Can Eulers product for $\sin x$ be generalised for any periodic function?

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$$\sin x=x\prod _{n=1}^{\infty }\left(1-\left(\frac{x}{n\pi }\right)^2\right)$$

Can Eulers product for $\sin x$ be generalised for any real periodic function with period p and a zero 'a'?

If so please explain the process for doing so.

Thanks ☺