OFFSET
0,1
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.
LINKS
Frank M. Jackson, Five a day (Letter to Editor), Mathematics Today 50-6 (2014) 321.
FORMULA
c = cos(Pi/(1+phi)) = cos(2*A180014).
EXAMPLE
0.3623748900804801199586466374749868993608655440055985464501567887401235062...
MATHEMATICA
N[Cos[Pi/(1+GoldenRatio)], 100]
PROG
(PARI) cos((3-sqrt(5))*Pi/2) \\ Charles R Greathouse IV, Jul 29 2011
CROSSREFS
KEYWORD
AUTHOR
Frank M Jackson, Jul 29 2011
STATUS
approved