Given complex number is $z=\frac{3+2i\mathrm{s}\mathrm{i}\mathrm{n}\theta }{1-2i\mathrm{s}\mathrm{i}\mathrm{n}\theta }$

$\Rightarrow z=\frac{(3+2i\mathrm{s}\mathrm{i}\mathrm{n}\theta )(1+2i\mathrm{s}\mathrm{i}\mathrm{n}\theta )}{(1+4{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta )}$

Since given complex number is purely imaginary

$\Rightarrow \mathrm{R}\mathrm{e}\left(z\right)=0$

$\Rightarrow 3-4{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta =0$

$\Rightarrow \mathrm{s}\mathrm{i}\mathrm{n}\theta =\pm \frac{\sqrt{3}}{2}$

$\Rightarrow \theta =\frac{-\pi }{3},\frac{\pi }{3},\frac{2\pi }{3}$

$\Rightarrow$ sum of all possible values of $\theta =\frac{2\pi }{3}$ .