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Nested Square Root of Two/Reflected Gray Code Series for Cos(X)

IP.com Disclosure Number: IPCOM000102428D
Original Publication Date: 1990-Nov-01
Included in the Prior Art Database: 2005-Mar-17
Document File: 2 page(s) / 45K

Publishing Venue

IBM

Related People

Brown, PJ: AUTHOR

Abstract

There exists an infinite series consisting of nested square roots (sqrt) of two that converges to Cos(X): Cos(X) = sqrt(2+-sqrt(2+-sqrt(2+- .........))) Cos(X/2) = sqrt( (1 + Cos(X)) / 2 ) {well known} = sqrt( (2 + 2Cos(X)) / 4 ) = sqrt( 2 + 2Cos(X) ) / 2 t therefore 2Cos(X/2) = sqrt( 2 + 2Cos(X) ) 2Cos(X) = sqrt( 2 + 2Cos(2X) ) 2Cos(2X) = sqrt( 2 + 2Cos(4X) ) etc. therefore 2Cos(X) = sqrt( 2 + 2Cos(2X) ) = sqrt( 2 + sqrt( 2 + 2Cos(4X) ) ) = sqrt( 2 + sqrt( 2 + sqrt( 2 + 2Cos(8X) ) ) ) etc. if 45o < X < 90o, Cos(2X) is negative, so - 2Cos(180 - 2X) can be used instead of + 2Cos(2X) therefore 2Cos(X) = sqrt(2+-sqrt(2+-sqrt(2+- . . . . .

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Nested Square Root of Two/Reflected Gray Code Series for

Cos

(X)

       There  exists  an infinite series consisting of nested
square roots (sqrt) of two that converges to Cos(X):
      Cos(X) = sqrt(2+-sqrt(2+-sqrt(2+- .........)))
            Cos(X/2) = sqrt( (1 +  Cos(X)) / 2 ) {well known}
                     = sqrt( (2 + 2Cos(X)) / 4 )
                     = sqrt(  2 + 2Cos(X) ) / 2 t
therefore 2Cos(X/2) = sqrt(  2 + 2Cos(X) )
           2Cos(X)   = sqrt(  2 + 2Cos(2X) )
           2Cos(2X)  = sqrt(  2 + 2Cos(4X) ) etc.
therefore 2Cos(X)   = sqrt( 2 + 2Cos(2X) )
                     = sqrt( 2 + sqrt( 2 + 2Cos(4X) ) )
                     = sqrt( 2 + sqrt( 2 + sqrt( 2 + 2Cos(8X) ) ) )
etc.  if 45o < X < 90o, Cos(2X) is negative,
   so  - 2Cos(180 - 2X)  can be used instead of  + 2Cos(2X)
therefore 2Cos(X) = sqrt(2+-sqrt(2+-sqrt(2+- . . . . .))) If the
units for angular measurement are selected such that a right angle =
2n units, then the signs in the series conform to the reflected gray
code.
The Algorithm
      For X where X less than or equal to 90o:
      2 Cos(X) = sqrt(2+-sqrt(2+-sqrt(2+- .........)))
The +-s are set to either + or - as follows:
   (Y is a binary integer).
1)   Compute  Y  = (X/90)x2n .  Using a larger value of n increases
the accuracy.
2)   Convert Y to reflected gray code. This is done by sh...