Trigonometry

Where sine and cosine come from

Take a circle of radius 1, centred at the origin, and a point that moves around its edge. The angle θ (theta) is measured from the positive x-axis. As the point moves, its horizontal position is called cosine of θ, and its vertical position is called sine of θ.

That's really the whole idea — sine and cosine are just coordinates of a point going around a circle. Tangent is the ratio of the two: tan θ = sin θ / cos θ, and it tells you the slope of the line from the origin to that point.

Switch to the Unit Circle tab to move the point yourself and watch the values update.

Core identities

Pythagorean identity

Follows directly from the unit circle — the point's coordinates always satisfy the circle's own equation x² + y² = 1.

Angle sum

Tangent

Undefined wherever cos θ = 0 — try dragging the point to 90° or 270° on the Unit Circle tab and watch the tangent value.