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208 geometry/trigonometry/euclidean.numbers.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

208 geometry/trigonometry/euclidean.numbers.p


For what numbers x is sin(x) expressible using only integers, +, -, *, / and
square root?

geometry/trigonometry/euclidean.numbers.s

Numbers generated by +, -, *, /, and sqrt from the integers are the
Euclidean numbers, so called because they are those for which line
segments can be constructed by use of straightedge and compass the
ratio of whose lengths has that value.

Using degrees, sin (360*M/N) (where (M,N)=1) is Euclidean if and only
if the regular polygon with N sides can be constructed by straightedge
and compass. This is true if (Gauss) and only if (easier) N is a power
of 2 times the product of different Fermat primes (3, 5, 17, 257, 65537
and probably no more). So sin (3/17) = sin (360/(2^3*3*5*17)) is
Euclidean, for example.

Some particular values:

sin(54) = (1 + sqrt(5))/4
sin(3) = sqrt(8 - sqrt(3) - sqrt(15) - sqrt(10 - 2*sqrt(5)))/4

 

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