Let's say we alter the definition of the radian, for example 1 radian = 1 degree and there are 360 radians in a circle, then one consequence that I can think of is that the Taylor expansions of trigonometric functions no longer work unless altered accordingly. $$\sin x = x-{x^3\over3!}+{x^5\over5!}-{x^7\over7!}+\cdots $$ should instead, in order to work, become $$\sin x= \left({\pi\over 180}\right)x-\left({\pi\over 180}\right)^3\left({x^3\over3!}\right)+\left({\pi\over 180}\right)^5\left({x^5\over5!}\right)-\cdots$$
My question is: did mathematicians recognize the elegance of defining $1 \operatorname{radian}={180^\circ/\pi}$ in Taylor expansions or also in other stuff (that it simplifies things nicely), or did the said definition come before anything else, or is there other significance to that? It is not important to question an already well-defined structure but I am just curious, thank you.