Reference Angle Chart
Type an angle to jump straight to its reference angle, or scroll the full chart below for every angle from 0° to 360° in 5° steps.
This chart is designed as a lookup, not a lesson. For the reasoning behind each formula, see reference angle formulas or how to find a reference angle. What's here is every 5° increment across a full rotation, with degree and radian measures, the quadrant, and the reference angle side by side, so a specific value can be confirmed in seconds without recalculating anything by hand.
Every angle, 0° to 360°
Highlighted rows mark the special angles, 30°, 45°, 60°, and 90°, and their reflections in each quadrant.
Scroll through slowly the first time and watch the reference-angle column: it climbs from 0° to 90° across Quadrant I, then counts back down to 0° across Quadrant II, climbs again through Quadrant III, and counts down once more through Quadrant IV. That up-down-up-down rhythm is the entire chart in miniature. Everything else is just the degree and radian labels attached to it.
The radian column follows the exact same logic as the degree column, just expressed as a fraction of π instead of a whole number. A row like 100° pairs with 5π/9 for the same reason 30° pairs with π/6. Both are the degree measurement divided by 180 and multiplied by π, then reduced to lowest terms. That consistency is what makes the two columns directly comparable rather than two unrelated ways of labeling the same angle.
The quadrant column is worth cross-referencing with the reference angle column directly above any row you're checking. Together they tell you everything needed to evaluate a trig function at that angle: the quadrant fixes the sign, and the reference angle fixes the magnitude, exactly as described on the trig functions page.
Bookmark this page for exam review rather than trying to memorize all 73 rows outright. Most exam questions draw from the highlighted special angles, 30°, 45°, 60°, and 90°, and their reflections in each quadrant, so those rows deserve the most attention, while the rest of the chart is here mainly to confirm the pattern holds everywhere else too.
| Angle ° | Angle rad | Quadrant | Ref ° | Ref rad |
|---|---|---|---|---|
| 0° | 0 | - | 0° | 0 |
| 5° | π/36 | I | 5° | π/36 |
| 10° | π/18 | I | 10° | π/18 |
| 15° | π/12 | I | 15° | π/12 |
| 20° | π/9 | I | 20° | π/9 |
| 25° | 5π/36 | I | 25° | 5π/36 |
| 30° | π/6 | I | 30° | π/6 |
| 35° | 7π/36 | I | 35° | 7π/36 |
| 40° | 2π/9 | I | 40° | 2π/9 |
| 45° | π/4 | I | 45° | π/4 |
| 50° | 5π/18 | I | 50° | 5π/18 |
| 55° | 11π/36 | I | 55° | 11π/36 |
| 60° | π/3 | I | 60° | π/3 |
| 65° | 13π/36 | I | 65° | 13π/36 |
| 70° | 7π/18 | I | 70° | 7π/18 |
| 75° | 5π/12 | I | 75° | 5π/12 |
| 80° | 4π/9 | I | 80° | 4π/9 |
| 85° | 17π/36 | I | 85° | 17π/36 |
| 90° | π/2 | - | 90° | π/2 |
| 95° | 19π/36 | II | 85° | 17π/36 |
| 100° | 5π/9 | II | 80° | 4π/9 |
| 105° | 7π/12 | II | 75° | 5π/12 |
| 110° | 11π/18 | II | 70° | 7π/18 |
| 115° | 23π/36 | II | 65° | 13π/36 |
| 120° | 2π/3 | II | 60° | π/3 |
| 125° | 25π/36 | II | 55° | 11π/36 |
| 130° | 13π/18 | II | 50° | 5π/18 |
| 135° | 3π/4 | II | 45° | π/4 |
| 140° | 7π/9 | II | 40° | 2π/9 |
| 145° | 29π/36 | II | 35° | 7π/36 |
| 150° | 5π/6 | II | 30° | π/6 |
| 155° | 31π/36 | II | 25° | 5π/36 |
| 160° | 8π/9 | II | 20° | π/9 |
| 165° | 11π/12 | II | 15° | π/12 |
| 170° | 17π/18 | II | 10° | π/18 |
| 175° | 35π/36 | II | 5° | π/36 |
| 180° | π | - | 0° | 0 |
| 185° | 37π/36 | III | 5° | π/36 |
| 190° | 19π/18 | III | 10° | π/18 |
| 195° | 13π/12 | III | 15° | π/12 |
| 200° | 10π/9 | III | 20° | π/9 |
| 205° | 41π/36 | III | 25° | 5π/36 |
| 210° | 7π/6 | III | 30° | π/6 |
| 215° | 43π/36 | III | 35° | 7π/36 |
| 220° | 11π/9 | III | 40° | 2π/9 |
| 225° | 5π/4 | III | 45° | π/4 |
| 230° | 23π/18 | III | 50° | 5π/18 |
| 235° | 47π/36 | III | 55° | 11π/36 |
| 240° | 4π/3 | III | 60° | π/3 |
| 245° | 49π/36 | III | 65° | 13π/36 |
| 250° | 25π/18 | III | 70° | 7π/18 |
| 255° | 17π/12 | III | 75° | 5π/12 |
| 260° | 13π/9 | III | 80° | 4π/9 |
| 265° | 53π/36 | III | 85° | 17π/36 |
| 270° | 3π/2 | - | 90° | π/2 |
| 275° | 55π/36 | IV | 85° | 17π/36 |
| 280° | 14π/9 | IV | 80° | 4π/9 |
| 285° | 19π/12 | IV | 75° | 5π/12 |
| 290° | 29π/18 | IV | 70° | 7π/18 |
| 295° | 59π/36 | IV | 65° | 13π/36 |
| 300° | 5π/3 | IV | 60° | π/3 |
| 305° | 61π/36 | IV | 55° | 11π/36 |
| 310° | 31π/18 | IV | 50° | 5π/18 |
| 315° | 7π/4 | IV | 45° | π/4 |
| 320° | 16π/9 | IV | 40° | 2π/9 |
| 325° | 65π/36 | IV | 35° | 7π/36 |
| 330° | 11π/6 | IV | 30° | π/6 |
| 335° | 67π/36 | IV | 25° | 5π/36 |
| 340° | 17π/9 | IV | 20° | π/9 |
| 345° | 23π/12 | IV | 15° | π/12 |
| 350° | 35π/18 | IV | 10° | π/18 |
| 355° | 71π/36 | IV | 5° | π/36 |
| 360° | 2π | - | 0° | 0 |
Why the chart repeats
The chart repeats the same 6 reference angles (0°, 30°, 45°, 60°, 75°, 90°) across all 4 quadrants. Once you notice this, the chart stops looking like 73 unrelated rows and starts looking like the same short list, mirrored four times with a sign change each time. That repetition is exactly what the reference angle concept is built to expose.
Reading the chart this way turns memorization into pattern recognition. Instead of treating 30°, 150°, 210°, and 330° as four unrelated angles to remember separately, it's more useful to see them as one reference angle, 30°, appearing once in each quadrant. The same is true for 45° (showing up as 45°, 135°, 225°, 315°) and 60° (as 60°, 120°, 240°, 300°). Every row in the chart between the highlighted special angles follows the same quadrant formula, so once the pattern clicks, the chart becomes a lookup you can reconstruct from a handful of values rather than something you need to memorize row by row.
This chart is most useful as a study aid alongside the reference angle calculator: use the calculator to check a specific angle quickly, and scan the chart to see how that angle fits into the larger repeating pattern across all four quadrants. Students preparing for exams often find it faster to memorize the six first-quadrant reference values and the ASTC sign rule than to memorize all 73 rows individually. The chart exists to make that shortcut visible.
The reference angle chart, visualized
The chart above shows the 4 reference angle formulas, one for each quadrant. The terminal side's quadrant determines which formula to use. All 4 return the same kind of result: a positive acute angle between 0° and 90°.
Frequently Asked Questions
What is a reference angle chart used for?
A reference angle chart lets you look up the quadrant and reference angle for any standard angle without recalculating it by hand. It's useful for homework, exams, and quick checks. The chart on this page covers every 5-degree increment from 0° to 360°, so almost any angle you encounter already has a row worked out.
What are the most common reference angles?
The most common reference angles are 30°, 45°, and 60°, because their sine, cosine, and tangent values are exact fractions involving simple square roots, and they appear repeatedly across all four quadrants. Memorizing just these three values, along with the quadrantal angles 0° and 90°, covers most angles that show up in typical trigonometry coursework.
What is the reference angle for 210°?
The reference angle for 210° is 30°. 210° is in Quadrant III, so θ̂ = 210° − 180° = 30°. Since 210° and 30° share the same reference angle, they also share the same tangent value, though sine and cosine differ in sign between the two quadrants.
What is the reference angle for 330°?
The reference angle for 330° is 30°. 330° is in Quadrant IV, so θ̂ = 360° − 330° = 30°. In Quadrant IV, cosine is positive and sine is negative, so cos(330°) equals cos(30°) while sin(330°) equals negative sin(30°) and tangent is negative too.
How do special angles appear in the chart?
Special angles, 30°, 45°, 60°, 90°, and their reflections in each quadrant, are highlighted in the chart because their exact trig values are worth memorizing, unlike arbitrary angles. Highlighting them makes it faster to spot the values most exam questions and textbook problems rely on, rather than scanning all 73 rows equally.