# 208 geometry/trigonometry/euclidean.numbers.p

## Description

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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