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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

71.0. "Trig identity w sqrt(n)" by HARE::STAN () Sat May 26 1984 05:11

The following trigonometric identities are true:

tan 2pi/3  + 4 sin  pi/3  = sqrt(3)

tan 5pi/7  + 4 sin 4pi/7  = sqrt(7)

tan 3pi/11 + 4 sin 2pi/11 = sqrt(11)

Do these generalize in any way?
T.RTitleUserPersonal
Name
DateLines
71.1TURTLE::GILBERTTue Jul 31 1984 16:291
Can someone supply proofs?
71.2AITG::DERAMODaniel V. {AITG,ZFC}:: D'EramoMon Nov 21 1988 21:5433
Newsgroups: sci.math
Path: decwrl!sun!pitstop!sundc!seismo!uunet!attcan!utgpu!watmath!watcgl!watmum!gjfee
Subject: trig expression for sqrt of a prime
Posted: 16 Nov 88 08:02:13 GMT
Organization: 
 
Richard Pavelle asks:
 
2) Is there some general scheme for generating these integer 
   identities?
 
    I was able to generate a trig. expression for sqrt(13)
by starting with the "anomalous factorization" of x^(2*13) - 13^13 .
Then substitute the known root  13^(1/2)*exp(2*Pi*I/26) into the
appropriate factor 
 
     12       11       10        9         8          7          6           5
   (x   - 13 x   + 91 x   - 507 x  + 2535 x  - 10985 x  + 41743 x  - 142805 x 
 
                4            3            2                       
      + 428415 x  - 1113879 x  + 2599051 x  - 4826809 x + 4826809)
 
of x^26 - 13^13 . Next solve for sqrt(13) in terms of cos functions
and rationalize the denominator to obtain
 
   -1 + 4*cos(1/13*Pi) + 4*cos(3/13*Pi) - 4*cos(4/13*Pi) = sqrt(13)
 
A similar procedure (starting with x^14+7^7) leads to
    2*sin(3/7*Pi) + 2*sin(2/7*Pi) - 2*sin(1/7*Pi) = sqrt(7)
 
and
    2*sin(1/11*Pi) + 2*sin(2/11*Pi) + 2*sin(3/11*Pi)
    -2*sin(4/11*Pi) + 2*sin(5/11*Pi) = sqrt(11)
71.3AITG::DERAMODaniel V. {AITG,ZFC}:: D'EramoMon Nov 21 1988 21:573
     There is a little more on this in notes 108.*.
     
     Dan