■正多面体の正多角形断面（その１３４）
Tn=sin(n+1)π/(N+1))sin(nπ/(N+1))/{sin(π/(N+1))}{sin(2π/(N+1))
α=π/(N+1)
Tn=sin(n+1)αsinnα/sinαsin2α
n=0のときTn=0
n=1のとき、Tn=sin(2π/(N+1))sin(π/(N+1))/{sin(π/(N+1))}{sin(2π/(N+1))＝1
n=2のとき、Tn=sin(3π/(N+1))sin(2π/(N+1))/{sin(π/(N+1))}{sin(2π/(N+1))＝-4{sin(π/(N+1))}^2+3
X=1+2cos(2π/(n+1))＝1+2（1-2{sin(π/(N+1))}^2）より
n=2のとき、Tn=sin(3π/(N+1))sin(2π/(N+1))/{sin(π/(N+1))}{sin(2π/(N+1))＝-4{sin(π/(N+1))}^2+3=X
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X=1+2cos2π/(N+1)
Tn=sin(n+1)π/(N+1))sin(nπ/(N+1))/{sin(π/(N+1))}{sin(2π/(N+1))
Tn=-1/2{cos(2n+1)α-cosα｝/sinαsin2α
次の課題は
Tncos2(n-1)α
Tnsin2(n-1)α
を求めることである。
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Tncos2(n-1)α=sin(n+1)αsinnαcos(2n-2)α/sinαsin2α
=1/4{-cos3x+cos(2n-3)x+cos(2n-1)x-cos(4n-1)x}/sinxsin2x
=1/4{-cos3x+cos2nxcos3x+sin2nxsin3x+cos2nxcosx+sin2nxsinx-cos4nxcosx-sin4nxsinx}/sinxsin2x
Tnsin2(n-1)α=sin(n+1)αsinnαsin(2n-2)α/sinαsin2α
=1/4{sin3x+sin(2n-3)x+sin(2n-1)x-sin(4n-1)x}/sinxsin2x
=1/4{sin3x+sin2nxcos3x-cos2nxsin3x+sin2nxcosx-cos2nxsinx-sin4nxcosx+cos4nxsinx}/sinxsin2x
Σsinrx=sinx+・・・+sinnx=sin((n+1)x/2)sin(nx/2)/sin(x/2)
Σcosrx=cosx+・・・+cosnx=cos((n+1)x/2)sin(nx/2)/sin(x/2)
Σsin2rx=sin((n+1)x)sin(nx)/sin(x)
Σcos2rx=cos((n+1)x)sin(nx)/sin(x)
Σsin4rx=sin(2(n+1)x)sin(2nx)/sin(2x)
Σcos4rx=cos(2(n+1)x)sin(2nx)/sin(2x)
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4sinxsin2xTncos2(n-1)α={-cos3x+cos(2n-3)x+cos(2n-1)x-cos(4n-1)x}
4sinxsin2xTnsin2(n-1)α={sin3x+sin(2n-3)x+sin(2n-1)x-sin(4n-1)x}
4sinxsin2xΣTncos2(n-1)α={-Σcos3x+Σcos(2n-3)x+Σcos(2n-1)x-Σcos(4n-1)x}
4sinxsin2xΣTnsin2(n-1)α={Σsin3x+Σsin(2n-3)x+Σsin(2n-1)x-Σsin(4n-1)x}
{-Σcos3x+Σcos(2n-3)x+Σcos(2n-1)x-Σcos(4n-1)x}^2+{Σsin3x+Σsin(2n-3)x+Σsin(2n-1)x-Σsin(4n-1)x}^2
=
(Σcos3x)^2+(Σsin3x)^2=n^2
(Σcos(2n-3)x)^2+(Σsin(2n-3)x)^2={sin(nx)cos3x/sin(x)}^2+{sin(nx)sin3x/sin(x)}^2=2{sin(nx)/sin(x)}^2
(Σcos(2n-1)x)^2+(Σsin(2n-1)x)^2=2{sin(nx)/sin(x)}^2
(Σcos(4n-1)x)^2+(Σsin(4n-1)x)^2=2{sin(2nx)/sin(2x)}^2
合わせると
n^2
4{sin(nx)/sin(x)}^2
2{sin(2nx)/sin(2x)}^2
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-2(Σcos3x)(Σcos(2n-3)x)=-2ncos((n+1)x)sin(nx)(cos3x)^2/sin(x)-2nsin((n+1)x)sin(nx)sin3xcos3x/sin(x)
+2(Σsin3x)(Σsin(2n-3)x)=+2nsin((n+1)x)sin(nx)sin3xcos3x/sin(x)-2ncos((n+1)x)sin(nx)(sin3x)^2/sin(x)
=-2ncos(n+1)xsin(nx)/sin(x)
-2(Σcos3x)(Σcos(2n-1)x)=-2ncos((n+1)x)sin(nx)cosxcos3x/sin(x)-2nsin((n+1)x)sin(nx)sinxcos3x/sin(x)
+2(Σsin3x)(Σsin(2n-1)x)=+2nsin((n+1)x)sin(nx)cosxsin3x/sin(x)-2ncos((n+1)x)sin(nx)sinxsin3x/sin(x)
=-2ncos(n+1)xsin(nx)cos2x/sin(x)+2nsin(n+1)xsin(nx)sin2x/sin(x)
=-2ncos(n+3)xsin(nx)/sin(x)
+2(Σcos3x)(Σcos(4n-1)x)=2ncos(2(n+1)x)sin(2nx)cosxcos3x/sin(2x)+2nsin(2(n+1)x)sin(2nx)sinxcos3x/sin(2x)
-2(Σsin3x)(Σsin(4n-1)x)=-2nsin(2(n+1)x)sin(2nx)cosxsin3x/sin(2x)+2ncos(2(n+1)x)sin(2nx)sinxsin3x/sin(2x)
=2ncos(2(n+1)x)sin(2nx)cos2x/sin(2x)-2nsin(2(n+1)x)sin(2nx)sin2x/sin(2x)
=2ncos(2n+4)xsin(2nx)/sin(2x)
合わせると
-2ncos(n+1)xsin(nx)/sin(x)
-2ncos(n+3)xsin(nx)/sin(x)
2ncos(2n+4)xsin(2nx)/sin(2x)
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+2(Σcos(2n-3)x)(Σcos(2n-1)x)
=2{cos((n+1)x)sin(nx)/sin(x)}^2・cos3xcox+{sin((n+1)x)sin(nx)/sin(x)}^2・sin3xsinx
+2cos((n+1)xsin(n+1)x{sin(nx)/sin(x)}^2(cos3xsinx+sin3xcosx)
=2{cos((n+1)x)sin(nx)/sin(x)}^2・cos3xcox+{sin((n+1)x)sin(nx)/sin(x)}^2・sin3xsinx
+2cos((n+1)xsin(n+1)x{sin(nx)/sin(x)}^2(sin4x)・・・訂正
+2(Σsin(2n-3)x)(Σsin(2n-1)x)
=2{sin((n+1)x)sin(nx)/sin(x)}^2・cos3xcox+{cos((n+1)x)sin(nx)/sin(x)}^2・sin3xsinx
-2cos((n+1)xsin(n+1)x{sin(nx)/sin(x)}^2(cos3xsinx+sin3xcosx)
=2{sin((n+1)x)sin(nx)/sin(x)}^2・cos3xcox+{cos((n+1)x)sin(nx)/sin(x)}^2・sin3xsinx
-2cos((n+1)xsin(n+1)x{sin(nx)/sin(x)}^2(sin4x)
合わせると
2{sin(nx)/sin(x)}^2・cos3xcox+2{sin(nx)/sin(x)}^2・sin3xsinx
=2{sin(nx)/sin(x)}^2・cos2x
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-2(Σcos(2n-3)x)(Σcos(4n-1)x)
(-2cos((n+1)x)cos3xcos(2(n+1)x)cosx-2cos((n+1)x)cos3xsin(2(n+1)x)sinx)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
(-2sin((n+1)x)sin3xcos(2(n+1)x)cosx-2sin((n+1)x)sin3xsin(2(n+1)x)sinx)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2(Σsin(2n-3)x)(Σsin(4n-1)x)
(-2sin((n+1)x)cos3xsin(2(n+1)x)cosx+2sin((n+1)x)cos3xcos(2(n+1)x)sinx)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
(+2cos((n+1)x)sin3xsin(2(n+1)x)cosx-2cos((n+1)x)sin3xcos(2(n+1)x)sinx)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+1)x)cos(2(n+1)x){cos3xcosx+sin3xsinx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2cos((n+1)x)sin(2(n+1)x){cos3xsinx-sin3xcosx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)cos(2(n+1)x){sin3xcosx-cos3xsinx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)sin(2(n+1)x){sin3xsinx+cos3xcosx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+1)x)cos(2(n+1)x){cos2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
+2cos((n+1)x)sin(2(n+1)x){sin2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)cos(2(n+1)x){sin2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)sin(2(n+1)x){cos2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+1)x){cos2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
+2sin((n+1)x){sin2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+3)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
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-2(Σcos(2n-1)x)(Σcos(4n-1)x)
-2cos((n+1)x)sin(nx)cosx/sin(x)cos(2(n+1)x)sin(2nx)cosx/sin(2x)-2cos((n+1)x)sin(nx)cosx/sin(x)sin(2(n+1)x)sin(2nx)sinx/sin(2x)
-2sin((n+1)x)sin(nx)sinx/sin(x)cos(2(n+1)x)sin(2nx)cosx/sin(2x)-2sin((n+1)x)sin(nx)sinx/sin(x)sin(2(n+1)x)sin(2nx)sinx/sin(2x)
-2(Σsin(2n-1)x)(Σsin(4n-1)x)
-2sin((n+1)x)sin(nx)cosx/sin(x)sin(2(n+1)x)sin(2nx)cosx/sin(2x)+2sin((n+1)x)sin(nx)cosx/sin(x)cos(2(n+1)x)sin(2nx)sinx/sin(2x)
+2cos((n+1)x)sin(nx)sinx/sin(x)sin(2(n+1)x)sin(2nx)cosx/sin(2x)-2cos((n+1)x)sin(nx)sinx/sin(x)cos(2(n+1)x)sin(2nx)sinx/sin(2x)
合わせると
-2cos((n+1)x)cos(2(n+1)x){cosxcosx+sinxsinx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2cos((n+1)x)sin(2(n+1)x){cosxsinx-sinxcosx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)cos(2(n+1)x){sinxcosx-cosxsinx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)sin(2(n+1)x){sinxsinx+cosxcosx}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+1)x)cos(2(n+1)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2sin((n+1)x)sin(2(n+1)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
合わせると
-2cos((n+1)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
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合わせると
n^2
4{sin(nx)/sin(x)}^2
2{sin(2nx)/sin(2x)}^2
-2ncos(n+1)xsin(nx)/sin(x)
-2ncos(n+3)xsin(nx)/sin(x)
2ncos(2n+4)xsin(2nx)/sin(2x)
2{sin(nx)/sin(x)}^2・cos2x
-2cos((n+3)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
-2cos((n+1)x)・sin(nx)/sin(x)・sin(2nx)/sin(2x)
n^2
4{sin(nx)/sin(x)}^2+2{sin(nx)/sin(x)}^2・cos2x
2{sin(2nx)/sin(2x)}^2
-n{cos(2n+4)x+cos2x}sin(nx)/sin(x)
2ncos(2n+4)xsin(2nx)/sin(2x)
-{cos(2n+4)x+cos2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
n^2
4{sin(nx)/sin(x)}^2+2{sin(nx)/sin(x)}^2・cos2x
2{sin(2nx)/sin(2x)}^2
-n{cos(2n+4)x+cos2x}sin(nx)/sin(x)
2ncos(2n+4)xsin(2nx)/sin(2x)
-{cos(2n+4)x+cos2x}・sin(nx)/sin(x)・sin(2nx)/sin(2x)
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そろそろ収束判定しておきたい。
x=π/(N+1)
n=N-1→∞
sin(nx)→sinπ=0
sin(2nx)→sin2π=0
n^2
-n{cos(2n+4)x+cos2x}sin(nx)/sin(x)→?
2ncos(2n+4)xsin(2nx)/sin(2x)→?
(4sinαsin2αΣTn)^2→4N^2(cosx)^2であるから
{4sinxsin2xΣTncos2(n-1)α}^2+{4sinxsin2xΣTncos2(n-1)α}^2→N^2(cosx)^2であればよい。
したがって、
-n{cos(2n+4)x+cos2x}sin(nx)/sin(x)+2ncos(2n+4)xsin(2nx)/sin(2x)→-N^2(sinx)^2であればよいことになる。
-2n{cos(2n+4)x+cos2x}sin(nx)cosx/sin(2x)+2ncos(2n+4)xsin(2nx)/sin(2x)
{-2ncos(2n+4){sin(nx)cosx+sin(2nx)}/sin(2x)}-2nsin(nx)cos2xcosx/sin(2x)
{-2ncos(2n+4)sin(nx){cosx+2cos(nx)}/sin(2x)}-2nsin(nx)cos2xcosx/sin(2x)
計算間違いがあるに違いない
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(Σcos(2n-3)x)=(Σcos2nxcos3x+Σsin2nxsin3x)=cos((n+1)x)sin(nx)cos3x/sin(x)+sin((n+1)x)sin(nx)sin3x/sin(x)
(Σsin(2n-3)x)=(Σsin2nxcos3x-Σcos2nxsin3x)=sin((n+1)x)sin(nx)cos3x/sin(x)-cos((n+1)x)sin(nx)sin3x/sin(x)
(Σcos(2n-1)x)=(Σcos2nxcosx+Σsin2nxsinx)=cos((n+1)x)sin(nx)cosx/sin(x)+sin((n+1)x)sin(nx)sinx/sin(x)
(Σsin(2n-1)x)=(Σsin2nxcosx-Σcos2nxsinx)=sin((n+1)x)sin(nx)cosx/sin(x)-cos((n+1)x)sin(nx)sinx/sin(x)
(Σcos(4n-1)x)=(Σcos4nxcosx+Σsin4nxsinx)=cos(2(n+1)x)sin(2nx)cosx/sin(2x)+sin(2(n+1)x)sin(2nx)sinx/sin(2x)
(Σsin(4n-1)x)=(Σsin4nxcosx-Σcos4nxsinx)=sin(2(n+1)x)sin(2nx)cosx/sin(2x)-cos(2(n+1)x)sin(2nx)sinx/sin(2x)
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