Unit Circle Explorer
Sine and cosine are not buttons on a calculator — they are the shadows of one point on one circle.
Tip: park the point at 45° and confirm both shadows match — then sync the wave and ride it.
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.
Trigonometry sticks once you have turned the circle yourself
This explorer is one small piece of how ManySpace teaches mathematics — every idea as something you can drag, spin, and read at a glance.