Radian
TrigonometryA radian is a unit of angle measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius.
Formula
\theta \text{ (radians)} = \frac{\text{arc length}}{\text{radius}}; \quad 2\pi \text{ radians} = 360 \text{ degrees}
Definition
A radian is a way to measure angles using the radius of a circle. One radian is the angle you get when the curved arc along the edge of a circle has the same length as the radius.
Example
On a circle with radius $5$ cm, if you walk $5$ cm along the edge, you have turned through exactly $1$ radian (about $57.3^\circ$). A full circle $= 2\pi \approx 6.28$ radians.
Key Insight
There are about $6.28$ radians in a full circle because the circumference ($2\pi r$) divided by the radius ($r$) $= 2\pi \approx 6.28$.
Definition
One radian is the central angle that intercepts an arc equal in length to the radius. Since circumference $= 2\pi r$, a full revolution $= 2\pi$ radians $= 360^\circ$. So $1$ radian $= 180^\circ/\pi \approx 57.296^\circ$.
Example
$\pi/6$ radians $= 30^\circ$, $\pi/4 = 45^\circ$, $\pi/3 = 60^\circ$, $\pi/2 = 90^\circ$, $\pi = 180^\circ$, $2\pi = 360^\circ$. Radians are preferred in calculus because they make derivative formulas for trig functions simpler.
Key Insight
In degrees, $d/dx[\sin(x)] = (\pi/180)\cos(x)$. In radians, $d/dx[\sin(x)] = \cos(x)$. Radians remove the messy conversion factor, which is why all of higher math uses radians.
Definition
Radian measure is the natural dimensionless angle unit defined by $\text{arc length}/\text{radius}$. It makes all calculus identities clean: the limit as $x \to 0$ of $\sin(x)/x = 1$ holds only in radians. Radian measure is the unique parametrization of $S^1$ that makes the exponential map $e^{ix}$ an isometry.
Example
The Taylor series $\sin(x) = x - x^3/6 + x^5/120 - \ldots$ requires $x$ in radians. If $x$ were in degrees, the series would need a factor of $(\pi/180)$ throughout, destroying the elegance of the formula.
Key Insight
Radian measure is "natural" in the sense that it emerges from the geometry of the circle without any arbitrary convention. The choice of $360$ degrees is historical (Babylonian base-$60$ system), but radians are dictated by the geometry itself.