Let us consider the set of complex numbers and the binary operation $\circ$ defined by
where $\Theta(z_b)$ is the argument of the complex number $z_b$.
Explain whether the set of complex numbers together with the binary operation $\circ$ forms a monoid.
I know I need to prove whether it satisfies closure, associativity and identity.
I'm not sure how to start with showing this, I've tried to put it into polar form which I get $z_a\circ z_b=|z_a|(\cos \Theta(z_b)+i\sin \Theta(z_b))$ but I have no idea what to do next.
Any help will be appreciated, thanks.