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

For a fixed $n\in \mathbb{N}$, prove that

$\cos(\frac{jr\pi}{n})\neq 1$ if and only if $\gcd(j,2n)=1$, where $1\leq j,r\leq (n-1)$.

share|cite|improve this question

$cos(\frac{jr\pi}{n}) = 1$ if and only if $\frac{jr\pi}{n} = 2k\pi$ for all $k \in \mathbb{Z}$. In particular, take k=1, So $\frac{jr}{n}=2$ i.e. $jr=2n$ $\Rightarrow$ j divides 2n. So, if j>1 $gcd(j,2n)\neq 1$. if j is 1 it is easy to prove.

share|cite|improve this answer
Thanks a lot.................... – user47929 Nov 3 '12 at 19:22

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.