The Unit Circle and Radians

Right-triangle ratios only handle angles below 90°. The unit circle removes that limit entirely.

The coordinates are the ratios

On a circle of radius 1, the terminal point has coordinates (cos θ, sin θ). Cosine is across and sine is up.

Any angle at all

Obtuse, reflex and negative angles all have terminal points, so all have sines and cosines. Nothing is out of reach.

Radians

One radian is the angle whose arc equals the radius. A full turn is 2π radians, and 180° is π.

Why radians

Arc length is simply rθ in radians. Degrees need an extra conversion factor everywhere, and calculus formulas assume radians.

Periodicity

Adding a full turn returns to the same point, so sine and cosine repeat every 2π. That repetition is what makes them model cycles.

Ratios freed from the triangle

On the unit circle, the point at angle θ has coordinates (cos θ, sin θ). This defines the functions for every angle — obtuse, reflex and negative — which a right triangle cannot.

Why radians

One radian subtends an arc equal to the radius, which makes arc length simply rθ. Every calculus result assumes radians, and using degrees inserts an awkward constant into every derivative.

Periodicity

Adding 2π returns to the same point, so sine and cosine repeat. That periodicity is why they model tides, sound, seasons and alternating current — anything driven by rotation or oscillation.

Signs come from the coordinates

The sign of sine is the sign of the y-coordinate and the sign of cosine that of x. Quadrant signs need no separate memorising once the coordinate reading is established.