why does $\sin\left(\sin x\ +\ \cos y\right)=\cos\left(\sin\left(x\cdot y\right)+\ \cos\ x\right)$ look so weird?

Question

recently, I was messing around on an online graphing calculator, and I came across this equation:

$\sin\left(\sin x\ +\ \cos y\right)=\cos\left(\sin\left(x\cdot y\right)+\ \cos\ x\right)$

the online calculator can't resolve the equation in as much detail as it would resolve any other questions.

I would like to know what causes things like that to happen on a graphing calculator.

equation: $\sin\left(\sin x\ +\ \cos y\right)=\cos\left(\sin\left(x\cdot y\right)+\ \cos\ x\right)$

Answer 1

No answer, but I made a higher accuracy plot in matlab Here a couple very interesting features are visible:

• The solution seems to be localized near the points $y=2k\pi$, $x = (2k+\tfrac 12)\pi$, $k\in\mathbb Z$

• Each cell seems to consist of a bunch of deformed circles

• In fact it seems that in each cell the deformed circles come in 2 groups that are slightly offset from each other.

• The number of deformed circles increases with $|k|$, in fact in the picture the number of circles per cell is (if I didn't miscount)

$$\begin{array}{}19&10&8&14\\ 18&8&3 &13\\18&10&8&14 \end{array}$$ Maybe a bit of asymptotic analysis near the central points can illuminate some of these questions.