A019810 Decimal expansion of sine of 1 degree.

0, 1, 7, 4, 5, 2, 4, 0, 6, 4, 3, 7, 2, 8, 3, 5, 1, 2, 8, 1, 9, 4, 1, 8, 9, 7, 8, 5, 1, 6, 3, 1, 6, 1, 9, 2, 4, 7, 2, 2, 5, 2, 7, 2, 0, 3, 0, 7, 1, 3, 9, 6, 4, 2, 6, 8, 3, 6, 1, 2, 4, 2, 7, 6, 4, 0, 5, 9, 7, 3, 8, 4, 2, 0, 3, 9, 2, 8, 0, 7, 0, 0, 4, 2, 0, 0, 1, 9, 2, 6, 7, 9, 1, 0, 2, 1, 3, 4, 6, 9, 1, 4, 4, 8, 8

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy
• changed

Comments

An algebraic number of degree 48. - Charles R Greathouse IV, Apr 14 2014

This algebraic number has denominator 2, the least integer k > 0 such that k times the number is an algebraic integer. - Charles R Greathouse IV, Nov 12 2014

The Fifteenth Century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) in his Risala al-Watar wa'l Jaib. - Mohammad K. Azarian, Jan 14 2017

The minimal polynomial is 281474976710656 *x^48 - 3377699720527872 *x^46 + 18999560927969280 *x^44 - 66568831992070144 *x^42 + 162828875980603392 *x^40 - 295364007592722432 *x^38 + 411985976135516160 *x^36 - 452180272956309504 *x^34 + 396366279591591936 *x^32 - 280058255978266624 *x^30 + 160303703377575936 *x^28 - 74448984852135936 *x^26 + 28011510450094080 *x^24 - 8500299631165440 *x^22 + 2064791072931840 *x^20 - 397107008634880 *x^18 + 59570604933120 *x^16 - 6832518856704 *x^14 + 583456329728 *x^12 - 35782471680 *x^10 + 1497954816 *x^8 - 39625728 *x^6 + 579456 *x^4 - 3456 *x^2 + 1 (WolframAlpha). - Rick L. Shepherd, Apr 12 2017

Examples

			0.01745240643728351281941897851631...
		

References

• Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414. Solution published in Vol. 37, No. 5, November 2006, pp. 394-395.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Mohammad K. Azarian, A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.
• Index entries for algebraic numbers, degree 48

Crossrefs

Cf. A110937, A280188.

Programs

Mathematica

Join[{0},RealDigits[N[Sin[Pi/180],200]][[1]]] (* and/or *)
Join[{0},RealDigits[N[Sin[1 Degree],200]][[1]]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

PARI

sin(Pi/180)

PARI

real((I^(89/90) - I^(91/90))/2) \\ (imaginary part is not exactly zero only because of finite precision) Rick L. Shepherd, Apr 12 2017

Formula

Equals sin(Pi/180) = cos(89*Pi/180) = (i^(89/90) - i^(91/90))/2 (the last from WolframAlpha, rearranged). - Rick L. Shepherd, Apr 12 2017

Extensions

More terms from James Sellers, Jan 19 2000