<s>[INST] Understanding proof of period of sin(ax/b) + cos(cx/d) I've been trying to understand this proof for determining that the period of $\sin(\frac{ax}{b}) + \cos(\frac{cx}{d})$ where $a, b, c, d$, are integers and $a \over b$, $c \over d$ are in lowest terms is $\frac{2 \pi *lcm(b,d)}{gcf(a, c)}$. The proof goes like this:


*

*Since $\sin (\frac {ax}{b})$ repeats every $\frac{2b\pi}{a}$ and $\cos (\frac {cx}{d})$ repeats every $\frac{2d\pi}{c}$, both functions complete an integral number of periods from $x = 0$ to $x = z$ if $\frac{z}{\frac{2b\pi}{a}}$ and $\frac{z}{\frac{2d\pi}{c}}$ are integers.

*Since $\frac{az}{2b\pi}$ and $\frac {cz}{2d\pi}$ are both integers, the numerator of $z$ must be divisible by $2\pi$, $b$, and $d$. Thus the numerator of $z$ is at least $2\pi$: times the least common multiple of $b$ and $d$.

*Since $az$ and $cz$ must both be integers, the denominator of z can be no greater than the greatest common factor of a and c. Putting this together, we find the minimum $z$ when the numerator is minimized and the denominator is maximized, or $z = \frac{2 \pi *lcm(b,d)}{gcf(a, c)}$.
I completely understand point 1. For point 2, I understand why $\frac{az}{2b\pi}$ and $\frac {cz}{2d\pi}$ are both integers, but what I don't understand is how it reasons about the numerator of z. I see that $z$ as a whole should be divisible by $2 \pi$, $b$, and $d$ but how does one know that that means $2\pi \times lcm(b,d)$ and not $\pi \times lcm(2, b, d)$? Point 3 is the most confusing; is the conclusion that $az$ and $cz$ must be integers because they are numerators of a fraction in lowest terms? More confusing is the conclusion that the denominator must be the greatest common factor of $a$ and $c$. Does anyone know of another way to explain the steps in this proof?
 [/INST] $\DeclareMathOperator{\lcm}{lcm}$It may help to work instead with the $1$-periodic functions
$$
C(x) = \cos(2\pi x),\qquad
S(x) = \sin(2\pi x),
$$
so that $S(\frac{a}{b}x) + C(\frac{c}{d}x)$ is $\ell$-periodic if and only if $\sin(\frac{a}{b}x) + \cos(\frac{c}{d}x)$ is $2\pi\ell$-periodic.
The question is, what is the (smallest positive) period $\ell$ of $S(\frac{a}{b}x) + C(\frac{c}{d}x)$? The steps of your proof become:


*

*Both $S(\frac{a}{b}x)$ and $C(\frac{c}{d}x)$ complete an integer number of periods for $0 \leq x \leq \ell$ if $m = \frac{a}{b}\ell$ and $n = \frac{c}{d}\ell$ are integers. Particularly, $\ell = \frac{mb}{a} = \frac{nd}{c}$ is rational. Write $\ell = \frac{p}{q}$.

*The numerator $p$ must be divisible by $b$ and by $d$, hence by $\lcm(b, d)$.

*Since $a\ell = mb$ and $c\ell = nd$ are integers, the denominator $q$ is no larger than $\gcd(a, c)$.
The smallest positive rational number satisfying these conditions is found by taking the smallest numerator divisible by $b$ and $d$ and the largest denominator dividing both $a$ and $c$, i.e., taking $p = \lcm(b, d)$, and $q = \gcd(a, c)$, so that $\ell = \frac{\lcm(b, d)}{\gcd(a, c)}$.
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