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 reasons are mostly the same as the fact that we usually use base $e$ exponentiation and logarithm. Radians are simply the natural units for measuring angles.


The length of a circle segment is $x\cdot r$, where $x$ is the measure and $r$ is the radius, instead of $x\cdot r\cdot \pi/180$.
The power series for sine is simply $\sin(x)=\sum_{i=0}^\infty(-1)^i{x}^{2i+1}/(2i+1)!$, not $\sin(x)=\sum_{i=0}^\infty(-1)^i(x\cdot \pi/180)^{2i+1}/(2i+1)!$.
The differential equation $\sin$ (and $\cos$) satisfies is $f+f''=0$, not $f+f''\pi^2/(180)^2=0$.
$\sin'=\cos$, not $\cos\cdot 180/\pi$.


You could add more and more to the list, but I think the point is clear.
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.