# Generalized Harmonic Addition Theorem: $a \cos n x + b \sin m x = ???$ for $m\neq n$

I just learned about the Harmonic Addition Theorem which applies to functions of the form: $$a\cos x+b\sin x = \operatorname{sgn}(a)\sqrt{a^2+b^2}\;\cos\left(x-\arctan\frac{b}{a}\right)$$

My question is:

Is there a way of generalizing the Theorem to $$a\cos nx + b\sin mx = \text{???}$$

I know, if $n=m$, then

$$a\cos nx +b\sin mx = \operatorname{sgn}(a)\sqrt{a^2+b^2}\;\cos\left(nx-\arctan\frac{b}{a}\right)$$

but that's common sense. I want to know when $n$ and $m$ are not equal. Thanks!

• 1) No simple formula, because when you add 2 different frequencies, you cannot await simple phenomena 2) Besides, nobody uses such a complicated term as "Harmonic addition theorem" : I would be curious to know on which book you have seen such a phraseology ... – Jean Marie Jan 18 '18 at 5:48
• In addition to Jean's comment, if $n$ and $m$ happen to be incommensurable, you can't even hope that the sum is periodic. – Robert Wolfe Jan 18 '18 at 8:17
• @JeanMarie What are the other names of the theorem are there? I was just doing some self-research. – Tom Himler Jan 20 '18 at 19:08
• 1) Most theorems have no name. 2) This result is at most a lemma, not a "theorem" .... 3) Besides, I understand now that you have chosen "harmonic" by thinking to the decomposition into harmonics. – Jean Marie Jan 20 '18 at 20:48
• @JeanMarie The only reason I called it a theorem is because that's what websites like Wolfram Alpha call it. But thanks for your words! – Tom Himler Jan 21 '18 at 0:00