Unit Circle Calculator
Every Trig Value Lives on This Circle
The unit circle — a circle of radius exactly 1 centered at the origin — is the single reference that defines sine and cosine for every possible angle, not just the acute angles that fit inside a right triangle. Any angle, positive or negative, maps to a unique point on that circle, and the point's x and y coordinates are cosine and sine by definition.
The Formula
x = cos(θ)
y = sin(θ)
tan(θ) = sin(θ) / cos(θ)
y = sin(θ)
tan(θ) = sin(θ) / cos(θ)
Tangent is undefined wherever cosine is zero — at 90° and 270° — since that step is a division by zero.
Common Reference Angles
| Angle | cos(θ) | sin(θ) |
|---|---|---|
| 0° | 1 | 0 |
| 30° | 0.8660 | 0.5 |
| 45° | 0.7071 | 0.7071 |
| 60° | 0.5 | 0.8660 |
| 90° | 0 | 1 |
| 180° | −1 | 0 |
| 270° | 0 | −1 |
Where This Matters
- Rotations in graphics and robotics — rotating a point or vector by an angle uses cosine and sine values read straight off the unit circle.
- Signal processing — sine and cosine waves that model periodic signals are generated by tracing the unit circle over time.
- Trigonometry education — the unit circle is the standard way to extend sine and cosine beyond acute triangle angles into negative angles and angles past 90°.
How to Use This Calculator
- Enter the Angle.
- Choose the unit: Degrees or Radians.
- Select Calculate to get the cosine, sine, and tangent values, plus the corresponding point on the unit circle.
Related Calculations
Working with coordinates instead of angles directly? See the Polar to Cartesian Converter, which uses the same cosine/sine relationship at any radius.