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.
@dlakelan I think he would be "right" under most definitions. Phase is only measured as a standalone when you can write the first piece as a 2*pi*x structure.
If you think of it as a translation in x, followed by a transformation in x, then transformation in y, then translation in y, that's the convention for taking a default trig function to a modified form. So you work in reverse to return it to standard form, then read off the A,B,C,D from A*trig(B*x + C) + D.
@dlakelan so in his case:
* sec(x)
* sec(x-2)
* sec(pi/4(x-2))
Fin.
There's no amplitude transformation or vertical translation. A=D=1.
@wsburr
i find this weird. like, suppose you have
sec(pi/4*x-pi/2)
put x=0 then the value is sec(-pi/2) showing that the function has been shifted by pi/2 to the right. this seems like the meaningful information you would want.
anyway I told him that if someone asks you what is the foo, you need to go find the definition of foo according to the book youre using, its not always the same for all books. so hes gonna get clarity from his teacher
@wsburr
if youre measuring on the scale of 1 period unit, with the period being 8 in this problem, then since the function is shifted by pi/2 and the period is 8, the answer would be pi/2/8 so it'd be pi/16?
no matter what I think of I can't make 2 be the obvious answer
@dlakelan I think the phase is 2 when considered on a standardized period of 2pi.
Then it's pi/4 as frequency (or (2pi)/(pi/4)=8 for period) and the 2 units of translation in the default domain turn into pi/2 units in the absolute domain.
So take pi/2 * (8/(2pi)) = 2. Or just pi/2 * (1/(pi/4)) if that's easier to understand.
Daniel Lakeland
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