There’s no oops there. Do you really think the problem I’m talking about disappears if you use radians? Do you think radians are magical and make the equations work out differently?
Any unit of measurement is intrinsically arbitrary, since we are the ones who assign them (except perhaps for quantum units). The math doesn’t change because you change the unit of measurement.
A complete revolution is 2π radians. It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.
It follows, via observation, that the 'Radian' relates to pi in a unique way ...