# Unit Circle Calculator

URL: https://speedycalculator.com/calculators/math/unit-circle/
Last updated: 2026-10-30

> The unit circle calculator finds the point (cos α, sin α) on the unit circle for any angle, with exact coordinates for multiples of 15°, the tangent, the quadrant and the reference angle.

## Inputs

- Angle unit
- Angle α

## Results

- x = cos α
- y = sin α
- tan α = y/x
- Quadrant
- Reference angle
- Angle in radians

## Formula

```
P(α) = (cos α, sin α),  x² + y² = 1,  tan α = y/x
```

## Explanation

The unit circle – radius 1, centred at the origin – defines sine and cosine for every angle, not just acute ones. Reading coordinates off it is the fastest way to learn the signs and special values of the trig functions.

### How to use the Unit Circle Calculator

- **Angle unit**
- **Angle α**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Enter the angle and its unit. The calculator places the point, reports its coordinates (exactly when possible), the quadrant, the reference angle and the angle in radians.

## Worked example

Inputs — Angle unit: Degrees (°); Angle α: 135.

Results — **x = cos α = −√2/2 ≈ −0.7071067812**; y = sin α = √2/2 ≈ 0.7071067812; tan α = y/x = −1; Quadrant = Quadrant II; Reference angle = 45°; Angle in radians = 2.35619449 rad (3π/4).

## Key terms

- **Unit circle**: The circle x² + y² = 1.
- **Reference angle**: The acute angle to the x-axis.

## FAQ

**Why is the radius 1?**
So the coordinates are exactly cos α and sin α with no scaling.

**What are the coordinates at 45°?**
(√2/2, √2/2).

**What is a reference angle?**
The acute angle between the terminal side and the x-axis.

**Do negative angles work?**
Yes – they go clockwise from the positive x-axis.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Unit circle](https://mathworld.wolfram.com/UnitCircle.html)