Unit Circle Calculator
Type an angle to find its sine, cosine, and tangent, plus the coordinates and quadrant on the unit circle.
How to use the calculator
- Enter an angle in degrees or radians.
- The calculator shows the sine, cosine, and tangent of the angle, plus the point's coordinates (x, y) on the unit circle and which quadrant the angle falls in.
What is the unit circle?
The unit circle is a circle with radius 1, centered at the origin of a coordinate system. For an angle v measured from the positive x-axis, the point on the circle has coordinates (cos v, sin v) — this is the very definition of sine and cosine in trigonometry. Tangent is sine divided by cosine, and is undefined where cosine is zero (at 90°, 270°, etc.).
Worked example: at 45°, sin(45°) = 0.7071 and cos(45°) = 0.7071 — exactly equal. That is no coincidence: 45° bisects the right angle, so the triangle is isosceles and the x and y coordinates must be identical. The value is exactly √2 ÷ 2. The tangent is then 0.7071 ÷ 0.7071 = 1.0000, a line rising exactly as much as it runs.
Frequently asked questions
Why does it say "undefined" for tangent?
Tangent is sine divided by cosine, and is mathematically undefined when cosine is zero — this happens at 90°, 270°, and equivalent angles.
How are the quadrants numbered?
Quadrant 1 is 0–90°, quadrant 2 is 90–180°, quadrant 3 is 180–270°, and quadrant 4 is 270–360° — standard mathematical convention, counterclockwise from the positive x-axis.
Can I use negative angles?
Yes, negative angles are interpreted as clockwise from the positive x-axis and are automatically normalized to 0–360° in the calculation.