Le mar. 8 déc. 2020 à 14:10, Bill Gosper <billgosper@gmail.com> a écrit :
Do they even teach the halfangle formulas in Trig anymore?
They're not even in Mathematica 12.2!
There's a difference in "not being taught" and not being "in"(?) a CAS. Often the reason for taking out stuff from the curricula is "nowadays this is done by (pocket) calculators". I don't use Mmca but would be very astonished if it didn't know about. Maybe there's just a new(?) syntax to "force" it to apply the formula?
From my experience with Maple and others, it was never very clear / logical to me what command does what, among "simplify", "rationalize", "expand", "combine", "convert" ... (often with 3rd argument like "exp", "trig", "sincos", ... to say what "transformation" you want), but I usually found what I needed through the online help.
Macsyma had them forever. And they were useful.
Of course! A must in a CAS. Personally I just recall that cos² oscillates twice as fast as cos, and between 1 and 0 instead of 1 and -1.
From there I easily get all half-angle formulas within seconds.
So here I was thinking the kids already knew them, and planning
to ask them why the Trig books never mention "thirdangle" formulas, and why don't the kids derive them for themselves. That's clearly a bit less "trivial", couldn't do that as quickly, would probably require scribbling down a few lines... But I admit I never was a big fan of trig formulas, I rederive most of them each time I need, though Re / Im exp(ix) etc. and I would recommend acquiring that technology rather than to memorize dozens of formulas (if one had to choose between the two options), even for young people who easily memorize. But if they try to invert the triple angle formulas, they hit the casus
irreducibilis bizarreness,
where they need to solve a cubic whose real roots cannot be expressed in
real terms.
Hm, not sure this is understandable. [edit : from the sequel I can guess what you mean]
Kids should meet this while still young enough to struggle against it.
And then they'd know why the Trig books never bring it up. But the books
have an excuse: Bringing it up requires kids to already know complex numbers.
To me that would be more a motivation than an excuse. Lots of prerequisites is not a good way to sell textbooks. e^ix = cos x + i sin x ( the picture of the unit circle ) and (a+ib)(c+id)=ac-bd +i(ad+bc) is not that much of a prerequisite... Certainly worth it -- will bring lots of economy. No more any need to memorize trig formulae (though young brains *should* do that kind of exercise).
But if the booksellers would accept the complex numbers prerequisite,
instead of explaining about the casus pain-in-the-asus, they could actually provide a nifty thirdangle formula!
I totally agree. But AFAIK it's not bookseller's choice, but that of politicians who fix the school programs (OK, that might vary depending on the country...(?)) Sin[x/3] == 1/2 Re[(1 + I Sqrt[3]) (I Cos[x] + Sin[x])^(1/3)],
Eeks! "I Sqrt", "I Cos" look like English phrases, but not like mathematics!!
Out[212]= Sin[π/18] == 1/2 Re[(1/2 + (I Sqrt[3])/2)^(1/3) (1 + I Sqrt[3])]
well, since π is a little less than 3 plus 5%, sin( π / 18 ) is a little less than 1/6 plus 5%, so I'd say 0.166 + 0.008 = 0.174 should do for practical applications... ;-) - M.