It appears that the sine transform can be computed from the Fourier transform.
Example with a simple function
f(t) = exp(-t)*heaviside(t);
Fourier transform of f(t)
F(xi) = subs(fourier(f(t),t,w),w,2*sym(pi)*xi)
F(xi) =

The sine transform is then
S(xi) = simplify(-F(xi) + F(-xi))/2i
S(xi) =

Compare to the sine transform computed from the defining integral
S(xi) = int(f(t)*sin(2*sym(pi)*xi*t),t,-inf,inf)
S(xi) =
