Basically wondering if it's possible to evaluate or approximate products like,
$$\prod_{j=1}^{m}\sin(\sqrt{j}x)$$ I thought that $$\sin^{m}(\sqrt{0.5m}x)$$ may be a reasonal approximation, since it uses the average value of $j$, however I'm not at all experienced with these kinds of sums so wouldn't know where to start to see if it is reasonable
Assuming $x$ is real, your product converges to $0$ as $m \to \infty$. How fast it converges, however, will depend on how close $\sqrt{j} x$ gets to integer multiples of $\pi$. This is too delicate for there to be a closed-form formula. If $(x/\pi)^2$ is rational, say $a/b$, then your product will be $0$ for $m \ge ab$.