N sin(π/N) の単調性を示したい
$ \sin \frac{\pi}{N} = \sin\left( \frac{\pi}{N+1} + \frac{\pi}{N(N+1)} \right) = \sin \frac{\pi}{N+1} \cos \frac{\pi}{N(N+1)} + \cos \frac{\pi}{N+1} \sin \frac{\pi}{N(N+1)}
これより
$ N \sin \frac{\pi}{N} = N \sin \frac{\pi}{N+1} \cos \frac{\pi}{N(N+1)} + N \cos \frac{\pi}{N+1} \sin \frac{\pi}{N(N+1)}
$ = (N+1) \sin \frac{\pi}{N+1} - (N+1) \sin \frac{\pi}{N+1} + N \sin \frac{\pi}{N+1} \cos \frac{\pi}{N(N+1)} + N \cos \frac{\pi}{N+1} \sin \frac{\pi}{N(N+1)}
$ = (N+1) \sin \frac{\pi}{N+1} + N\left[ \left( \cos \frac{\pi}{N(N+1)} - (1+\frac1N) \right)\sin \frac{\pi}{N+1} + \left( \sin \frac{\pi}{N(N+1)} \right) \cos \frac{\pi}{N+1} \right]
$ = (N+1) \sin \frac{\pi}{N+1} + N K \sin\left( \frac{\pi}{N+1} + \theta \right)
ここで
$ \left( \cos\frac{\pi}{N(N+1)} - ( 1 + \frac1N ) ,\ \sin\frac{\pi}{N(N+1)} \right) = K (\cos\theta,\sin\theta) と置いた
これは
$ \left( \cos\frac{\pi}{N(N+1)} ,\ \sin\frac{\pi}{N(N+1)} \right) - \left( 1 + \frac1N ,\ 0 \right) より第2象限の角
これと
$ \left( \sin\frac{\pi}{N+1} ,\ \cos\frac{\pi}{N+1} \right) のなす角が直角よりも大きい?
$ \left( -\cos\frac{\pi}{N+1} ,\ \sin\frac{\pi}{N+1} \right) よりもちょっと下?