This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
For what numbers x is sin(x) expressible using only integers, +, -, *, / and
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