Why do we require radians in calculus? I think this is just something I've grown used to but can't remember any proof.
When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?


Radians make it possible to relate a linear measure and an angle
  measure. A unit circle is a circle whose radius is one unit. The one
  unit radius is the same as one unit along the circumference. Wrap a
  number line counter-clockwise around a unit circle starting with zero
  at (1, 0). The length of the arc subtended by the central angle
  becomes the radian measure of the angle.

From Why Radians? | Teaching Calculus
We are therefore comparing like with like the length of a radius and and the length of an arc subtended by an angle $L = R \cdot \theta$ where $L$ is the arc length, $R$ is the radius and $\theta$ is the angle measured in radians.
We could of course do calculus in degrees but we would have to introduce awkward scaling factors.
The degree has no direct link to a circle but was chosen arbitrarily as a unit to measure angles: Presumably its $360^o$ because 360 divides nicely by a lot of numbers. 
