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That is precisely what the article is talking about: math library functions that seem at first glance to be superfluous, until you happen to need exactly the thing they are designed to do.

It's much, much more than that - these are math library functions that do what you want, and are such that the obvious implementations using the obvious math functions are dangerously unstable or inaccurate.

Things would be somewhat more obvious still if we had one function which returned [sin(θ), cos(θ)] (called say, tan2) and another function which did atan2. To be honest, there are undoubtedly more intuitive names for these, but whatever.

Then atan2(tan2(θ)) = θ, and tan2(atan2(y, x)) = [y, x]

I've long thought that the traditional/current form of our trig functions makes it harder for people to learn and use than they should be. Sadly, basic and widespread abstractions are almost impossible to change.

Well its similar in the sense that you lose information that you might sometimes need by doing atan(y/x) to find an angle. The difference is that its about range rather than precision...