Radian Measure

Right-triangle trigonometry, built on degree measure, is genuinely limited: a right triangle can only ever contain acute angles, so this approach alone can never define $\sin\theta$ or $\cos\theta$ for an angle of, say, $200^\circ$ or $-40^\circ$. Historically, $360^\circ$ was chosen for a full rotation — a convention tracing back thousands of years to the Babylonians, quite possibly linked to the number of days in a year — and while $360$ divides conveniently into $30^\circ,45^\circ,60^\circ,90^\circ,180^\circ$, the choice is ultimately arbitrary and man-made.

When, in the 17th century, trigonometry was extended into physics and chemistry, those subjects needed trigonometric functions whose domain was the set of real numbers, not a set of angles measured in an arbitrary human unit. The bridge that makes this possible is to measure an angle not in degrees, but by the length of arc it sweeps out on a circle — a radian measure.

Definition 3.1 (Radian). The radian measure of an angle is the ratio of the arc length it subtends, to the radius of the circle in which it is the central angle.

Concretely: take a circle of radius $r$, and let an angle $\theta$ at the centre cut off an arc of length $s$ on the circle. Then

A few important observations:

• It doesn't depend on which circle is chosen. All circles are similar, so for a given central angle, the ratio (arc length)/(radius) comes out the same regardless of the circle's size — exactly why radian measure is well defined as a property of the angle alone.
• One radian, concretely. When the arc length exactly equals the radius ($s=r$), the angle is, by definition, $1$ radian. More generally, $\theta = k$ radians precisely when $s = kr$ — the radian measure counts how many "radius-lengths" of arc the angle sweeps out. …