삼각함수의 배각공식 표

$\cos n\theta$ 는 $x= \cos \theta$ 의 다항식으로 표현되며 체비셰프 다항식 이 사용됨
사용된 매쓰매티카 명령어
1. S:=Table[n,{n,0,20}]
  Do[Print["cos ",n,"t=",ExpandAll[ChebyshevT[n,x]]],{n,S}]
목록

cos 0t=1
cos 1t=x
cos 2t=-1+2 x^2
cos 3t=-3 x+4 x^3
cos 4t=1-8 x^2+8 x^4
cos 5t=5 x-20 x^3+16 x^5
cos 6t=-1+18 x^2-48 x^4+32 x^6
cos 7t=-7 x+56 x^3-112 x^5+64 x^7
cos 8t=1-32 x^2+160 x^4-256 x^6+128 x^8
cos 9t=9 x-120 x^3+432 x^5-576 x^7+256 x^9
cos 10t=-1+50 x^2-400 x^4+1120 x^6-1280 x^8+512 x^10
cos 11t=-11 x+220 x^3-1232 x^5+2816 x^7-2816 x^9+1024 x^11
cos 12t=1-72 x^2+840 x^4-3584 x^6+6912 x^8-6144 x^10+2048 x^12
cos 13t=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13
cos 14t=-1+98 x^2-1568 x^4+9408 x^6-26880 x^8+39424 x^10-28672 x^12+8192 x^14
cos 15t=-15 x+560 x^3-6048 x^5+28800 x^7-70400 x^9+92160 x^11-61440 x^13+16384 x^15
cos 16t=1-128 x^2+2688 x^4-21504 x^6+84480 x^8-180224 x^10+212992 x^12-131072 x^14+32768 x^16
cos 17t=17 x-816 x^3+11424 x^5-71808 x^7+239360 x^9-452608 x^11+487424 x^13-278528 x^15+65536 x^17
cos 18t=-1+162 x^2-4320 x^4+44352 x^6-228096 x^8+658944 x^10-1118208 x^12+1105920 x^14-589824 x^16+131072 x^18
cos 19t=-19 x+1140 x^3-20064 x^5+160512 x^7-695552 x^9+1770496 x^11-2723840 x^13+2490368 x^15-1245184 x^17+262144 x^19
cos 20t=1-200 x^2+6600 x^4-84480 x^6+549120 x^8-2050048 x^10+4659200 x^12-6553600 x^14+5570560 x^16-2621440 x^18+524288 x^20

$\sin n\theta$ 는 $x= \sin \theta$ 의 다항식으로 표현되며 체비셰프 다항식 이 사용됨
사용된 매쓰매티카 명령어
1. S:=Table[n,{n,0,19,2}]
  Do[Print["sin ",n+1,"t=",ExpandAll[x*ChebyshevU[n,Sqrt[1-x^2]]]],{n,S}]
목록

sin 1t=x
sin 3t=3 x-4 x^3
sin 5t=5 x-20 x^3+16 x^5
sin 7t=7 x-56 x^3+112 x^5-64 x^7
sin 9t=9 x-120 x^3+432 x^5-576 x^7+256 x^9
sin 11t=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11
sin 13t=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13
sin 15t=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15
sin 17t=17 x-816 x^3+11424 x^5-71808 x^7+239360 x^9-452608 x^11+487424 x^13-278528 x^15+65536 x^17
sin 19t=19 x-1140 x^3+20064 x^5-160512 x^7+695552 x^9-1770496 x^11+2723840 x^13-2490368 x^15+1245184 x^17-262144 x^19

$\sin n\theta$ 는 $x= \sin \theta$ 의 다항식과 $y=\cos\theta$ 의 곱으로 표현되며 체비셰프 다항식 이 사용됨
사용된 매쓰매티카 명령어
1. S:=Table[n,{n,1,19,2}]
  Do[Print["sin ",n+1,"t=",y*Expand[x*ChebyshevU[n,Sqrt[1-x^2]]/Sqrt[1-x^2]]],{n,S}]
목록

sin 2t=2 x y
sin 4t=(4 x-8 x^3) y
sin 6t=(6 x-32 x^3+32 x^5) y
sin 8t=(8 x-80 x^3+192 x^5-128 x^7) y
sin 10t=(10 x-160 x^3+672 x^5-1024 x^7+512 x^9) y
sin 12t=(12 x-280 x^3+1792 x^5-4608 x^7+5120 x^9-2048 x^11) y
sin 14t=(14 x-448 x^3+4032 x^5-15360 x^7+28160 x^9-24576 x^11+8192 x^13) y
sin 16t=(16 x-672 x^3+8064 x^5-42240 x^7+112640 x^9-159744 x^11+114688 x^13-32768 x^15) y
sin 18t=(18 x-960 x^3+14784 x^5-101376 x^7+366080 x^9-745472 x^11+860160 x^13-524288 x^15+131072 x^17) y
sin 20t=(20 x-1320 x^3+25344 x^5-219648 x^7+1025024 x^9-2795520 x^11+4587520 x^13-4456448 x^15+2359296 x^17-524288 x^19) y