"삼각함수의 배각공식"의 두 판 사이의 차이

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* <math>x= \sin \theta</math> 로 이해<br>
 
* <math>x= \sin \theta</math> 로 이해<br>
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* <math>\sin n\theta</math> 는 <math>x= \sin \theta</math>의 다항식으로 표현되며 [[체비셰프 다항식]]과 관계<br>
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* <math>\sin n\theta</math> 는 <math>x= \sin \theta</math>의 다항식으로 표현되며 [[체비셰프 다항식]] 이 사용됨<br>
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*  사용된 매쓰매티카 명령어<br>
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*#  S:=Table[n,{n,0,19,2}]<br> Do[Print["Sin ",n+1,"\[Theta]=",ExpandAll[x*ChebyshevU[n,Sqrt[1-x^2]]]],{n,S}]<br>
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*  목록<br> Sin 1\[Theta]=x<br> Sin 3\[Theta]=3 x-4 x^3<br> Sin 5\[Theta]=5 x-20 x^3+16 x^5<br> Sin 7\[Theta]=7 x-56 x^3+112 x^5-64 x^7<br> Sin 9\[Theta]=9 x-120 x^3+432 x^5-576 x^7+256 x^9<br> Sin 11\[Theta]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11<br> Sin 13\[Theta]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13<br> Sin 15\[Theta]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15<br> Sin 17\[Theta]=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<br> Sin 19\[Theta]=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<br>
  
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Sin 1\[Theta]=x<br> Sin 3\[Theta]=3 x-4 x^3<br> Sin 5\[Theta]=5 x-20 x^3+16 x^5<br> Sin 7\[Theta]=7 x-56 x^3+112 x^5-64 x^7<br> Sin 9\[Theta]=9 x-120 x^3+432 x^5-576 x^7+256 x^9<br> Sin 11\[Theta]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11<br> Sin 13\[Theta]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13<br> Sin 15\[Theta]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15<br> Sin 17\[Theta]=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<br> Sin 19\[Theta]=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
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<h5 style="margin: 0px; line-height: 2em;"><math>\sin (2n+1)\theta</math></h5>
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* <math>x=\sin\theta</math>, <math>\sqrt{1-x^2}=\cos\theta</math>로 이해<br>
 +
 
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* <math>\sin n\theta</math> 는 <math>x= \sin \theta</math>의 다항식으로 표현되며 [[체비셰프 다항식]] 이 사용됨<br>
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*  사용된 매쓰매티카 명령어<br>
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*#  S:=Table[n,{n,0,19,2}]<br> Do[Print["Sin ",n+1,"\[Theta]=",ExpandAll[x*ChebyshevU[n,Sqrt[1-x^2]]]],{n,S}]<br>
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*  목록<br> Sin 1\[Theta]=x<br> Sin 3\[Theta]=3 x-4 x^3<br> Sin 5\[Theta]=5 x-20 x^3+16 x^5<br> Sin 7\[Theta]=7 x-56 x^3+112 x^5-64 x^7<br> Sin 9\[Theta]=9 x-120 x^3+432 x^5-576 x^7+256 x^9<br> Sin 11\[Theta]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11<br> Sin 13\[Theta]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13<br> Sin 15\[Theta]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15<br> Sin 17\[Theta]=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<br> Sin 19\[Theta]=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<br>
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xU_0[Sqrt[1-x]]=x<br> xU_1[Sqrt[1-x]]=2 x Sqrt[1-x^2]<br> xU_2[Sqrt[1-x]]=3 x-4 x^3<br> xU_3[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-1+2 x^2)<br> xU_4[Sqrt[1-x]]=x (5-20 x^2+16 x^4)<br> xU_5[Sqrt[1-x]]=2 x Sqrt[1-x^2] (3-16 x^2+16 x^4)<br> xU_6[Sqrt[1-x]]=7 x-56 x^3+112 x^5-64 x^7<br> xU_7[Sqrt[1-x]]=-8 x Sqrt[1-x^2] (-1+10 x^2-24 x^4+16 x^6)<br> xU_8[Sqrt[1-x]]=x (9-120 x^2+432 x^4-576 x^6+256 x^8)<br> xU_9[Sqrt[1-x]]=2 x Sqrt[1-x^2] (5-80 x^2+336 x^4-512 x^6+256 x^8)<br> xU_10[Sqrt[1-x]]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11<br> xU_11[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-3+70 x^2-448 x^4+1152 x^6-1280 x^8+512 x^10)<br> xU_12[Sqrt[1-x]]=x (13-364 x^2+2912 x^4-9984 x^6+16640 x^8-13312 x^10+4096 x^12)<br> xU_13[Sqrt[1-x]]=2 x Sqrt[1-x^2] (7-224 x^2+2016 x^4-7680 x^6+14080 x^8-12288 x^10+4096 x^12)<br> xU_14[Sqrt[1-x]]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15<br> xU_15[Sqrt[1-x]]=-16 x Sqrt[1-x^2] (-1+42 x^2-504 x^4+2640 x^6-7040 x^8+9984 x^10-7168 x^12+2048 x^14)<br> xU_16[Sqrt[1-x]]=x (17-816 x^2+11424 x^4-71808 x^6+239360 x^8-452608 x^10+487424 x^12-278528 x^14+65536 x^16)<br> xU_17[Sqrt[1-x]]=2 x Sqrt[1-x^2] (9-480 x^2+7392 x^4-50688 x^6+183040 x^8-372736 x^10+430080 x^12-262144 x^14+65536 x^16)<br> xU_18[Sqrt[1-x]]=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<br> xU_19[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-5+330 x^2-6336 x^4+54912 x^6-256256 x^8+698880 x^10-1146880 x^12+1114112 x^14-589824 x^16+131072 x^18)<br> xU_20[Sqrt[1-x]]=x (21-1540 x^2+33264 x^4-329472 x^6+1793792 x^8-5870592 x^10+12042240 x^12-15597568 x^14+12386304 x^16-5505024 x^18+1048576 x^20)
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2009년 11월 22일 (일) 13:01 판

이 항목의 스프링노트 원문주소

 

 

간단한 소개

 

\(\sin (2n+1)\theta\)
  • \(x= \sin \theta\) 로 이해
  • \(\sin n\theta\) 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,"\[Theta]=",ExpandAll[x*ChebyshevU[n,Sqrt[1-x^2]]]],{n,S}]
  • 목록
    Sin 1\[Theta]=x
    Sin 3\[Theta]=3 x-4 x^3
    Sin 5\[Theta]=5 x-20 x^3+16 x^5
    Sin 7\[Theta]=7 x-56 x^3+112 x^5-64 x^7
    Sin 9\[Theta]=9 x-120 x^3+432 x^5-576 x^7+256 x^9
    Sin 11\[Theta]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11
    Sin 13\[Theta]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13
    Sin 15\[Theta]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15
    Sin 17\[Theta]=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 19\[Theta]=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

 

 

\(\sin (2n+1)\theta\)
  • \(x=\sin\theta\), \(\sqrt{1-x^2}=\cos\theta\)로 이해
  • \(\sin n\theta\) 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,"\[Theta]=",ExpandAll[x*ChebyshevU[n,Sqrt[1-x^2]]]],{n,S}]
  • 목록
    Sin 1\[Theta]=x
    Sin 3\[Theta]=3 x-4 x^3
    Sin 5\[Theta]=5 x-20 x^3+16 x^5
    Sin 7\[Theta]=7 x-56 x^3+112 x^5-64 x^7
    Sin 9\[Theta]=9 x-120 x^3+432 x^5-576 x^7+256 x^9
    Sin 11\[Theta]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11
    Sin 13\[Theta]=13 x-364 x^3+2912 x^5-9984 x^7+16640 x^9-13312 x^11+4096 x^13
    Sin 15\[Theta]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15
    Sin 17\[Theta]=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 19\[Theta]=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

 

 

xU_0[Sqrt[1-x]]=x
xU_1[Sqrt[1-x]]=2 x Sqrt[1-x^2]
xU_2[Sqrt[1-x]]=3 x-4 x^3
xU_3[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-1+2 x^2)
xU_4[Sqrt[1-x]]=x (5-20 x^2+16 x^4)
xU_5[Sqrt[1-x]]=2 x Sqrt[1-x^2] (3-16 x^2+16 x^4)
xU_6[Sqrt[1-x]]=7 x-56 x^3+112 x^5-64 x^7
xU_7[Sqrt[1-x]]=-8 x Sqrt[1-x^2] (-1+10 x^2-24 x^4+16 x^6)
xU_8[Sqrt[1-x]]=x (9-120 x^2+432 x^4-576 x^6+256 x^8)
xU_9[Sqrt[1-x]]=2 x Sqrt[1-x^2] (5-80 x^2+336 x^4-512 x^6+256 x^8)
xU_10[Sqrt[1-x]]=11 x-220 x^3+1232 x^5-2816 x^7+2816 x^9-1024 x^11
xU_11[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-3+70 x^2-448 x^4+1152 x^6-1280 x^8+512 x^10)
xU_12[Sqrt[1-x]]=x (13-364 x^2+2912 x^4-9984 x^6+16640 x^8-13312 x^10+4096 x^12)
xU_13[Sqrt[1-x]]=2 x Sqrt[1-x^2] (7-224 x^2+2016 x^4-7680 x^6+14080 x^8-12288 x^10+4096 x^12)
xU_14[Sqrt[1-x]]=15 x-560 x^3+6048 x^5-28800 x^7+70400 x^9-92160 x^11+61440 x^13-16384 x^15
xU_15[Sqrt[1-x]]=-16 x Sqrt[1-x^2] (-1+42 x^2-504 x^4+2640 x^6-7040 x^8+9984 x^10-7168 x^12+2048 x^14)
xU_16[Sqrt[1-x]]=x (17-816 x^2+11424 x^4-71808 x^6+239360 x^8-452608 x^10+487424 x^12-278528 x^14+65536 x^16)
xU_17[Sqrt[1-x]]=2 x Sqrt[1-x^2] (9-480 x^2+7392 x^4-50688 x^6+183040 x^8-372736 x^10+430080 x^12-262144 x^14+65536 x^16)
xU_18[Sqrt[1-x]]=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
xU_19[Sqrt[1-x]]=-4 x Sqrt[1-x^2] (-5+330 x^2-6336 x^4+54912 x^6-256256 x^8+698880 x^10-1146880 x^12+1114112 x^14-589824 x^16+131072 x^18)
xU_20[Sqrt[1-x]]=x (21-1540 x^2+33264 x^4-329472 x^6+1793792 x^8-5870592 x^10+12042240 x^12-15597568 x^14+12386304 x^16-5505024 x^18+1048576 x^20)

 

 

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