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From: steve@anasazi.UUCP (Steve Villee)
Newsgroups: net.math
Subject: Re: transcendental numbers
Message-ID: <486@anasazi.UUCP>
Date: Fri, 17-Jan-86 15:42:24 EST
Article-I.D.: anasazi.486
Posted: Fri Jan 17 15:42:24 1986
Date-Received: Mon, 20-Jan-86 05:40:25 EST
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> Is tan 1 transcendental? Does there exist a direct proof that it
> is (without using a big name theorem)?
> 
> 
> -- 
>  gross (Howard Gross)	{decvax,hplabs,ihnp4,sdcrdcf}!trwrb!trwspp!spp2

You may rest assured that tan(1) is transcendental.  In fact, given that

	e**(2*i*x) = (1 - i * tan(x)) / (1 + i * tan(x))

and applying the Lindemann-Weierstrass theorem, it should be clear that
tan(x) is transcendental whenever x is algebraic and nonzero.

I don't know of any way to prove this without using something like
Lindemann-Weierstrass, though.


--- Steve Villee (ihnp4!terak!anasazi!steve)
    International Anasazi, Inc.
    7500 North Dreamy Draw Drive, Suite 120
    Phoenix, Arizona 85020
    (602) 870-3330