Why radians are natural

Remark — Why radians are "natural"

The radian is the only unit of angular measure for which the length of an arc equals the product of the angle and the radius: $\ell = \alpha\cdot R$ (with $\alpha$ in radians). If we used degrees, we would have to write $\ell = \frac{\pi}{180}\,\alpha_{\text{gradi}}\cdot R$, with a conversion factor always present. With radians the factor disappears. For the same reason, all the formulae of analysis (derivatives of sine and cosine, Taylor series, integrals) take their simplest form when angles are in radians.

Links

Topics: Trigonometry
Concepts: Radian