Choose a quadrant tool
Select an option, enter values, and view steps below.
Understanding quadrant calculations
Point quadrants
The coordinate plane is split by the x-axis and y-axis. A point belongs to a quadrant only when both coordinates are nonzero. Positive x and positive y create Quadrant I. Negative x and positive y create Quadrant II. Negative x and negative y create Quadrant III. Positive x and negative y create Quadrant IV.
Angle quadrants
An angle can be larger than one full turn or negative. First reduce it to a coterminal angle between 0° and 360°. Then compare it with the main axis angles. Angles exactly at 0°, 90°, 180°, and 270° lie on axes, not inside quadrants.
Reference angles
A reference angle is the acute angle made with the nearest x-axis. It helps transfer known trigonometric values across quadrants. The value stays positive as an angle measure. The signs of sine, cosine, and tangent then come from the actual quadrant.
Trig sign rules
Sine follows the y-coordinate. Cosine follows the x-coordinate. Tangent follows sine divided by cosine. This creates the classic ASTC rule. All ratios are positive in Quadrant I. Sine is positive in Quadrant II. Tangent is positive in Quadrant III. Cosine is positive in Quadrant IV.
Vectors, polar points, and complex numbers
Vectors, polar coordinates, and complex numbers all use the same plane logic. Components, rectangular coordinates, or real and imaginary parts determine the quadrant. The atan2 function is important because it uses both signs to place the direction angle correctly.
FAQs
What does this quadrant calculator do?
It classifies points, angles, terminal points, vectors, polar coordinates, complex numbers, line samples, and reflections by quadrant or axis.
Are axis points in a quadrant?
No. Points on the x-axis or y-axis are boundary points. They are not inside any of the four quadrants.
Can I enter radians?
Yes. Angle fields support radians. You can type decimals, fractions, pi, pi/3, or similar values.
What is a coterminal angle?
A coterminal angle shares the same terminal side. Add or subtract 360° or 2π to get another coterminal angle.
How is the reference angle found?
First normalize the angle. Then measure the acute angle from the terminal side to the nearest x-axis.
Why use atan2 instead of arctangent?
atan2 uses both x and y signs. It places the direction angle in the correct quadrant automatically.
What does ASTC mean?
ASTC means All, Sine, Tangent, Cosine. It summarizes which trig ratios are positive in each quadrant.
Can tangent be undefined?
Yes. Tangent is undefined when cosine equals zero, usually on the positive or negative y-axis.
How are polar coordinates classified?
Convert them to rectangular coordinates using x = r cos θ and y = r sin θ. Then classify the signs.
What happens with negative polar radius?
A negative radius places the point opposite the given terminal side. The calculator also shows the equivalent direction.
Can complex numbers use this calculator?
Yes. The real part is x. The imaginary part is y. The quadrant and argument follow from that point.
Does the graph update by tool?
Yes. It draws points, rays, vectors, segments, reflections, lines, or quadrant highlights based on the selected mode.