How to prove periodicity of a trigonometric function

Question

$f(x)= \sin(2x)+3\cos(8x)$

Is the function periodic ?

What I did is equalize $f(x)=f(x+T)$ and after noting that $\sin(2x)=\sin(2T)=\sin(8x)=0$ and $\cos(2x)=\cos(2T)=\cos(8x)=1$ we get that both sides of the equation equal 3.

Is that enough to show that the function is periodic ?

Thanks.

Answer 1

no need to be too complicated here:

both $sin \;x$ and $cos \;x$ are perodic, each with period $2\pi$

so $sin \; 2x$ is periodic, with period $\frac{2\pi}{2} = \pi$

and $cos \; 8x$ is periodic, with period $\frac{2\pi}{8} = \frac{\pi}4$

can you find the least common multiple of $\pi$ and $\frac{\pi}4$?

do you see why that must be the period of $\sin(2x)+3\cos(8x)$?

Answer 2

$$\begin{align}f(x+\pi)&=\sin(2(x+\pi))+3\cos(8(x+\pi))\\ &=\sin(2x+2\pi)+3\cos(8x+8\pi)\\ &=\sin(2x)\cos(2\pi)+\cos(2x)\sin(2\pi)+3\cos(8x)\cos(8\pi)-3\sin(8x)\sin(8\pi)\\ &=\sin(2x)+3\cos(8x)\\ &=f(x)\end{align}$$

Answer 3

periodic says that for some $T$ ,given function repeats its value after $T$,or

$f(x)=f(x+T)$

now in your case we know that $sin(x)=sin(x+2*\pi)$ and $cos(x)=cos(x+2*\pi)$

but for some of periodic function,please read this

If the periods of two periodic functions do not have a common multiple, then their sum is not periodic. Perhaps the simplest example is $\sin(x) + \sin(\pi x),$ whose terms have least periods 2π and 2 respectively.

in your case period of first function is $2*\pi/2=\pi$ and period of second function is $2*\pi/8=\pi/4$

can you continue?