Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
what would be the correct hypothesis based on the question then ? – Comic Book Guy 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.