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Well...

sage: z=QQbar(3+2*i);z
2*I + 3
sage: arg(z)
arctan(0.6666666666666667?)

Note the "?" printed at the end of the decimal approximation. That means that this number is "exact", in the sense that this "quantity" represents an algorithm able to approximate the underlying complex to any precision. At least, that's what Sage says :

sage: arg(z).is_exact()
True

Note that, while there exists a radical expression for z (the original 3+2*i), there cannot be one for arctan(z)`, unless this one happens to be _known_ to be rational, in which case it can be coerced to be rational and a radical expression can be searched. As far as Sage can say, this is not the case...