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?

The answer is simple, it's a distance measure. When you move in a straight line you use inches or metres, in circles it is radians.  
If you are at Disneyland and ask how far it was to Anaheim Stadium [go, Angels!] and I tell you that from my house it's about 45º, you are probably not going to be happy.  
You want the distance traveled, at 1 mile out from my house, from one point to another. This distance is pi/4 * 1 mi = about 0.8 miles. Enjoy the game.
