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From: lambert@boring.UUCP (Lambert Meertens)
Newsgroups: net.math
Subject: Re: transcendental numbers
Message-ID: <6729@boring.UUCP>
Date: Wed, 15-Jan-86 22:11:24 EST
Article-I.D.: boring.6729
Posted: Wed Jan 15 22:11:24 1986
Date-Received: Fri, 17-Jan-86 06:43:34 EST
References: <846@spp2.UUCP>
Reply-To: lambert@boring.UUCP (Lambert Meertens)
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Organization: CWI, Amsterdam
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> Is tan 1 transcendental? Does there exist a direct proof that it
> is (without using a big name theorem)?

If we put t = tan 1 and u = exp i, then t = (u^2-1)/(u^2+1).  So the
transcendence of t follows from that of u, which, I think, is an immediate
consequence of a theorem of Gel'fond.  However, that is a big name.

If we let s = sin 1, then t = s/sqrt(1-s^2).  So proving that s is
transcendental should suffice.  This can probably be done by "replaying"
Hermite's proof for e.  That proof is not simple at all, unfortunately.
(The proof that s is irrational is in comparison trivial.)
-- 

     Lambert Meertens
     ...!{seismo,okstate,garfield,decvax,philabs}!lambert@mcvax.UUCP
     CWI (Centre for Mathematics and Computer Science), Amsterdam