# Is the sine function periodic in $x$ or $y$?

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?

Yes your function is periodic on the interval $$2\pi$$.
• Yes your function $y(x)$ is periodic in $y$ on the interval $x \in [ 0, 2 \pi ]$, since $y(0)=y(2\pi)$. – ConvexHull Aug 23 '20 at 16:28
• 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. – Abdullah Ali Sivas Aug 23 '20 at 17:53
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 $$x$$).