A323601 -id:A323601

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A019829

Decimal expansion of sine of 20 degrees.

+10
10

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Index entries for algebraic numbers, degree 6

FORMULA

Equals cos(7*Pi/18) = 2F1(13/12,-1/12;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008

Root of the equation 64*x^6 - 96*x^4 + 36*x^2 - 3 = 0. - Vaclav Kotesovec, Jan 19 2019

Equals sqrt(8 - 2^(4/3)*(1 + i*sqrt(3))^(2/3) + i*2^(2/3)*(1 + i*sqrt(3))^(1/3)*(i + sqrt(3)))/4, where i is the imaginary unit. - Vaclav Kotesovec, Jan 19 2019

Equals 2*A019819 *A019889. - R. J. Mathar, Jan 17 2021

EXAMPLE

0.34202014332566873304409961468225958076308336751416062846504849768471476...

MATHEMATICA

RealDigits[ Sin[Pi/9], 10, 111][[1]] (* Robert G. Wilson v *)

PROG

(PARI) /* for x = 20 degrees, sin(9x) = 0 */

/* so sin(x) is a zero of this polynomial */

sin_9(x)=9*x-120*x^3+432*x^5-576*x^7+256*x^9

x=34; y=100; print(3); print(4);

for(digits=1, 110, {d=0; y=y*10; while(sin_9((10*x+d)/y) > 0, d++);

d--; /* while loop overshoots correct digit */

print(d); x=10*x+d})

\\ Michael B. Porter, Jan 27 2010

CROSSREFS

Cf. A323601.

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

A232736

Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

+10
10

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The corresponding real part is in A232735.

Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..1000

FORMULA

Equals cos(3*Pi/7). - G. C. Greubel, Sep 04 2022

EXAMPLE

0.222520933956314404288902564496794759466355568764544955311987...

MATHEMATICA

RealDigits[Cos[3*Pi/7], 10, 120][[1]] (* G. C. Greubel, Sep 04 2022 *)

PROG

(Magma) R:= RealField(120); Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022

(SageMath) numerical_approx(cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022

CROSSREFS

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).