Reference Angle Calculator
A reference angle calculator reduces any angle, in degrees or radians, to the acute angle between its terminal side and the x-axis. The one below also reports the quadrant, the positive and negative coterminal angles, the exact quadrant formula it applied, and the signed sine, cosine and tangent values worked out from the reference angle, all in one pass.
Reference angle
Quick Answer
For the pre-filled angle of 150°: the coterminal angle inside one turn is 150° itself, which sits in Quadrant II, so the reference angle is 30° (180° − 150°), or 0.523599 rad. The signed trig values are sin = +0.5000, cos = −0.8660, tan = −0.5774. The calculator above is pre-filled with this exact angle.
Key takeaways
- A reference angle is always acute (0° to 90°). It carries the size of the trig value; the quadrant carries the sign.
- Reduce first: any angle, however large or negative, is reduced to a coterminal angle between 0° and 360° before the quadrant formula is applied.
- Results stay hidden until you click Calculate. Reset restores the worked example (150°).
- This tool also shows the negative coterminal angle and a one-line trace of the formula it used, so you can copy the working into your homework.
How to use the reference angle calculator
Type any angle and pick its unit. The tool accepts negative angles, angles beyond a full turn, and decimal radians, and results stay hidden until you click Calculate.
- Angle: enter any number. 150 is pre-filled. A value like 725 or −45 is fine; the tool reduces it to its coterminal angle first.
- Unit: choose Degrees or Radians. In radian mode, type a plain decimal such as 2.618, not a fraction of pi like 5pi/6; the tool does not parse pi text.
- Calculate: read the reference angle in degrees and radians, the quadrant, both coterminal angles, the signed sin, cos and tan, and the working line showing the formula applied.
- Reset: clears the result and restores the 150° worked example.
If you need the trig values themselves for inverse work, the inverse sine calculator and inverse cosine calculator go the other direction, from a value back to an angle.
How the reference angle is worked out
Every run follows the same three steps: reduce the angle to one full turn, read off the quadrant, then apply that quadrant's formula.
Step 2, locate: Q I (0°–90°), Q II (90°–180°), Q III (180°–270°), Q IV (270°–360°)
Step 3, formulas: Q I: ref = θ | Q II: ref = 180° − θ | Q III: ref = θ − 180° | Q IV: ref = 360° − θ
In radians: Q I: ref = θ | Q II: ref = π − θ | Q III: ref = θ − π | Q IV: ref = 2π − θ
| Quadrant of the reduced angle | Reference angle formula | Signs: sin, cos, tan |
|---|---|---|
| I (0°–90°) | ref = θ | +, +, + |
| II (90°–180°) | ref = 180° − θ | +, −, − |
| III (180°–270°) | ref = θ − 180° | −, −, + |
| IV (270°–360°) | ref = 360° − θ | −, +, − |
The sign table is the classic ASTC rule (All Students Take Calculus) that OpenStax's precalculus text uses when it defines a reference angle as the smallest acute angle between the terminal side and the horizontal axis, and then reads the sine and cosine signs off the quadrant. The calculator applies those signs to the actual decimal trig values, which is why you see entries like cos = −0.8660 rather than just "negative".
Worked example: 150°, and three more runs
The pre-filled run is 150°. Here is everything the calculator returns for it, verified in the tool itself, plus three variations that show how negative angles, radians and axis-adjacent angles behave.
| Input | Quadrant / coterminal | Reference angle | sin, cos, tan |
|---|---|---|---|
| 150° (pre-filled) | Quadrant II, coterminal 150° | 30° (0.523599 rad) | +0.5000, −0.8660, −0.5774 |
| 225° | Quadrant III, coterminal 225° | 45° (0.785398 rad) | −0.7071, −0.7071, +1.0000 |
| −45° | Quadrant IV, coterminal 315° (also −45°) | 45° (0.785398 rad) | −0.7071, +0.7071, −1.0000 |
| 3.665191429 rad (210°) | Quadrant III, coterminal 210.00° | 30° (0.523599 rad) | −0.5000, −0.8660, +0.5774 |
Two honest notes about the tool's conventions, both kept as the tool has always behaved and stated here so nothing surprises you. First, angles that land exactly on an axis are assigned to the quadrant above the boundary: 90° reads as Quadrant II (reference angle 90°) and 270° as Quadrant IV, following the tool's less-than comparisons at 90, 180 and 270. Second, a trig value that is mathematically zero still carries its quadrant sign in the display, so you will see sin = +0.0000 at 0° and tan = undefined at 90° where cosine passes through zero.
What competing calculators miss
Several pages rank for this keyword, and most stop at the reference angle itself. The gaps are consistent:
- Vedantu's reference angle calculator: clean formulas and a steps table, but it returns no trig values at all, so you still have to look up sin, cos and tan elsewhere.
- OnlyCalculators: shows the quadrant formula and the trig function signs per quadrant, but only as signs, never as the signed decimal values, and it shows no coterminal angle.
- referenceanglecalculator.com: the strongest rival, with pi-fraction input, decimal-place options and an optional angle visualisation. Its trig display is an optional extra, and its working is less direct: this page prints the exact formula line (ref = 180° − θ = 30°) in the result itself.
- Omni Calculator: a long reference-angle lookup table that is good for quick answers, but the tool behind it is basic: no negative angles beyond the simple case and no worked trace.
- CalculatorsHub's standard-position tool: only accepts angles from 0° to 360°, so negative angles and multi-turn angles are rejected instead of reduced.
- The gap this page fills: negative and multi-turn angles in degrees or radians, both the positive and the negative coterminal angle, the exact formula used, and the signed decimal trig values, all in one pass.
For angles measured at the centre of a circle rather than in standard position, the central angle calculator handles arc-based work, and the pitch angle calculator covers slope-style angles in the applied direction.
Frequently asked questions
A reference angle is the smallest positive acute angle between the terminal side of an angle and the x-axis, always between 0° and 90°. It gives you the size of the sine, cosine and tangent values; the quadrant tells you their signs.
150° lies in Quadrant II (between 90° and 180°), so use ref = 180° − θ: 180° − 150° = 30°. Since sine is positive and cosine negative in Quadrant II, sin(150°) = +0.5000 and cos(150°) = −0.8660.
Reduce it to a positive coterminal angle first by adding 360°. For −45°: −45° + 360° = 315°, which is in Quadrant IV, so ref = 360° − 315° = 45°. The calculator above does this reduction automatically and also shows the negative coterminal angle.
The same way, with pi-based formulas: Q I: ref = θ, Q II: ref = π − θ, Q III: ref = θ − π, Q IV: ref = 2π − θ. For example, 3.665191429 radians is 210°, in Quadrant III, giving a reference angle of 30° (0.523599 rad). Enter a plain decimal number of radians in the tool.
It is a convention choice, stated openly. The tool compares the reduced angle with less-than boundaries at 90, 180 and 270, so an angle landing exactly on an axis is assigned to the quadrant above it: 90° reads as Quadrant II, 270° as Quadrant IV. The reference angle it returns (90°) is the same under either convention.
Evaluate the trig function of the acute reference angle, then apply the quadrant sign. For 150°, the reference angle is 30°: sin(150°) = +sin(30°) = +0.5000, cos(150°) = −cos(30°) = −0.8660, tan(150°) = −tan(30°) = −0.5774. That is exactly what the tool's signed trig line prints.
A coterminal angle shares the same terminal side as your angle but can be any size, positive or negative (for example, 150°, 510° and −210° are all coterminal). A reference angle is always the acute angle between that terminal side and the x-axis (30° for all three). This calculator shows both.
Add This Calculator to Your Website
Embed this calculator on your own site with the code below. Pick a size to preview it live, then copy the code.
Powered by Calculator Zilo
References
- OpenStax. Precalculus 2e, Section 5.1: Angles (coterminal and reference angles, degree-radian conversion). Accessed October 23, 2026.
- OpenStax. Precalculus 2e, Section 5.2: Unit Circle: Sine and Cosine Functions (finding reference angles, quadrant signs of sine and cosine). Accessed October 23, 2026.
- Mathematics LibreTexts. 5.02: Angles (coterminal and reference angles with worked examples). Accessed October 23, 2026.
- Mathematics LibreTexts. 1.7: Trigonometric Functions of Any Angle (reference angles used to determine trig function values). Accessed October 23, 2026.
- Mathematics LibreTexts. 19.2.01: Degree and Radian Measure (converting between radians and degrees). Accessed October 23, 2026.