- 501 名前：１３２人目の素数さん mailto:sage [2007/02/03(土) 22:48:18 ]
- >494
2sin(x) = {1/sin(π/2n)} {cos(x - π/2n) - cos(x + π/2n)} (← 積和公式)
L = Σ[k=1,n-1] 2sin(kπ/n)
= {1/sin(π/2n)} Σ[k=1,n-1] {cos[(k-1/2)π/n] - cos[(k+1/2)π/n]}
= {1/sin(π/2n)} {cos(π/2n) - cos((n-1/2)π/n)}
= 2/tan(π/2n).
>500
最大の正５角形の外接円の半径Rは
R = 1/{cos(3π/20) + cos(π/20)} = 1/{2cos(π/10)cos(π/20)} = 0.5322844…
(正５角形の1頂点から対辺への垂線が正方形の対角線上に来るように置く)
sin(2π/5) = √{(5+√5)/2} = 0.9510565… より
S = (5/2)sin(2π/5)･R^2 = 2.377641… R^2 = 0.67364926…
