§1What is a reference angle?
The reference angle of θ is the positive acute angle between the terminal side of θ and the x-axis. It always measures between 0° and 90° (between 0 and π/2 in radians). Every angle that is not on an axis has exactly one reference angle, and many different angles share the same one: 30°, 150°, 210° and 330° all have a reference angle of 30°.
That is what makes reference angles useful. They let you reduce any angle to one in Quadrant I, where the values are easy to remember, and then fix the sign using the quadrant.
§2The rule for each quadrant
First bring the angle into one turn: between 0° and 360°, or between 0 and 2π. Then use the rule for the quadrant it lands in.
| Quadrant | θ between | Degrees | Radians |
|---|---|---|---|
| I | 0° and 90° | θ | θ |
| II | 90° and 180° | 180° − θ | π − θ |
| III | 180° and 270° | θ − 180° | θ − π |
| IV | 270° and 360° | 360° − θ | 2π − θ |
A quick way to remember it: in every quadrant you subtract whichever of 0°, 180° or 360° the terminal side is closest to, and keep the result positive.
§3Worked examples
225°. 180° < 225° < 270°, so Quadrant III → 225° − 180° = 45°
−130°. −130° + 360° = 230° → Quadrant III → 230° − 180° = 50°
410°. 410° − 360° = 50° → Quadrant I → the reference angle is 50°
5π/3. 3π/2 < 5π/3 < 2π → Quadrant IV → 2π − 5π/3 = π/3 (60°)
2.618 rad. 2.618 × 180°/π ≈ 150.0004° → Quadrant II → 180° − 150.0004° ≈ 29.9996°
2.618 is a rounded value of 5π/6, which is why the answer is almost exactly 30°.
§4Using reference angles for trig values
The sine, cosine and tangent of an angle have the same size as those of its reference angle. Only the sign can change, and the quadrant tells you which sign: sine is positive in Quadrants I and II, cosine in I and IV, and tangent in I and III.
sin 150°. Reference angle 30°, Quadrant II, sine positive → sin 150° = sin 30° = 1/2
cos 150°. Same reference angle, cosine negative in Quadrant II → cos 150° = −√3/2
The result panel above lists sin θ, cos θ and tan θ next to the reference angle so you can see this connection for any angle. The interactive unit circle shows the same idea visually: the triangle in every quadrant is a mirror image of the one in Quadrant I.
§5Angles on the axes
Angles like 0°, 90°, 180° and 270° are called quadrantal angles: their terminal side lies on an axis, not inside a quadrant. The terminal side then makes an angle of 0° with the x-axis (for 0° and 180°) or 90° (for 90° and 270°), and this calculator reports those values. Some textbooks instead say quadrantal angles have no reference angle, so follow the convention your class uses.
§6Common mistakes
- Measuring to the y-axis. The reference angle is always measured to the x-axis, even when the terminal side is closer to the y-axis.
- Skipping the coterminal step. For −130° or 410°, find the angle between 0° and 360° first; the coterminal angle calculator shows that step on its own.
- Mixing units, such as computing π − 150°. Work entirely in degrees or entirely in radians.
- Giving a negative answer. A reference angle is never negative; if you get one, you used the wrong rule.
§7FAQ
Can a reference angle be bigger than 90°?
No. By definition it is acute, between 0° and 90°. If your answer is larger, the angle was not brought into one turn first or the wrong quadrant rule was used.
What is the reference angle of 300°?
300° is in Quadrant IV, so the reference angle is 360° − 300° = 60°.
How do I find a reference angle in radians?
Use π in place of 180°: π − θ in Quadrant II, θ − π in Quadrant III and 2π − θ in Quadrant IV. For 7π/6, the reference angle is 7π/6 − π = π/6.
Do coterminal angles have the same reference angle?
Yes. They end on the same terminal side, so 30°, 390° and −330° all have a reference angle of 30°.