Proving a function is peridoic

[trap101]

Show that cos(x) + cos($\alpha$x) is periodic if $\alpha$ is a rational number.


Ok So I don't think I have ever done a question proving periodicity. But by definition a function f is periodic if:

f(x + p) = f(x)

so then:

f(x+p) = cos(x +p) + cos($\alpha$(x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos($\alpha$x)cos($\alpha$p)-sin($\alpha$x)sin($\alpha$p)


and this is where I am stuck, so from what I've learned I have to solve for p is some fashion, I'm just not sure how,

 

[tiny-tim]

hi trap101!

hint: cos(0) + cos($\alpha$0) = 2

 

[pasmith]

Show that cos(x) + cos($\alpha$x) is periodic if $\alpha$ is a rational number.

The first thing to do with a rational number is write it as $\alpha = r/s$ for co-prime integers r and s.

Ok So I don't think I have ever done a question proving periodicity. But by definition a function f is periodic if:

f(x + p) = f(x)

so then:

f(x+p) = cos(x +p) + cos($\alpha$(x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos($\alpha$x)cos($\alpha$p)-sin($\alpha$x)sin($\alpha$p)


and this is where I am stuck, so from what I've learned I have to solve for p is some fashion, I'm just not sure how,

You already know that $\cos x$ has period $2\pi$ and $\cos(\alpha x)$ has period $(2\pi/\alpha)$. So you need to find $p$ such that $x + p = x + 2n\pi$ and $x + p = x + 2m\pi/\alpha$ for integers n and m.

 

[UltrafastPED]

You have expressions for f(x) [given], and f(x+p) [derived above].
Now assume that alpha is rational, m/n, and then see if there is any period p which satisfies the condition f(x)=f(x+p).