Radians. Degrees is just base 60 autism and should only be kept in seconds.

Radians because series expansions.

>>7964321
I use degrees when constructing frames and what not because the numbers are easier and I use radians when analyzing rotational motion because it makes the calculus easier.

Arcseconds

>>7964321
I personally am partial to radians farenheit.

Radians, so that for a triangle with angles α,β and γ, the area is [math]\pi -( \alpha + \beta + \gamma )[/math]