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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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Yes your function is periodic on the interval $2\pi$.

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  • $\begingroup$ It's the lingo not the understanding: is it "periodic in y" since the function values y repeat? Thanks! $\endgroup$ – Amanda Aug 23 '20 at 15:58
  • $\begingroup$ Yes your function $y(x)$ is periodic in $y$ on the interval $x \in [ 0, 2 \pi ]$, since $y(0)=y(2\pi)$. $\endgroup$ – ConvexHull Aug 23 '20 at 16:28
  • $\begingroup$ I think $y(x)$ is periodic in $x$. For example, if you had a function $z(x,y)$ and $z(0,y)=z(2\pi,y)$ for all $y$ then you would say $z$ is $2\pi$-periodic in $x$. $y(x)$ being periodic in $y$ is a weird idea unless it is somehow implicitly defined and $y$ is a function of itself. $\endgroup$ – Abdullah Ali Sivas Aug 23 '20 at 17:53
  • $\begingroup$ @AbdullahAliSivas Maybe you are right. I am not an native english speaker. $\endgroup$ – ConvexHull Aug 23 '20 at 18:21
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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.

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