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

= January 12 =

The sign of harmonic addition theorem
✅

has the following equations

given that $$a \ne 0$$. In addition, the following equation can be found in the

which implies

Now, consider the case

According to $$ and $$, $$c$$ should be positive. According to $$, $$ and $$, $$c$$ should be negative. Aforementioned results of $$c$$ seem to be inconsistent. So what mistakes have I made? - Justin545 (talk) 17:36, 12 January 2022 (UTC)
 * Equation $$ is accompanied by $$\varphi = \arctan(-b/a)$$. The function arctan is in principle multivalued, but its value is conventionally restricted to $$-\pi/2 < \varphi < \pi/2$$. The sign of c comes from that convention. In your equation $$ you break that convention, and if you insist on breaking it, then you must switch the sign in $$. --Wrongfilter (talk) 18:17, 12 January 2022 (UTC)
 * Thanks you for the help. Indeed, signs of $$a$$ and $$c$$ are the same if $$-\pi/2 < \varphi < \pi/2$$. And it seems signs of $$a$$ and $$c$$ are different if $$\pi/2 < \varphi < 3\pi/2$$. So, if I understand correctly, how about rewrite $$ as


 * in order to make sure $$ and $$ are consistent for all angles. - Justin545 (talk) 20:04, 12 January 2022 (UTC)
 * Or maybe replace $$a$$ in $$ by the RHS of $$:


 * given that $$\cos \varphi \ne 0$$ - 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 $$\varphi = \arctan(-b/a)$$ (or $$\tan\varphi = -b/a$$, if you prefer). You need two equations to find $$(c, \varphi)$$. --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)