Daniel LakelandFeb 23

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?

Show threadWesley Burr

@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.

Feb 23 at 12:46pmMastodon for Android
Show threadWesley BurrFeb 23

@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.

Show threadWesley BurrFeb 23
@dlakelan (and of course, period is actually 2*pi/B, not straight B)
Show threadDaniel LakelandFeb 23

@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

Show threadWesley BurrFeb 23
@dlakelan Totally fair. As you said, foo. Relative phase and absolute phase are different things. I believe the default in these calculus courses is relative phase, chosen so you're measuring the something on the scale of one "period unit" rather than an absolute offset.
Show threadDaniel LakelandFeb 23

@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

Show threadWesley BurrFeb 23

@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.