$f(x)=\sin(43x)+\cos(2x)$ is periodic function? $f(x)=\sin(43x)+\cos(2x)$ is periodic function.
I got the period of $\sin(43x)$ is $\frac{2\pi/}{43}$ and period of $\cos(2x)$ is $\pi$. Then the period of $f(x)$ is $2\pi$. Am I right? Any comment? Thank you.
 A: The period is the minimum $T$ which is a multiple of $\pi$ and of $\frac{2\pi}{43}$.
And indeed, $T=2\pi$.
A: $43x$ goes from $0$ to $2\pi$ as $x$ goes from $0$ to $2\pi/43$.
$2x$ goes from $0$ to $2\pi$ as $x$ goes from $0$ to $\pi$.
Some number of copies of $2\pi/43$ add up to some number of copies of $\pi$.
Specifically
$$
\underbrace{\frac{2\pi}{43}+\cdots+\frac{2\pi}{43}}_{\text{43 terms}} = \underbrace{\pi+\pi}_{\text{2 terms}}.
$$
That number, $2\pi$, is the smallest common integer multiple of $2\pi/43$ and $\pi$.
With some pairs of numbers it's more complicated.  For example, suppose you have $\cos(42x) + \cos(30x)$.  You have $\gcd(42,30)=6$, so
\begin{align}
42 = \text{something}\cdot6 = 7\cdot 6, & & 7\cdot30 = 5\cdot 42\quad (=210) \\
30 = \text{something}\cdot6= 5\cdot 6, & &
\end{align}
Hence
$$
\underbrace{\frac{2\pi}{30}+\cdots+\frac{2\pi}{30}}_{\text{5 terms}} = \underbrace{\frac{2\pi}{42}+\cdots+\frac{2\pi}{42}}_{\text{7 terms}} \quad = \frac \pi 3.
$$
So $\pi/3$ is the smallest common integer multiple of the two periods, and is therefore the period of $\cos(42x) + \cos(30x)$.
