0

Consider the function $$f(x)=\cos(3x)+\sin\left(\frac{x}{2}\right)$$ It's quite easy to find, that $\cos(3x)$ has minimal positive period $\frac{2}{3}\pi$, and $\sin\left(\frac{x}{2}\right)$ has the period $4\pi$. So the least common mutiple of these two period $4\pi$, must be the period of $f(x)$.

But, how can we proof $4\pi$ is mimimal positive period, or find a smaller one? Maybe the question is easy to solve with computer programming, but what I try to find out is the solution without using technology.

One possible way that I have tried is let $x=0$. With the period $T$, the formula can be written, $$f(T)=f(0)$$ $$\cos(3T)+\sin\left(\frac{T}{2}\right)=1$$ Then I am stuck.

N. Owad
  • 6,822
ZhouYang
  • 53
  • 6
  • https://math.stackexchange.com/questions/897987/principal-period-of-sin-frac3x4-cos-frac2x5 – lab bhattacharjee Sep 22 '23 at 12:05
  • @labbhattacharjee thank you! But after looking what you provided, I'm still confused, that why the lcm of the minimal peroiods of individual parts in the function is the LEAST period of the whole function? How can we proof that there's no smaller one? – ZhouYang Sep 22 '23 at 12:17

0 Answers0