Rotate the angle θ from 0° to 360°. The coordinates of the point on the circle become exactly (cos θ, sin θ), and you can also watch how the sign changes every time you cross into a new quadrant.
The sin and cos you learn with a right triangle only work for angles between 0° and 90°. But to handle things that keep rotating, like a Ferris wheel or a pendulum, you need angles like 120° or 200° too. So we rebuild the definition using a point P on the unit circle (a circle with radius 1) — P's x-coordinate is cos θ, and its y-coordinate is sin θ.
This definition doesn't throw away the trig ratios you already know — it extends them across the full 360°. Every time the angle crosses into a different quadrant, the sign of the coordinates flips, and the signs of sin, cos, and tan change in a fixed pattern too.
Use the slider to rotate the angle θ from 0° all the way to 360°, and watch in real time how point P's coordinates, the sin/cos/tan values, and which quadrant you're currently in all change.
Point P's x-coordinate is cos θ, and its y-coordinate is sin θ. You can think of tan θ as the y-coordinate of the point where the circle's radius, extended as a ray, meets the tangent line at x=1. Move the slider and the wave below is drawn along with it.
| sin θ | 0.643 |
| cos θ | 0.766 |
| tan θ | 0.839 |
The height of the point on the circle (its y-coordinate, sin θ) gets carried straight over to the right, tracing the wave as far as the point has traveled. Press "Auto-rotate" to watch it happen.