Question

Будет ли функция y=cos2x + cos((√3)x) периодической?

Answer 1

нет.

т.к. период cos2x равен П, а cos((√3)x) равен √3П

отношение периодов является иррациональным числом.

Answer 2

To determine if the function $y = \cos(2x) + \cos(\sqrt{3}x)$ is periodic, we need to check if there exists a positive constant $P$ such that the function repeats itself after every $P$ units of $x$.

Let's analyze each term of the function:

1. $y = \cos(2x)$: The function $\cos(2x)$ is periodic with a period of $\frac{\pi}{2}$. This means that it repeats itself every $\frac{\pi}{2}$ units of $x$.

2. $y = \cos(\sqrt{3}x)$: The function $\cos(\sqrt{3}x)$ is also periodic. Its period can be found by setting $\sqrt{3}x$ equal to $2\pi$ (or any multiple of $2\pi$) and solving for $x$: $\sqrt{3}x = 2\pi$ $x = \frac{2\pi}{\sqrt{3}}$

So, the period of $\cos(\sqrt{3}x)$ is $\frac{2\pi}{\sqrt{3}}$.

Now, to check if the combined function $y = \cos(2x) + \cos(\sqrt{3}x)$ is periodic, we need to find the smallest positive constant $P$ that satisfies both $\frac{\pi}{2}$ and $\frac{2\pi}{\sqrt{3}}$ as periods.

The smallest common period $P$ can be found by taking the least common multiple (LCM) of the individual periods: $P = \text{LCM}\left(\frac{\pi}{2}, \frac{2\pi}{\sqrt{3}}\right)$

Since $\frac{2\pi}{\sqrt{3}}$ cannot be expressed as a multiple of $\frac{\pi}{2}$ (and vice versa), the LCM of these two periods is their product: $P = \frac{\pi}{2} \cdot \frac{2\pi}{\sqrt{3}} = \frac{\pi^2}{\sqrt{3}}$

So, the function $y = \cos(2x) + \cos(\sqrt{3}x)$ is periodic with a period of $\frac{\pi^2}{\sqrt{3}}$.