arctan( k tan(x) )

I recently had to work with the function

f(x; k) = arctan( k tan(x) )

The semicolon between x and k is meant to imply that we’re thinking of x as the argument and k as a parameter.

This function is an example of the coming full circle theme I’ve written about several times. Here’s how a novice, a journeyman, and an expert might approach studying our function.

  • Novice: arctan( k tan(x) ) = kx.
  • Journeyman: You can’t do that!
  • Expert: arctan( k tan(x) ) ≈ kx for small x.

Novices often move symbols around without thinking about their meaning, and so someone might pull the k outside (why not?) and notice that arctan( tan(x) ) = x.

Someone with a budding mathematical conscience might conclude that since our function is nonlinear in x and in k that there’s not much that can be said without more work.

Someone with more experience might see that both tan(x) and arctan(x) have the form x + O(x³) and so

arctan( k tan(x) ) ≈ kx

should be a very good approximation for small x.

Here’s a plot of our function for varying values of k.

Plot of atan( k tan(x) ) for varying k

Each is fairly flat in the middle, with slope equal to its value of k.

As k increases, f(x; k) becomes more and more like a step function, equal to -π/2 for negative x and π/2 for positive x, i.e.

arctan( k tan(x) ) ≈ sgn(x) π/2

for large k. Here again we might have discussion like above.

  • Novice: Set k = ±∞. Then ±∞ tan(x) = ±∞ and arctan(±∞) = ±π/2.
  • Journeyman: You can’t do that!
  • Expert: Well, you can if you interpret everything in terms of limits.

Related posts

One thought on “arctan( k tan(x) )

  1. Hi John,
    I discovered something interesting involving the tangent-constant commutator atan(k*tan(a)) form. It is this: one can represent an arbitrary rotated ellipse in polar coordinates r(a) using a formula that includes atan(k·tan(a)), where k is the ratio of the long and short axes of the un-rotated ellipse; further, one can represent the tangent angle of that ellipse (which is the derivative of r(a)) as atan(k²·tan(a)).
    I have always found commutators interesting, ever since I learned about them in quantum mechanics. As you may know, another sort of commutator can be used in computer graphics to “zoom” to an arbitrary point in an image.

Comments are closed.