About

Two angles are coterminal when they share the same terminal side on the unit circle. The general formula is θ_coterminal = θ + 360° ⋅ n, where n ∈ Z. A common error in trigonometry problems is confusing coterminal angles with supplementary or complementary angles. This tool computes all coterminal angles for a given input, determines the quadrant of the terminal side, and extracts the reference angle α. It handles arbitrarily large positive or negative inputs and works in both degrees and radians.

The reference angle is always the acute angle between the terminal side and the nearest portion of the x-axis. This matters because trigonometric function values repeat across coterminal angles. An incorrect reference angle propagates errors through every subsequent calculation. Note: this tool normalizes angles assuming standard position (vertex at origin, initial side on positive x-axis). Non-standard coordinate systems require manual adjustment.

Formulas

The general coterminal angle formula expresses all angles sharing the same terminal side:

θ_coterminal = θ + 360° ⋅ n, n ∈ Z

In radians the equivalent expression is:

θ_coterminal = θ + 2π ⋅ n

To find the normalized (standard) angle θ_std within [0°, 360°):

θ_std = θ − 360° ⋅ floor(θ360°)

The reference angle α depends on the quadrant of θ_std:

{

α = θ_std if Q Iα = 180° − θ_std if Q IIα = θ_std − 180° if Q IIIα = 360° − θ_std if Q IV

Degree-radian conversion uses the identity:

θ_rad = θ_deg ⋅ π180

Where: θ = input angle, n = any integer (positive generates larger coterminals, negative generates smaller), α = reference angle (always acute, 0° ≤ α ≤ 90°), π ≈ 3.14159265.

Reference Data

Quadrant	Angle Range (Degrees)	Angle Range (Radians)	Reference Angle α	sin Sign	cos Sign	tan Sign
I	0° - 90°	0 - π2	θ	+	+	+
II	90° - 180°	π2 - π	180° − θ	+	−	−
III	180° - 270°	π - 3π2	θ − 180°	−	−	+
IV	270° - 360°	3π2 - 2π	360° − θ	−	+	−
Common Angle Reference Values
-	0°	0	0°	0	1	0
I	30°	π6	30°	12	√32	1√3
I	45°	π4	45°	√22	√22	1
I	60°	π3	60°	√32	12	√3
-	90°	π2	90°	1	0	undef
II	120°	2π3	60°	√32	−12	−√3
II	135°	3π4	45°	√22	−√22	−1
II	150°	5π6	30°	12	−√32	−1√3
-	180°	π	0°	0	−1	0
III	210°	7π6	30°	−12	−√32	1√3
III	225°	5π4	45°	−√22	−√22	1
IV	315°	7π4	45°	−√22	√22	−1
IV	330°	11π6	30°	−12	√32	−1√3
-	360°	2π	0°	0	1	0

Frequently Asked Questions

Infinitely many. Since coterminal angles are defined as θ ± 360°·n for all integers n, the set is unbounded in both positive and negative directions. This calculator generates a user-specified count (n = 1 through n = 5 by default) in each direction.

Yes. A negative angle represents clockwise rotation from the positive x-axis in standard position. For example, −45° is coterminal with 315° because −45° + 360° = 315°. Both share the same terminal side in Quadrant IV. The calculator normalizes any input to the [0°, 360°) range before determining quadrant and reference angle.

Quadrantal angles do not belong to any quadrant. The terminal side lies on an axis. The reference angle is either 0° or 90°. Trigonometric functions at these angles produce exact values (0, 1, −1, or undefined for tan at 90°/270°). The tool labels these as "Quadrantal" rather than assigning Q I - Q IV.

The reference angle α maps any angle to Q I, where all trig functions are positive. You then apply the correct sign based on the actual quadrant (ASTC rule: All, Sine, Tangent, Cosine). Without the correct reference angle, sign errors cascade through identities and equations. For example, sin(150°) = +sin(30°) = 0.5 because 150° is in Q II where sine is positive.

The normalization formula θ_std = θ − 360°·floor(θ/360°) handles arbitrarily large values. An input of 1,110° normalizes to 1,110 − 360·3 = 30°. The coterminal list then builds from the original input, not the normalized form. This preserves the user's frame of reference while still identifying quadrant and reference angle from the normalized result.

Yes. JavaScript uses IEEE 754 double-precision floating point, so π is approximated to about 15 significant digits. For most academic and engineering purposes this is sufficient. However, exact symbolic fractions of π (like 2π/3) cannot be represented perfectly. The tool displays radian results rounded to 6 decimal places and also shows the approximate fractional-π form when the input is close to a common multiple.