It seems quite arbitrary until you look into the basic principles of differentiation. You are looking to find the slope of a curve, a sine wave say, by making a small triangle and working out the lengths of the sides. Then you find the value as the triangle size approaches zero. The simple answer that d(Sin θ)/dθ = Cos θ relies on using radians to measure that angle.
"www.mash.dept.shef.ac.uk/Resources/sincosfirstprinciples.pdf" presents it more or less as I remember being taught in medieval times by dear old Mr Worthington. Half way through, they make the point that the angle is in radians. If you don't measure the angle in radians, every time you differentiate or integrate a trig function, you get a bizarre constant, depending on which angle units you chose.
The inhabitants of Planet Zog, who use 413 zogdegrees in their circles will also be using radians and not zogdegrees for this reason.