I know this might sound like a silly question at first. Let me elaborate.
What I mean by 'line of reasoning; here is what the person who defined radians the way they are defined thought to arrive at that definition for radians.
I know about some advantages of using radians. For example, it makes the calculation of arc lengths easier and when we enter the value of $\alpha$ in radians, then $\dfrac{d(\sin\alpha)}{d\alpha} = \cos\alpha$ but when we enter the value of degrees, $\dfrac{d(\sin\alpha)}{d\alpha} = \dfrac{\pi}{180^o}\cos\alpha$ and the former seems simpler.
So, I'd like to know what steps of reasoning could have lead to the present definition of a radian i,e. the angle extended by an arc of length equal to the radius of a circle at the centre of that circle.
Thanks!