The unit circle is the circle of radius $1$ centered at the origin:

$$x^2+y^2=1.$$

An angle $\theta$ in standard position begins on the positive $x$-axis. Its terminal side meets the unit circle at a point whose coordinates are

$$(\cos\theta,\sin\theta).$$

This gives sine and cosine a geometric meaning for angles beyond the acute angles of a right triangle.

Signs in the four quadrants

The signs of cosine and sine are the signs of the $x$- and $y$-coordinates. In quadrant I both are positive; in quadrant II cosine is negative and sine positive; in quadrant III both are negative; and in quadrant IV cosine is positive and sine negative.

Common angles

Right-triangle geometry gives exact coordinates for important angles. For example,

$$\cos\frac{\pi}{3}=\frac12,\qquad \sin\frac{\pi}{3}=\frac{\sqrt3}{2}.$$

Symmetry then gives corresponding values elsewhere on the circle. For example,

$$\cos\frac{2\pi}{3}=-\frac12,\qquad \sin\frac{2\pi}{3}=\frac{\sqrt3}{2}.$$

Periodicity

Adding a full revolution returns to the same point, so

$$\cos(\theta+2\pi)=\cos\theta,$$

$$\sin(\theta+2\pi)=\sin\theta.$$

The unit circle therefore connects angle, coordinates, symmetry and periodic behavior in a single picture.