Find the (x, y) coordinates on the unit circle — a circle of radius 1 centered at the origin — for any angle.
How it works
On the unit circle, x = cos(θ) and y = sin(θ). At 60°, the coordinates are (0.5, 0.866); at 210°, they’re (−0.866, −0.5).
What this does not include
This works for any angle, not just the commonly memorized special angles (30°, 45°, 60°, and their multiples) — enter any value and get the exact coordinates.
How to use this calculator
- Enter an angle in degrees.
A worked example
At 60° on the unit circle: coordinates = (cos60°, sin60°) = (0.5, 0.866025).
At 210°: coordinates = (−0.866025, −0.5) — negative in both directions, since 210° falls in the third quadrant.
What the variables mean
| Variable | Meaning |
|---|---|
| Angle | Angle measured counterclockwise from the positive x-axis |
| x, y | Coordinates of the point on a circle of radius 1 |
Edge cases worth knowing
Every point on the unit circle satisfies x² + y² = 1 — a quick way to verify a result, since the radius is fixed at exactly 1 by definition.
The quadrant determines the sign pattern. Both coordinates go negative in the third quadrant (180°–270°), which is exactly why the 210° example above shows two negative values.
Why is the unit circle useful in trigonometry?
It provides a direct geometric picture of sine and cosine as coordinates, making it easier to see why these functions repeat, why they’re bounded between −1 and 1, and how they relate to each other.
Why do x and y coordinates always satisfy x² + y² = 1?
Because the radius of the unit circle is exactly 1, and the Pythagorean theorem applied to any point on the circle gives exactly that relationship — this is also the origin of the identity sin²θ + cos²θ = 1.
What happens at 90°, 180°, and 270°?
These land exactly on the axes: 90° gives (0, 1), 180° gives (−1, 0), and 270° gives (0, −1), each a useful reference point for checking your work.
