If angles are unitless, what are radians and degrees?

Question

In $s=r\theta$, is $\theta$ unitless? Isn't it measured in radians? If angles are unitless, what are radians and degrees? Please clarify.

Answer 1

Angles are usually measured in degrees, or radians.

By convention the angle corresponding to the full circle is divided in 360 equal parts, of degrees. The convention goes back to ancient times and 360 is convenient because it has a lot of divisors. Thus it is measure that it is useful for practical purposes.

Mathematicians do measure angles in radians. By definition the radians measure of an angle is the length of the radius $1$ arc defined by the angle itself.

Thus, the full circle is an angle of $2\pi$ radians as that is the circumference of a circle of radius $1$. Mathematicians prefer radians because the formulae

$$ \frac d{dx}\sin(x)=\cos(x),\qquad \frac d{dx}\cos(x)=-\sin(x) $$ hold only when $x$ is measured in radians.