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I'm aware that the angle in radians is 1 when the arc length is equal to the radius. However, I wonder if this is exactly where radians come from. Have we set radians like this (like we set $i^2$ to $-1$) or is there something more to it?

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    Amongst other things, the derivatives of the $\sin$ and $\cos$ functions become much easier when the arguments are radians rather than degrees or other unit. – Thomas Andrews May 04 '21 at 22:26

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According to Earliest Known Uses of Some of the Words of Mathematics, the term "radian" was coined independently by James Moore and William Thompson (Lord Kelvin) in the 19th century.

Robert Israel
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