My attempt:

$\displaystyle Arg\left ( \frac{-1}{i\omega+1} \right )=Arg(-1)-Arg(i\omega+1)$

I looked at

this Wikipedia article under the section "Computation" where it says that:

$\displaystyle Arg(x+iy)=\pi/2-arctan(x/y)$

for $\displaystyle y>0$

I'd assume this is the definition I need to look at because in the formula, it says that $\displaystyle \omega>0$ and the omega is the imaginary part.

So that leaves me with:

$\displaystyle \varphi(\omega)=\pi/2-arctan(1/\omega)$

So I've figured out this much. I'm a bit confused as what to do next. What about $\displaystyle Arg(-1)$?