where
a= u/(u^2+v^2)
b= -v/(u^2+v^2)
then
tan(angle)= b/a = -v/u
as said this is optional (and only done as demonstration), we can get angle easier way if we just keep in mind that it will be negative if we use denominator components.
the other way to look at it is that complex number can be expressed as
q= e^j*angle
but we take reciprocal, angle will be negative.
EDIT:
i see that you added comment about circuit, you STILL don't describe it fully. YOUR description does NOT indicate where the output is taken from - capacitor or resistor (that makes the difference).
in case of low pass filter, output is taken from capacitor (which is what I have derived in this exercise).