Raised-Cosine Filter - calculating phase response
Just like the sinc pulse, a raised cosine pulse with zero delay (centered around $t=0$) has a real Fourier transform. You can find its expression in section 11.4 of "Telecommunications Breakdown" by Johnson and Sethares (a free draft version is available here).

For a time-shifted pulse, you can apply the time-shift property of the Fourier Transform.