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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