Why is radian so common in maths? I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)
My question: Is it just mathematical convention that radians are much more used in higher maths than degrees or do radians have some intrinsic advantage over degrees?
For me personally it doesn't matter if I write $\cos(360°)$ or $\cos(2\pi)$. Both equals 1, so why bother with two conventions?

It is natural to define 
$$
\sin x=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}\tag1
$$
rather than
$$
\sin x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}\left (\frac {2\pi x}{360}\right)^{2n+1}\tag2.
$$
The function defined by (1) has many nice properties, for example it is the solution of the ODE:
$$f''(x)=-f (x), \quad f (0)=0,f'(0)=1.
$$
The same reason is the background of a special role of the Euler number $e $.
