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Inspired by this question

The methodology used in the answers to that question can be used to prove that $\cos(A\pi)$ is an algebraic number for all rational numbers $A$. This got me thinking: if A is an irrational algebraic, is $\cos(A\pi)$ algebraic?

Some playing around on Wolfram Alpha told me that for $A=\sqrt{2}, \sqrt{3}, \sqrt{5},\sqrt[\leftroot{2}\uproot{-1}3]{2}$, and $\sqrt[\leftroot{2}\uproot{-1}3]{5}$ are all transcendental numbers, but no reasoning for this is provided. Looking at Wikipedia, numbers of this type are not listed as known transcendental numbers.

This leaves me with two questions:

  1. Can anyone provide a source that $\cos(\sqrt{2}\pi)$, for example, is transcendental?
  2. Are there any irrational algebraics for which $\cos(A\pi)$ is algebraic? This is equivalent to the question of if for all algebraic numbers X between -1 and 1, is $\frac{\cos^{-1}(X)}{\pi}$ either a rational number or a transcendental number, or can it also be an irrational algebraic number?
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  • $\begingroup$ @LeeMosher: Your suggested proof (if I follow correctly) is that there exist algebraic numbers B for which $\cos(A\pi)=B$ returns a value of A which is irrational. However, it is possible that for all of these, A is a transcendental number. For example, $\frac{cos^{-1}(\frac{1}{5})}{\pi}$ is an irrational number, but it is quite possible that it is transcendental $\endgroup$
    – Moko19
    Aug 4, 2020 at 18:03

1 Answer 1

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Gelfond-Schneider theorem
If $a,b$ are algebraic; $a \ne 0,1$; $b$ irrational then (any value of) $a^b$ is transcendental.

Let $A$ be algebraic irrational. We claim that $\cos(A\pi)$ transcendental. Assume not: $\cos(A\pi)$ is algebraic. Then $\sin(A\pi) = \pm\sqrt{1-\cos^2(A\pi)}$ is algebraic. Then $e^{iA\pi}=\cos(A\pi)+i\sin(A\pi)$ is algebraic. Also $e^{iA\pi} \ne 0$ and $e^{iA\pi} \ne 1$. Now $2/A$ is algebraic irrational. Note $$ \big(e^{iA\pi}\big)^{(2/A)} = e^{2\pi i} = 1 $$ is algebraic. This contradicts Gelfond-Schneider.

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