# Why do we study the projective representations of SO(3) in the context of the theory of angular momentum? [duplicate]

I have been studying the group theoretic formalism of quantum mechanics and I have yet to find a satisfying explanation for the need for projective representations in the theory of angular momentum. I understand the connection between SO(3) and angular momentum, and I understand the the projective representation of SO(3) is given by the usual representation of the double cover, SU(2). The thing that I am missing is $\textit{why}$ the projective representation is important in the first place.

Furthermore, I understand the consequences of studying SU(2) rather than SO(3), namely that for half-integer spin particles a rotation of $2\pi$ under SU(2) transforms the wavefunction to its negative, but for integer spin particles the rotation is equal to the identity, but as I said, I see this a a consequence rather than motivation for studying SU(2) over SO(3). Is there a more satisfying answer to my question other than "we do it this way because it works"? Thanks!