I was playing around in Desmos, when I came across this graph: $$\cos(\sin x-\sin y)=\frac12.$$
My goal was to come up with reasons for why an equation leads to a graph, but I was stuck on this one. I could not come up with a reasonable explanation for how this equation leads to weird rounded shapes across the plane. If someone could explain I would be grateful.
First visualize the 3D graphs of $f(x,y)=\sin x$ and $g(x,y)=-\sin y$. These graphs form "waves" in the $x$ direction and in the $y$ direction, respectively.
Now consider the 3D graph of their sum, $h(x,y)=f(x,y)+g(x,y)=\sin x-\sin y$. Adding the vertical and horizontal waves together gives us a regular pattern of smooth bumps. Also, since the $\sin$ function's range is $[-1,1]$, the values of $h$ lie in $[-2,2]$.
Now consider your equation, $\cos(\sin x-\sin y)=\frac12$. When $\theta\in[-2,2]$, the equation $\cos\theta=\frac12$ is equivalent to $\theta=\pm\alpha$, where $\alpha=\cos^{-1}\frac12=\frac\pi3\approx1.047$. (You can see this roughly by looking at the graph of the cosine function.) This means your equation is equivalent to $h(x,y)=\pm\alpha$.
In other words, your equation finds the places where the 3D graph of $h$ intersects the planes $z=\alpha$ and $z=-\alpha$. This amounts to a pattern of rings that encircle our (upward and downward) bumps.