard position starts on the positive x-axis and turns counterclockwise for positive values and clockwise for negative ones. Where it stops is its terminal side. Every extra full turn brings it back to the same terminal side, which is why 30°, 390° and −330° look identical on the unit circle and share every trigonometric value.

The coterminal angle formula

For an angle θ and any whole number n:

The floor brackets ⌊ ⌋ round down to the nearest whole number, so ⌊1,230 ÷ 360⌋ = ⌊3.42⌋ = 3 and ⌊−1,000 ÷ 360⌋ = ⌊−2.78⌋ = −3. Rounding down rather than toward zero keeps negative angles right: −1,000° − 360° × (−3) = 80°.

Homework often asks for the coterminal angle in some other window, such as between 360° and 720° or between −360° and 0°. Reduce to the 0° to 360° value first, then add or subtract the turns that land you in the window: 80° + 360° = 440° for the first window and 80° − 360° = −280° for the second.

Worked example 1: reducing 1,230°

Dividing by 360 gives 3.42, so the angle makes 3 full turns plus a remainder:

150° lies between 90° and 180°, in Quadrant II, so the reference angle is 180° − 150° = 30°. Sine is positive there while cosine and tangent are negative, so sin 1,230° = sin 30° = 0.5 and cos 1,230° = −cos 30°. In radians the angle is 41π/6, which reduces to 5π/6. The coterminal angles for n = −3 to 3 are −930°, −570°, −210°, 150°, 510°, 870° and 1,230°, so the smallest positive coterminal angle is 150° and the largest negative one is −210°.

Worked example 2: a negative angle in radians, −17π/4

Negative angles turn clockwise. To bring −17π/4 into the interval from 0 to 2π, add whole turns written in quarters. One turn is 2π = 8π/4, so:

7π/4 is 315°, in Quadrant IV, so the reference angle is 2π − 7π/4 = π/4, or 45°. Between −π and π the same angle is −π/4 (−45°). Converting the original value shows how far it turned: −17π/4 × 180/π = −765°, two full clockwise turns plus another 45°. Cosine is the only positive function in Quadrant IV, so cos(−17π/4) = cos(π/4) ≈ 0.7071.

Worked example 3: decimal radians, 2.5 rad

A plain decimal such as 2.5 has no exact π fraction, so the calculator converts it with the full value of π: 2.5 × 180/π = 143.239449°, or 143° 14′ 22.02″. It is already between 0 and 2π (about 6.283185), so no turns are removed. It lies in Quadrant II, and the reference angle is π − 2.5 = 0.641593 rad, or 36.760551°. Its largest negative coterminal angle is 2.5 − 2π = −3.783185 rad. As a check, sin 2.5 and sin 0.641593 both equal 0.598472.

Reference angles quadrant by quadrant

The reference angle is the acute angle between the terminal side and the x-axis, never the y-axis. Each quadrant has its own rule, and each has its own pattern of signs:

The sign column matches Table 4.16.1 of the NIST Digital Library of Mathematical Functions. Together, the reference angle and the sign turn any trig value into a first-quadrant lookup: 2,026° reduces to 226° in Quadrant III, where tangent is positive, so tan 2,026° = tan 46° ≈ 1.0355.

Computed table: the unit-circle angles

For the fifteen angles that appear on every unit circle, this table lists the exact radian measure, one positive and one negative coterminal angle, the location and the reference angle. All values come from the calculator's exact arithmetic.

Computed table: large and negative angles reduced

Homework and code tend to produce angles far outside one turn. These ten examples show how many turns the calculator removes and where each angle ends up.

A negative number of turns means turns were added, not removed. 405° and 1,845° are coterminal, since both reduce to 45°, while −765° reduces to 315°: it shares their reference angle of 45° without being coterminal with them. And −450° ends on the negative y-axis, a quadrantal angle with no acute reference angle.

Degree calculator: degrees, radians and DMS

One degree is π/180 radian, one minute of arc is 1/60 of a degree and one second is 1/60 of a minute; NIST lists exactly these values for the angle units accepted for use with the SI. The radian itself is a ratio of two lengths, arc length over radius, which is why a full circle with circumference 2πr measures 2π radians. The calculator reads degree input with minutes and seconds, such as 45°30′15″ (45.504167°), and shows every result in decimal degrees, degrees-minutes-seconds, exact π fractions and decimal radians. Familiar pairs: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, and 1 rad ≈ 57.29578° = 57° 17′ 44.81″.

How to use the coterminal angle calculator

Type the angle and choose Degrees or Radians. Plain numbers follow that switch, anything containing π or pi is read as radians, and a degree sign forces degrees, so 7pi/4, −11π/6, 3pi, 2.5 and 45°30′15″ all work; the π button inserts the symbol on a phone. The blue card shows the coterminal angle between 0° and 360° (or 0 and 2π), the reference angle, the quadrant, the angle between −180° and 180°, and the smallest positive and largest negative coterminal angles. Below it, the unit circle shades the reference angle, a table lists n = −3 to 3, and the step-by-step card shows the arithmetic.

Who uses coterminal and reference angles

Trigonometry and precalculus students use them to evaluate functions of large or negative angles and to sketch angles in standard position. Calculus students rely on the periodicity behind them: sin(θ + 2πn) = sin θ for every whole number n. Electrical engineers wrap phase angles the same way, so a phase of 390° is the same as 30°. Robotics, navigation and game code keep headings in one range so that a turn from 350° to 10° reads as 20° rather than 340°, and surveyors switch between decimal degrees and degrees, minutes and seconds.

Key terms

• Standard position: the vertex at the origin and the initial side on the positive x-axis.
• Terminal side: the ray where the angle stops turning.
• Coterminal angles: angles with the same terminal side; they differ by multiples of 360° or 2π.
• Reference angle: the acute angle between the terminal side and the x-axis.
• Quadrantal angle: an angle whose terminal side lies on an axis, a multiple of 90°.
• Radian: the angle that cuts off an arc as long as the radius; 2π radians make one full turn.

What changes the result, and limitations

• The unit. 2.5 read as degrees is 2.5°; read as radians it is 143.24°. Check the switch before trusting a plain number.
• The range convention. The interval (−180°, 180°] includes 180° but not −180°, so both 180° and −180° are reported as 180°. Some books and software use [−180°, 180°) instead.
• Quadrantal angles. Conventions differ on whether they have a reference angle, as explained in the FAQ.
• Limits. The calculator does not evaluate sine, cosine or tangent, accepts angles up to ±1,000,000,000°, and reads π expressions written as simple fractions such as 7pi/4 or (7/4)π, not sums such as π + 1.

Related calculators

• For the sine, cosine and tangent of the reduced angle, use the Trigonometry Calculator.
• For the arc an angle cuts from a circle, which needs the angle in radians, use the Arc Length Calculator.
• To convert the resulting arc or chord lengths between inches, feet, centimeters and meters, use the Unit Converter.
• To solve a triangle from its angles and sides, use the Triangle Calculator.
• For triangles given two angles and a side, the Law of Sines Calculator goes step by step.

Sources

Coterminal and reference angles are exact definitions applied to your input, and the unit relationships come from these primary references:

• NIST Special Publication 811, Chapter 5, Table 6: 1° = (π/180) rad, 1′ = (1/60)° = (π/10 800) rad and 1″ = (1/60)′ = (π/648 000) rad.
• NIST Special Publication 811, Chapter 4, Table 3: the radian as a derived unit equal to m/m = 1.
• NIST Digital Library of Mathematical Functions, §4.14: periodicity, sin(z + 2kπ) = sin z, cos(z + 2kπ) = cos z and tan(z + kπ) = tan z (equations 4.14.8 to 4.14.10).
• NIST Digital Library of Mathematical Functions, §4.16: Table 4.16.1, the signs of the trigonometric functions in the four quadrants.
• Common Core State Standards for Mathematics, HSF-TF.A.1 and HSF-TF.A.2: radian measure as arc length on the unit circle and the extension of trigonometric functions to all real numbers.