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I am trying to show that it is not possible, unless $\theta$ is an integer, that $\sin n\pi\theta$ or $\cos n\pi\theta$ should be nearly equal, for all large $n$, to one or other of two values $a$, $b$.

I am unsure if the wording of the question is fully clear to me. Can the hypothesis be stated as $$\cos n\pi\theta - \sin n\pi\theta \rightarrow 0$$ only when $\theta \in Z$.

Any help or clues would be highly appreciated.

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2 Answers 2

Take $$e^{-in\theta}+ie^{in\theta}=\cos(n\theta)-i\sin(n\theta)-\sin(n\theta)+i\cos(n\theta)$$ $$e^{-in\theta}+ie^{in\theta}=\cos(n\theta)-\sin(n\theta) +i(\cos(n\theta)-\sin(n\theta)).$$

If you supposed that your expression was converging towards zero then the above expression would vanish. This is equivalent to saying$$e^{2in\theta}\to -i$$ which is absurd.

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what would be the correct hypothesis based on the question then ? –  Hardy Apr 22 '12 at 4:50
    
Shouldn't it be $+i$? –  anon Apr 22 '12 at 7:56

You might want to use a trig id on THIS equation (not sure if you want a more general equation, or not) multiply by $\cos(\pi/4)=\sin(\pi/4)$ and divide by the same, then your function is $\cos(n\pi\theta+\pi/4)$. I hope this helps.

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