My Favorite One-Liners: Part 67

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here are a couple of similar problems that arise in Precalculus:

  1. Convert the point (5,-5) from Cartesian coordinates into polar coordinates.
  2. Convert the complex number 5 - 5i into trigonometric form.

For both problems, a point is identified that is 5 steps to the right of the origin and then 5 steps below the x-axis (or real axis). To make this more kinesthetic, I’ll actually walk 5 paces in front of the classroom, turn right face, and then walk 5 more paces to end up at the point.

I then ask my class, “Is there a faster way to get to this point?” Naturally, they answer: Just walk straight to the point. After some work with the trigonometry, we’ll establish that

  1. (5,-5) in Cartesian coordinates is equivalent to (5\sqrt{2}, -\pi/4) in polar coordinates, or
  2. $5-5i$ can be rewritten as 5\sqrt{2} [ \cos(-\pi/4) + i \sin (-\pi/4)] in trigonometric form.

Once this is obtained, I’ll walk it out: I’ll start at the origin, turn clockwise by 45 degrees, and then take 5\sqrt{2} \approx 7 steps to end up at the same point as before.

Continuing the lesson, I’ll ask if the numbers 5\sqrt{2} and -\pi/4, or if some other angle and/or distance could have been chosen. Someone will usually suggest a different angle, like 7\pi/4 or 15\pi/4. I’ll demonstrate these by turning 315 degrees counterclockwise and walking 7 steps and then turning 675 degrees and walking 7 steps (getting myself somewhat dizzy in the process).

Finally, I’ll suggest turning only 135 degrees clockwise and then taking 7 steps backwards. Naturally, when I do this, I’ll do a poor man’s version of the moonwalk:

For more information, please see my series on complex numbers.

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