# How to sketch $\cos (xy)$ by hand

I've been practicing sketching maps $\mathbb R^2 \to \mathbb R$ and mostly I can do it but this one has me stuck completely.

I want to sketch $\cos (xy)$ by hand.

Here is what I have so far:

For $x=0$ and $y=0$ respectively, we get the constant $1$ map.

Along $x=y$ we get $\cos z^2$ which gets smaller and smaller periods towards infinity.

In fact, somehow along $x=$constant and $y=$constant we get normal cosine.

But now I don't see how I can work out how to draw this map in between the points I know. (even though I think I seem to kind of know it at every point)

How to "guess" what it should look like? How to draw this (on paper)?

Please note that I do know how to use graphic tools to draw functions. This question is exclusively about how to draw it without using software tools.

• Fix $y$, then you have $z=cos(kx)$ along the line $y = k$, now plot $z$ as a function of $x$ for different values of $y$, this will give you an idea of how z changes with $x$. Now fix $x$ and do the same. This should give you a rough picture of what the graph looks like. – Burrrrb Jun 2 '16 at 3:19
• Draw the hyperbolic level curves for the peaks and troughs in a flat plane first. Then try to get those drawn neatly on the xy plane in 3d space at some perspective. Practice this first. Then do the same thing but have the hyperbolic curves appropriately shifted to $z=\pm1$. Practice at different perspectives. Then practice drawing a few cosine curves connecting them. I'll draw a picture and pay it as an answer later. – jdods Jun 2 '16 at 12:03

## 1 Answer

Think about when $xy$ is some constant $k$. Note that the graph $xy=k$ is a rectangular hyperbola. So $z=\cos(xy)$ is going to be a cosine wave with a hyperbolic wave front. In fact, the peaks of this function should occur when $xy=2nπ$ ($n$ is an integer). The valleys occur at $xy=2nπ+π$. (I know the image below is computer generated, but I just wanted to provide a visual.) • Wow, you drew that graph by hand on a piece of paper? I am impressed. – Rudy the Reindeer Jun 2 '16 at 3:43
• @RudytheReindeer Sorry, I just wanted to provide a visual (I can delete it if you want). – Hrhm Jun 2 '16 at 11:45
• No, not at all. This is not my question, this is up to the OP. I am not down/up voting here either. – Rudy the Reindeer Jun 2 '16 at 23:38
• Nice, thanks, I can do it now (I figured it out yesterday). – self-learner Jun 4 '16 at 3:31