Wikipedia:Reference desk/Archives/Mathematics/2016 January 13

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January 13

Trigonometry identities

Does the following trigonometry identity holds

$\sin ^{2}A+\cos ^{2}A=1\$

if A is 30i degrees? (Thirty imaginary degrees) 175.45.116.66 (talk) 01:46, 13 January 2016 (UTC)

Yes. Sin and Cos are entire, so $\sin ^{2}z+\cos ^{2}z-1$ is also entire. For every real z $\sin ^{2}z+\cos ^{2}z-1=0$, so it follows that $\sin ^{2}z+\cos ^{2}z-1=0$ for every complex z as well.

You could also show it directly by starting with $\sin z={\frac {e^{iz}-e^{-iz}}{2i}},\ \cos z={\frac {e^{iz}+e^{-iz}}{2}}$ and doing some algebra. -- Meni Rosenfeld (talk) 01:59, 13 January 2016 (UTC)

I'm not so sure it's as simple as appealing to the entirity of sin and cos (but the direct proof is legit). How do you know that the "actual" formula isn't $\sin(A)*{\overline {\sin(A)}}+\cos(A)*{\overline {\cos(A)}}=1$? Robinh (talk) 02:07, 13 January 2016 (UTC)

The function $z\mapsto {\overline {z}}$ is not analytic. --Trovatore (talk) 04:03, 13 January 2016 (UTC)

Actually you can just appeal to sin and cos being entire. From the Identity theorem, two entire functions (sin²+cos² and 1) which agree on the real line also agree on the complex plane. Kind of like using a sledge hammer to swat a mosquito but it gets the job done. --RDBury (talk) 05:13, 13 January 2016 (UTC)

I am not making myself clear. Suppose we have $f(z)=\sin(A)*{\overline {\sin(A)}}+\cos(A)*{\overline {\cos(A)}}$, and then observe that $f(z)=1$ for z on the real axis (this would be the observation that $\sin ^{2}x+\cos ^{2}x=1$ for reals). Then cos and sin might be analytic but f(z) isn't, and one cannot deduce that f(z)=1 for general complex z. I'm just making the point that there is more than one way to generalize $\sin ^{2}x+\cos ^{2}x=1$ from the reals to the complex plane. Robinh (talk) 08:43, 13 January 2016 (UTC)

(ec) You ment to write $f(z)=\sin(z)*{\overline {\sin(z)}}+\cos(z)*{\overline {\cos(z)}}$. Bo Jacoby (talk) 10:19, 13 January 2016 (UTC).

But your f isn't analytic because of you're using conjugation. You can extend many ways from the reals to the imaginaries non-analytically, but in those cases the identity theorem doesn't apply. --RDBury (talk) 10:09, 13 January 2016 (UTC)

User:Robinh, is it possible that you missed that "entire" means "analytic everywhere"? Meni wikilinked the term, but you might have thought you knew what it meant so you didn't follow the link. The cosine and sine functions are entire, so cos^2+sin^2 is also entire, but your f from your message of 18:43 is not. --Trovatore (talk) 14:22, 13 January 2016 (UTC)

I think Robinh's point is that the original question is ambiguous and might have been about this other function (and in that case the answer is no). Although x² usually means squared norm only where there's no ambiguity. -- BenRG (talk) 20:25, 13 January 2016 (UTC)

Oh, I see. I don't think it's ambiguous. It seems completely well-specified to me. I am unfamiliar with any notational convention interpreting x² as |x|². --Trovatore (talk) 00:53, 14 January 2016 (UTC)

Yes, there may be several reasons for that, including just laziness or a perceived elegance by reducing the number of symbols used in an equation. For the latter reason, I use x^□ for the squared norm and even for x² when x is real and the purpose of the operation is just to map the result to ℝ_≥0. — Sebastian 22:12, 13 January 2016 (UTC)

Relative quantity

Ground (electricity)#Circuit ground versus earth currently reads: "Voltage is a differential quantity." Of course, this has nothing to do with any of the meanings of differential (mathematics). (It's an annoying fad to call difference "differential", even as a noun; presumably because it sounds more educated.) What's the correct term? I thought it was "relative quantity", but that's neither a redirect nor mentioned in physical quantity. — Sebastian 18:12, 13 January 2016 (UTC)

It has nothing to do with fads, but with dictionary definitions: "differential = of, or relating to a difference." (of course, this is consistent with the use of the -al suffix everywhere else in the English language. The conversion of ce to ti is the same as in confidential, credential, substantial, essential...). Voltage is a quantity relating to a difference, so calling it a differential quantity is appropriate. The fact that mathematicians have borrowed the word for a number of specific concepts is their own problem... -- Meni Rosenfeld (talk) 18:36, 13 January 2016 (UTC)

Good points, Meni, you convinced me about the use in the voltage example. (I still think it's a fad to use it as a noun instead of "difference", but that's off topic here.) My bigger question is where such differential quantities are covered. (A purely mathematical instance of such a quantity is the antiderivative, where the fact that it is differential is conventionally expressed by "+C". That you would get a differential quantity through "anti-differentiation" deepens the counterintuitiveness.) I want to link from Ground (electricity) to the appropriate article so readers won't get confused. Is there any article that covers such differential quantities? — Sebastian 21:45, 13 January 2016 (UTC)

You may be right about usage as a noun specifically. Anyway, this is closely related to interval scales, perhaps that's the appropriate term to use/link. -- Meni Rosenfeld (talk) 21:51, 13 January 2016 (UTC)

Thank you, that is indeed the same concept; I'll link to that. — Sebastian 22:12, 13 January 2016 (UTC)