# Unit Circle Explorer

Canonical: https://manyspace.aporia.systems/tools/unit-circle
Updated: 2026-10-13

Sine and cosine are not buttons on a calculator — they are the shadows of one point on one circle.

## Try this

- Land the classics. Hop through the canonical chips 0° → 90° → 180° → 270°. Watch the readout flip between exact values and watch the shadows trade places.
- Chase the signs. Drag slowly from Q1 into Q2. Cosine crosses zero and goes negative while sine stays positive — the quadrant readout narrates the whole journey.
- Unroll the wave. Reveal the synced sine wave and drag. The marker rides the curve exactly as the point rides the circle: the wave is the circle, unrolled in time.

## How to read the circle

Fix a radius of 1 and an angle θ from the positive x-axis. The point lands at (cos θ, sin θ) — so the horizontal shadow is the cosine and the vertical shadow is the sine. No tables needed: quadrants tell you the signs (+/+ in Q1, −/+ in Q2, and so on), and the axes give you the clean 0s, 1s, and −1s.

The canonical angles (30°, 45°, 60° and friends) earn exact values — √2/2, √3/2 — because they come from halved squares and equilateral triangles, not from memorization. Turn the point slowly through a full rotation and feel sine rise, peak, fall, and bottom out exactly once.

## Related tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative)
- [Integral Explorer](https://manyspace.aporia.systems/tools/integral)
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane)
- [Probability Lab](https://manyspace.aporia.systems/tools/probability)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series)