rapidly converging series for cos (n * atan(3/2)) ?

Many thanks to all who helped with the previous question.

I wondered whether there is a nice rapidly converging series (NRCS) for
the real and imaginary parts of a power such as, say, (3+2i)^n.

(3+2i)^n = (13^(n/2), n * atan(3/2))
        =  13^(n/2) cis (n * atan(3/2))

So the q boils down to, are there NRCSs for

cos (n * atan(3/2))      and      sin (n * atan(3/2))

TIA

Chris