Combination of function's roots question

Question

I have a trig function $$\sin(\frac{\pi}{3}x)$$ with roots at 0, 3, 6, 9. I also have a function $$\sin(\frac{\pi}{4}x)$$ with roots at 0, 4, 8, 12. I am looking for a generalized way to combine them as one function in a way that preserves their roots (and doesn't add any), regardless of what happens everywhere else. This is more of a question of the method rather than a specific answer. All creative answers are much appreciated!

Answer 1

Just multiply the two functions. You will get roots only when you have a root of either (or both) of those functions. Also see my comment about the roots of the cosine function.