Find the reference angle for any angle in standard position. Enter degrees or radians to see the quadrant, reference angle, ASTC sign pattern, standard trig values, and a live unit circle diagram.
A reference angle is the positive acute angle (between 0° and 90°) formed between the terminal ray of a given angle and the nearest x-axis.
Every angle in standard position has a reference angle. Use the formula for the quadrant containing the normalized angle:
| Location | Formula | Example |
|---|---|---|
| Quadrant I | ref = θ | θ = 40° → ref = 40° |
| Quadrant II | ref = 180° − θ | θ = 150° → ref = 30° |
| Quadrant III | ref = θ − 180° | θ = 210° → ref = 30° |
| Quadrant IV | ref = 360° − θ | θ = 330° → ref = 30° |
| On an axis | ref = 0° | θ = 180° → ref = 0° |
The mnemonic ASTC (read quadrant by quadrant, starting in Q I and going counterclockwise) tells you which trig functions are positive in each quadrant. All others are negative.
One-on-one Trigonometry tutoring makes the unit circle click — we work through quadrants, reference angles, and ASTC together until it becomes second nature.