A019812 Decimal expansion of sine of 3 degrees.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 02 2012

The minimal polynomial is p(x) = 65536*x^16 - 262144*x^14 + 430080*x^12 - 372736*x^10 + 182784*x^8 - 50176*x^6 + 7040*x^4 - 384*x^2 + 1 and p(x/2) = x^16 - 16*x^14 + 105*x^12 - 364*x^10 + 714*x^8 - 784*x^6 + 440*x^4 - 96*x^2 + 1. - Joerg Arndt, Apr 23 2024

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) from sine of 3 degrees in his Risala al-Watar wa'l Jaib. - Mohammad K. Azarian,Jan 14 2017

Examples

			0.052335956242943832722118629609...
		

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.
• Dr. Rob, The Math Forum at Drexel, The Exact Value of the Sine of 1 Degree
• Scott Surgent, Exact values of the sine and cosine functions in increments of 3 degrees, 2012
• Wikipedia, Exact trigonometric values
• Index entries for algebraic numbers, degree 16

Crossrefs

Cf. A019896 (cosine), A019901 (tangent), A019985 (cotangent): for 3 degrees.

Cf. A019810, A019811, A019897, A019898.

Programs

Mathematica

First[RealDigits[Sin[Pi/60], 10, 100, -1]] (* Paolo Xausa, Jun 19 2024 *)

PARI

sin(Pi/60) \\ Charles R Greathouse IV, Aug 27 2017

Formula

Equals cos(87 degrees) = cos(29*Pi/60) = sin(Pi/60) = sqrt(8-sqrt(3)-sqrt(15)-sqrt(10-2*sqrt(5)))/4 (an intermediate calculation by Dr. Rob - see Math Forum link). - Rick L. Shepherd, Jul 03 2006

Equals A019811*A019898 + A019810*A019897. - R. J. Mathar, Jan 27 2021