A related interesting fact is that small angular motions compose almost like vectors, order does not matter (i.e. they are commutative). This makes differential kinematics easier to deal with when dealing with polar or cylindrical coordinate systems.

Large angular deflections while being linear transforms, do not in general commute.

It will spoil the linear relation in your elegant expression, but a slightly better approximation for cos for small θ is

    1 - 0.5θ²