radian

The radian looks like an inconvenient choice until you differentiate something. Define angle as arc length divided by radius and the small-angle limit $\sin\theta \approx \theta$ becomes exact as $\theta \to 0$, which makes $\frac{d}{dx}\sin x = \cos x$ come out clean. In degrees the same derivative is $\frac{\pi}{180}\cos x$, and that stray factor propagates through every series expansion, every oscillation solution and every Fourier transform. Radians are not a convention with a nicer feel; they are the only angle measure in which calculus does not carry luggage.

The definition also makes arc length free. An angle $\theta$ in radians on a circle of radius $r$ subtends an arc of exactly $r\theta$, and sweeps an area of $\tfrac{1}{2}r^2\theta$. A robot arm 0.8 m long rotating 1.2 rad moves its tip 0.96 m, with no conversion. The same identity is why angular velocity in rad/s multiplied by radius gives linear velocity in m/s directly, the relation $v = \omega r$ that underpins every gear, belt and wheel calculation. Because the radian is a ratio of lengths it is formally dimensionless, so it appears and disappears from unit checks at will, which is either elegant or maddening depending on how the sheet is going.