$f(x)=\sin(43x)+\cos(2x)$ is periodic function?

Question

$f(x)=\sin(43x)+\cos(2x)$ 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.

Answer 1

$43x$ 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 $\cos(42x) + \cos(30x)$.

Answer 2

The period is the minimum $T$ which is a multiple of $\pi$ and of $\frac{2\pi}{43}$.

And indeed, $T=2\pi$.