was helping my kid with trig, he had a problem like sec(pi/4*x -pi/2) and it wanted the period and the phase shift.
now words aren't always uniformly defined even in math, but to my mind this thing has period 8, and phase shift pi/2 but he wanted to say phase shift was 2 based on rewriting the argument as pi/4*(x-2)
which would you expect to be the definition of phase shift in a precalculus text?
Hey! Have you taught your kid(s) that pi is the wrong circle constant and tau is the correct one?
If not, you should. Sure, it's a fun joke, but tau (since tau "means" one full cycle, whereas pi "means" half a cycle) simplifies a lot of basic trig to the point that it immediately makes sense.
For example, e^ipi = -1 is stupid: since pi is half way around, it leaves you at the left side of the circle, but e^itau = 1, since it brings you back to the start.
I have discussed this with the younger one, but I should mention it to the older one who is doing the trig stuff right now.
My extension of the tau joke is: consider the Voyager interstellar probes. We put all sorts of messages on them to tell aliens how smart we are, including pi to a ridiculous number of decimal places.
So an advanced alien civilization finds it and has trouble figuring out what 3.14159 is. Finally they realize it's half the circle constant, and can't believe any civilization, advanced or otherwise, would waste their time on such a useless thing, so they nuke us.
Daniel Lakeland
David, a Bostonian in Tokyo.