Testwiki:Reference desk/Archives/Mathematics/2022 January 12

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January 12

The sign of harmonic addition theorem

Template:Plain link has the following equations Template:NumBlk Template:NumBlk given that $a\ne 0$. In addition, the following equation can be found in the Template:Plain link Template:NumBlk which implies Template:NumBlk Now, consider the case Template:NumBlk Template:NumBlk According to Template:EquationNote and Template:EquationNote, $c$ should be positive. According to Template:EquationNote, Template:EquationNote and Template:EquationNote, $c$ should be negative. Aforementioned results of $c$ seem to be inconsistent. So what mistakes have I made? - Justin545 (talk) 17:36, 12 January 2022 (UTC)

Equation Template:EquationNote is accompanied by $\phi =\arctan (-b/a)$. The function arctan is in principle multivalued, but its value is conventionally restricted to $-\pi /2<\phi <\pi /2$. The sign of c comes from that convention. In your equation Template:EquationNote you break that convention, and if you insist on breaking it, then you must switch the sign in Template:EquationNote. --Wrongfilter (talk) 18:17, 12 January 2022 (UTC)

Thanks you for the help. Indeed, signs of $a$ and $c$ are the same if $-\pi /2<\phi <\pi /2$. And it seems signs of $a$ and $c$ are different if $\pi /2<\phi <3\pi /2$. So, if I understand correctly, how about rewrite Template:EquationNote as

in order to make sure Template:EquationNote and Template:EquationNote are consistent for all angles. - Justin545 (talk) 20:04, 12 January 2022 (UTC)

Or maybe replace $a$ in Template:EquationNote by the RHS of Template:EquationNote:

given that $\cos \phi \ne 0$ - Justin545 (talk) 20:13, 12 January 2022 (UTC)

I'm not sure what you're really trying to achieve and for some reason you always omit the second equation $\phi =\arctan (-b/a)$ (or $\tan \phi =-b/a$, if you prefer). You need two equations to find $(c,\phi )$. --Wrongfilter (talk) 20:31, 12 January 2022 (UTC)

Okay, those are thoughtless replies. I am sorry for bothering you with the replies. - Justin545 (talk) 20:51, 12 January 2022 (UTC)