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Homework Statement
Prove that
[ltex]
f\left(x\right) = \cos(x) + \cos\left(\alpha x \right)
[/ltex]
where alpha is a rational number, is a periodic function.
EDIT: Also, what is it's period?
Homework Equations
[ltex]
f\left(x\right + p) = f\left(x\right)
[/ltex]
trig identities
The Attempt at a Solution
First, I used the definition of periodicity, then trig identities and term collection to get
[ltex]
\cos x \cos p - \sin x \sin p - \cos x = \cos \alpha x - \cos \alpha x \cos \alpha p + \sin \alpha x \sin \alpha p
[/ltex]
Since p is a constant (if it exists) and the left side is periodic by definition, the right side and hence the function must be periodic as well, yes?
EDIT: Now for the period, it would seem to need to be greater than 2 pi due to the naked cosine out in front. So, I would suppose that it would be something like [ltex]2 \pi + \frac{2 \pi}{\alpha}[/ltex], right?
EDIT2: WTH is going on, the board keeps eating my latex! (much later) and then it starts working again. How bizarre.
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