Quadrant Calculator
Enter any angle to find its quadrant, the sign of each trig function there, and its reference angle, updated as you type.
The quadrant an angle falls in decides two things at once: which reference angle formula applies, and which trig functions come out positive or negative. This page covers both, the four quadrant ranges, the ASTC sign rule, the special quadrantal angles that sit on the axes, and a full sign table for quick lookup.
4 quadrant definitions and angle ranges
0° to 90°
x > 0, y > 0
90° to 180°
x < 0, y > 0
180° to 270°
x < 0, y < 0
270° to 360°
x > 0, y < 0
Quadrants are numbered counterclockwise starting from the upper right, where both coordinates are positive. This numbering is a fixed convention in mathematics. It never changes regardless of what the angle represents.
The quadrant system comes directly from the signs of the x- and y-coordinates. In Quadrant I, both x and y are positive. Moving counterclockwise into Quadrant II, x becomes negative while y stays positive. Continuing into Quadrant III, both x and y are negative. Finally, in Quadrant IV, x turns positive again while y stays negative. Since cosine is defined as the x-coordinate and sine as the y-coordinate on the unit circle, this sign pattern is exactly what drives the sign of every trigonometric function.
Knowing the quadrant of an angle is often more useful on its own than knowing the exact angle. In physics, the quadrant tells you the direction a vector points without needing the precise angle. In navigation and surveying, bearings are frequently described relative to a quadrant before being refined to an exact degree measurement. In trigonometric equations, the quadrant is what narrows an infinite family of possible solutions down to the one that actually fits the original problem.
The ASTC rule
ASTC (All Students Take Calculus) names which functions are positive in each quadrant.
All
Quadrant I
All trig functions are positive.
Sine
Quadrant II
Only sine (and cosecant) is positive.
Tangent
Quadrant III
Only tangent (and cotangent) is positive.
Cosine
Quadrant IV
Only cosine (and secant) is positive.
Quadrantal angles
Quadrantal angles, exactly 0°, 90°, 180°, 270°, and 360°, sit directly on an axis rather than inside any of the four quadrants. Their terminal side lies exactly on the x-axis or y-axis, so they aren't assigned to Quadrant I, II, III, or IV at all. Most tools, including this calculator, label these as "Quadrantal" instead of forcing them into a quadrant that doesn't actually contain them.
Quadrantal angles are special because one of sine or cosine equals exactly 0 there, which makes tangent or cotangent undefined at certain points (tangent is undefined at 90° and 270°; cotangent is undefined at 0°, 180°, and 360°).
These five angles act as the boundary markers between quadrants. As an angle sweeps from 0° to 360°, it passes through each quadrantal angle exactly once, and crossing one always means moving into the next quadrant. This is also why quadrantal angles come up constantly in graphing: they mark the x-intercepts and y-intercepts of the sine and cosine curves, and they're the points where those curves reach their maximum or minimum values.
Sign table for all trig functions
| Function | Q I | Q II | Q III | Q IV |
|---|---|---|---|---|
| sin | + | + | − | − |
| cos | + | − | − | + |
| tan | + | − | + | − |
| cot | + | − | + | − |
How to determine the quadrant from any angle
- 1
Normalize the angle
Add or subtract 360° until the angle is between 0° and 360°.
- 2
Compare to the ranges
Check which 90° band the angle falls in: I, II, III, or IV.
- 3
Watch for axis angles
0°, 90°, 180°, 270°, and 360° are quadrantal, not inside any quadrant.
This same three-step process works regardless of unit. In radians, just swap the boundaries: 0 to π/2 is Quadrant I, π/2 to π is Quadrant II, π to 3π/2 is Quadrant III, and 3π/2 to 2π is Quadrant IV, with 0, π/2, π, 3π/2, and 2π marking the quadrantal boundaries instead.
Frequently Asked Questions
What are the 4 quadrants?
The coordinate plane is divided into four quadrants by the x-axis and y-axis. Quadrant I is 0° to 90°, Quadrant II is 90° to 180°, Quadrant III is 180° to 270°, and Quadrant IV is 270° to 360°, numbered counterclockwise starting from the upper right.
How do you determine which quadrant an angle is in?
Reduce the angle to a coterminal value between 0° and 360°, then compare it to the four quadrant ranges: 0°–90° is Quadrant I, 90°–180° is Quadrant II, 180°–270° is Quadrant III, and 270°–360° is Quadrant IV. Angles exactly on a boundary, like 90° or 270°, are quadrantal rather than inside a quadrant.
What quadrant is 150° in?
150° is in Quadrant II, since it falls between 90° and 180°. In Quadrant II, sine is positive while cosine and tangent are both negative, and its reference angle is 180° − 150° = 30°, following the standard Quadrant II formula.
What quadrant is 225° in?
225° is in Quadrant III, since it falls between 180° and 270°. In Quadrant III, tangent is positive while sine and cosine are both negative, and its reference angle is 225° − 180° = 45°, following the standard Quadrant III formula.
What quadrant is 315° in?
315° is in Quadrant IV, since it falls between 270° and 360°. In Quadrant IV, cosine is positive while sine and tangent are both negative, and its reference angle is 360° − 315° = 45°, following the standard Quadrant IV formula.