Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that $\cos(nt) =P_n(cos(t))$ for all $t \in \mathbb{R}$ , one can prove that $\cos(n)$ is irrational for all $n$. Is there an 'easy' way to prove this without relying on (1).

share|improve this question
1  
You can always just prove (1) by using Lindemann's theorem that if $\alpha$ is algebraic, $e^\alpha$ is transcendental. Then using that $i$ is algebraic and $e^{i}=\cos 1+i\sin 1$, you get (1). –  Adam Hughes Jul 5 at 19:17

1 Answer 1

You can mimic the proof of the irrationality of $e$ relying on the Taylor series of the cosine function, for which: $$\cos(n)=\sum_{k=0}^{+\infty}\frac{(-1)^k n^{2k}}{(2k)!}.$$

share|improve this answer
    
Oh, I was not able to mimic the proof but something along the lines of using the taylor series helped me solve it. –  Panda Bear Jul 6 at 14:41

Your Answer

 
discard

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.