What this calculator does
Trigonometric values repeat every 360 degrees, so any angle can be reduced to an equivalent one in a single turn. The reference angle then measures the acute gap to the nearest x-axis, which is what the tables are built on.
An angle of −750 degrees reduces to 330 degrees, sits in quadrant IV, and has a reference angle of 30.00 degrees. Its trigonometric values are those of 30 degrees with signs set by the quadrant.
The formula
The angle is reduced modulo 360 into the range 0 to 360. The quadrant follows from which quarter it falls in, and the reference angle is the distance to the nearest horizontal axis.
| Term | Meaning |
|---|---|
| Coterminal angle | An angle sharing the same terminal side, differing by whole turns. |
| Reference angle | The acute angle to the nearest x-axis, always between 0 and 90. |
| Quadrant | Which quarter of the plane the angle lands in, which sets the signs. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle (degrees, any value) | The angle in degrees. Any value works, including negative ones and several full turns. |
When to use it
Evaluating a trigonometric function
The reference angle gives the magnitude and the quadrant gives the sign.
Simplifying a large angle
Reducing modulo 360 makes any angle manageable.
Handling negative angles
Clockwise rotations reduce to an equivalent anticlockwise one.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does each quadrant give?
Four angles and their reductions.
Questions
Why is the reference angle always acute?
Because it measures the distance to the nearest x-axis, and no point can be more than 90 degrees from one. That is what makes it useful: every trigonometric value reduces to one for an angle between 0 and 90.
How does the quadrant affect the sign?
By the usual rule: all functions are positive in quadrant I, only sine in II, only tangent in III and only cosine in IV. The reference angle gives the magnitude and the quadrant supplies the sign.
What happens on an axis?
The reference angle is zero and no quadrant applies, since the angle lies on the boundary. Trigonometric values at those points are the special cases of 0, 1 and undefined.
Do negative angles work the same way?
Yes. Adding full turns until the result is positive gives the coterminal angle, and everything proceeds from there. A negative angle simply means rotating clockwise instead.
For the clock hand version of angle arithmetic, see the clock angle calculator. For trigonometric values, see the trigonometry calculator.