The reference angle lets you evaluate trigonometric functions of any angle using only acute angles: sin 150° = sin 30°, with the sign fixed by the quadrant. It is a core step in unit-circle problems.
How to use the Reference Angle Calculator
- Angle α
Results update as you type, and everything is calculated in your browser.
How it works
The angle is first reduced to [0°, 360°); then the formula for its quadrant gives the reference angle.
Tips
- Angles are measured counter-clockwise from the positive x-axis by default.
- Add or subtract 360° (2π) to move between coterminal angles.
- Complementary angles add to 90°, supplementary angles to 180°.
- Use the unit menu to switch between degrees, radians and gradians.
Common mistakes to avoid
- Mixing up complementary and supplementary angles.
- Forgetting that a reference angle is always between 0° and 90°.
- Leaving an angle in radians when the question asks for degrees.
What each result means
- Coterminal angle in [0°, 360°) = mod([Angle α], 360)
- Reference angle in radians = rad([Reference angle])
Example values
How the main result changes when Angle α changes and everything else stays at the example values above:
| Angle α (°) | Reference angle (°) |
|---|---|
| 75 | 75 |
| 112.5 | 67.5 |
| 150 | 30 |
| 187.5 | 7.5 |
| 225 | 45 |
| 300 | 60 |
Formula
QI: α · QII: 180° − α · QIII: α − 180° · QIV: 360° − α (after reducing α to [0°, 360°))Worked example
Inputs — Angle α: 150 °.
Results — Coterminal angle in [0°, 360°) = 150 °; Reference angle = 30 °; Quadrant = Quadrant II; Reference angle in radians = 0.52359878 rad.
Key terms
- Reference angle
- The acute angle to the x-axis.
Frequently asked questions
What is the reference angle of 210°?
30°.
Is a reference angle ever negative?
No – it is always between 0° and 90°.
Why are reference angles useful?
Trig values of any angle equal ± those of its reference angle.
What is the reference angle of 90°?
90° – it lies on an axis.
What formula does the Reference Angle Calculator use?
QI: α · QII: 180° − α · QIII: α − 180° · QIV: 360° − α (after reducing α to [0°, 360°)).