A093819 Algebraic degree of sin(2*Pi/n).

1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, 12, 6, 8, 4, 16, 6, 18, 2, 12, 10, 22, 4, 20, 12, 18, 3, 28, 8, 30, 8, 20, 16, 24, 3, 36, 18, 24, 8, 40, 12, 42, 5, 24, 22, 46, 8, 42, 20, 32, 6, 52, 18, 40, 12, 36, 28, 58, 4, 60, 30, 36, 16, 48, 20, 66, 8, 44, 24, 70, 12, 72, 36, 40, 9, 60, 24

Offset: 1

Author

Eric W. Weisstein, Apr 16 2004

Keywords

• nonn

Comments

The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011

This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011

Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011

See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013

References

• I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
• Eric Weisstein's World of Mathematics, Trigonometry Angles

Crossrefs

Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.

Cf. A181872/A181873, A181871, A231188. - Wolfdieter Lang, Nov 30 2013

Programs

Mathematica

a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

Formula

a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)=A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - Wolfdieter Lang, Jan 09 2011

a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - Wolfdieter Lang, Nov 30 2013