An angle measures rotation from one direction to another. Counterclockwise rotation is usually taken as positive and clockwise rotation as negative.

Degrees and radians

One complete revolution is

$$360^\circ=2\pi\text{ radians},$$

so

$$180^\circ=\pi\text{ radians}.$$

This gives the conversion rules

$$\theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180},$$

$$\theta_{\text{deg}}=\theta_{\text{rad}}\frac{180}{\pi}.$$

For example,

$$60^\circ=\frac{\pi}{3}.$$

Coterminal angles

Angles that differ by a whole number of revolutions point in the same final direction. Thus

$$30^\circ,\qquad390^\circ,\qquad-330^\circ$$

are coterminal.

Why radians matter

Radians measure an angle through the arc it cuts from a circle. If a circle has radius $r$ and an angle of $\theta$ radians subtends an arc of length $s$, then

$$s=r\theta.$$

In particular, an angle of one radian subtends an arc whose length equals the radius. This direct relation between angle and length is why radians are the natural angular unit in calculus and physics.