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

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, 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, 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

Offset: 0

Author

Frank M Jackson, Jul 29 2011

Keywords

• easy
• nonn
• cons

Comments

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

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.

Examples

			0.3623748900804801199586466374749868993608655440055985464501567887401235062...
		

Links

• Frank M. Jackson, Five a day (Letter to Editor), Mathematics Today 50-6 (2014) 321.

Programs

Mathematica

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

PARI

cos((3-sqrt(5))*Pi/2) \\ Charles R Greathouse IV, Jul 29 2011

Formula

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