Arctangents and showmanship

This story comes from Fall 1996, my first semester as a college professor. I was teaching a Precalculus class, and the topic was vectors. I forget the exact problem (believe me, I wish I could remember it), but I was going over the solution of a problem that required finding \tan^{-1}(7). I told the class that I had worked this out ahead of time, and that the approximate answer was 82^o. Then I used that angle for whatever I needed it for and continued until obtaining the eventual solution.

(By the way, I now realize that I was hardly following best practices by computing that angle ahead of time. Knowing what I know now, I should have brought a calculator to class and computed it on the spot. But, as a young professor, I was primarily concerned with getting the answer right, and I was petrified of making a mistake that my students could repeat.)

After solving the problem, I paused to ask for questions. One student asked a good question, and then another.

Then a third student asked, “How did you know that \tan^{-1}(7) was 82^o?

Suppressing a smile, I answered, “Easy; I had that one memorized.”

The class immediately erupted… some with laughter, some with disbelief. (I had a terrific rapport with those students that semester; part of the daily atmosphere was the give-and-take with any number of exuberant students.) One guy in the front row immediately challenged me: “Oh yeah? Then what’s \tan^{-1}(9)?

I started to stammer, “Uh, um…”

“Aha!” they said. “He’s faking it.” They start pulling out their calculators.

Then I thought as fast as I could. Then I realized that I knew that \tan 82^o \approx 7, thanks to my calculation prior to class. I also knew that \displaystyle \lim_{x \to 90^-} \tan x = \infty since the graph of y = \tan x has a vertical asymptote at x = \pi/2 = 90^o. So the solution to \tan x = 9 had to be somewhere between 82^o and 90^o.

So I took a total guess. “84^o,” I said, faking complete and utter confidence.

Wouldn’t you know it, I was right. (The answer is about 83.66^o.)

In stunned disbelief, the guy who asked the question asked, “How did you do that?”

I was reeling in shock that I guessed correctly. But I put on my best poker face and answered, “I told you, I had it memorized.” And then I continued with the next example. For the rest of the semester, my students really thought I had it memorized.

To this day, this is my favorite stunt that I ever pulled off in front of my students.

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  1. My Favorite One-Liners: Part 69 | Mean Green Math

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