it seems at least superficially plausible that for real $x \ne 0$ then at least one of $\{x, \cos\;x\}$ is transcendental over $\mathbb{Q}$. has this assertion been proved to be true or false?
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6The http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem implies that if $x$ is algebraic, then $\cos(x)$ is transcendental. – Prahlad Vaidyanathan Dec 29 '13 at 05:30
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thank you Prahlad - that reference leads to some interesting reading – David Holden Dec 29 '13 at 06:16
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Prahlad Vaidyanathan's comment answers the question:
The Lindemann–Weierstrass theorem implies that if $x$ is algebraic, then $\cos(x)$ is transcendental.
So your conjecture is correct.
MJD
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