The Unit Circle and Radians

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

The coordinates are the ratios

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

The quadrants set the signs

Sine follows y and cosine follows x, so each is negative wherever its coordinate is.

Radians

One radian is the angle whose arc equals the radius. A full turn is 2π and a half turn is π.

Reference angles

The acute angle to the x-axis gives the size of the ratio, and the quadrant gives the sign.

Periodicity

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

Trigonometry beyond the triangle

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

Signs come from the coordinates

Sine takes the sign of the y-coordinate and cosine that of x. Quadrant sign rules need no separate memorising once the coordinate reading is established.

Why radians

One radian subtends an arc equal to the radius, making arc length simply rθ. Every calculus result assumes radians, so they become the default measure from here onwards.

Periodicity is why they model repetition

Going round again returns the same values, so sine and cosine repeat forever. That is what makes them the natural functions for tides, sound, seasons and alternating current.