Show that cos(x) + cos([itex]\alpha[/itex]x) is periodic if [itex]\alpha[/itex] 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([itex]\alpha[/itex](x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos([itex]\alpha[/itex]x)cos([itex]\alpha[/itex]p)-sin([itex]\alpha[/itex]x)sin([itex]\alpha[/itex]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,
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([itex]\alpha[/itex](x+p))
= [cos(x)cos(p)-sin(x)sin(p)] + [cos([itex]\alpha[/itex]x)cos([itex]\alpha[/itex]p)-sin([itex]\alpha[/itex]x)sin([itex]\alpha[/itex]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,
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