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A193537 Decimal expansion of cos(Pi/(1+phi)), where phi is the golden ratio. 0


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

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

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

%N Decimal expansion of cos(Pi/(1+phi)), where phi is the golden ratio.

%C cos(Pi/(1+phi)) is the first term in the identity:

%C cos(Pi/(1+phi))+cos(Pi/phi)=0 which when converted to the exponential form gives: e^(i*Pi/(1+phi))+e^(-i*Pi/(1+phi))+e^(i*Pi/phi)+e^(-i*Pi/phi)=0. In this form it is known as the phi identity because it combines the golden ratio phi with the five fundamental mathematical constants Pi, e, i, 1, 0 that are found in Euler's identity e^(i*Pi) + 1 = 0.

%H Frank M. Jackson, <a href="http://www.researchgate.net/publication/292983417">Five a day (Letter to Editor)</a>, Mathematics Today 50-6 (2014) 321.

%F c = cos(Pi/(1+phi)) = cos(2*A180014).

%e 0.3623748900804801199586466374749868993608655440055985464501567887401235062...

%t N[Cos[Pi/(1+GoldenRatio)],100]

%o (PARI) cos((3-sqrt(5))*Pi/2) \\ _Charles R Greathouse IV_, Jul 29 2011

%K easy,nonn,cons

%O 0,1

%A _Frank M Jackson_, Jul 29 2011

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Last modified August 17 17:00 EDT 2019. Contains 326059 sequences. (Running on oeis4.)