## Feeling Functions in Your Bones: Arctangent

We have a lot of intuition about the things we do every day. If a mathematical function describes one of these habits, all that intuition applies to understanding the mathematics.

Imagine standing at the edge of a road. Perhaps you’re waiting for a bus or for the traffic light to switch to walk. Cars are zipping by you, cruising at some constant speed. One of these cars catches your eye. Maybe it’s a brand new Tesla, or, even more exotic, a Transition flying car. As you follow it, your head swivels to track it.

We must have experienced this situation or something very similar countless times in our lives. You can probably imagine the situation so well that you don’t even need real cars to simulate your neck’s rotation.

The arctangent!

The arctangent function is usually met for the first time in Algebra 2 or Trigonometry after sine, cosine, and tangent are thoroughly discussed. Students often regard it as somewhat esoteric: It’s not even a “regular” trigonometric function; it’s the inverse of one! But you can use your intuition about watching cars pass by to understand the arctangent function’s salient features.

For definiteness, let’s say that the car starts in the distance to your right. When you look it, your neck will be swiveled almost, but not quite, 90 degrees to the right. As the car approaches, your neck turns, slowly at first, but with increasing speed. As you can imagine, your head will be turning fastest just as the car passes directly in front of you, and then your neck will start slowing its turn rate, mirroring what it did during the car’s approach. As the car continues off into the distance, your head will swivel closer and closer to 90 degrees, but never quite reach it.

Here is a graph of $\arctan(t)$:

(Recall that $\frac{\pi}{2}$ radians corresponds to 90 degrees.)

The fact that your neck turns to almost 90 degrees as the car drives off into the distance corresponds to the mathematical fact $\lim_{t \to \infty} \arctan(t) = \frac{\pi}{2}$. The fact that your head swivels fastest when the car passes just before you corresponds mathematically to the first derivative of the arctangent function (which is $\frac{1}{1+t^2}$) being maximal at $t = 0$. The fact that your head’s turn rate mirrors itself after the car passes before you corresponds to the symmetry $\arctan(t) = -\arctan(-t)$.

Different car speeds and distances from the curb will correspond to stretching the graph out in the horizontal direction. If the car goes the other way, the graph flips about the vertical axis. Different time calibration will correspond to shifting the graph left or right. And if you regard looking directly forward as something other than 0 degrees, the graph shifts up or down correspondingly. Mathematically, these differences generalize the situation to functions of the form $c + \arctan(at + b)$ where $a$, $b$, and $c$ are constants.

If you feared the arctangent function before, I hope you don’t anymore. You’ve already got as much familiarity with it as you do with watching cars pass by!