I know sine is $2\pi$-periodic, but some people say the motion is "periodic in..."
Would it be periodic in $y$?
If we move along the $x$-axis, the function values $y = \sin(x)$ will repeat, so I think it's "periodic in $y$".
Is this correct?
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Don't get hung up on these details of language. One might argue that because $x$ is the independent variable, it moves in arbitrary ways (can be chosen arbitrarily) and so it would make sense to say "the sine function is periodic in $y$". But one could also say "the sine function is a periodic function of $x$" (because it is a function of $x$).
In the end, whatever you say, people will understand what you're trying to say: "the sine function is a periodic function" omits all details and yet, everyone understand what you are trying to say.