Is it possible to write $\prod_{x=1}^5 \sin(x) $ as a factorial?

Question

$$\prod_{x=1}^5 \sin(x) $$ is actually equal as writing $$\sin(5)\cdot\sin(4)\cdot\sin(3)\cdot\sin(2)\cdot\sin(1)$$ I think there is no way of writing it as factorial? Or is there any?

Please help, Thanks

Answer 1

Using Pochhammer symbols, you have

$$P_n=\prod_{x=1}^n \sin(x)=2^{-(n+1)} e^{-\frac{1}{2} i n (n-\pi +1)} \left(-1;e^i\right){}_{n+1} \left(e^i;e^i\right){}_n$$ and so, you look for $k$ such that $$k_n!=P_n \qquad \text{or better} \qquad \Gamma(k_n+1)=P_n$$

This is a difficult numerical problem since, as @K.defaoite already commented, the solution is not unique. But this is doable (do not wait for a closed form solution).