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三角関数 - 9倍角の公式

カテゴリー[ 昆虫| 田園| | | 数学・幾何学| 寺院| | 祭り| 鉄道| | 風力発電]

イメージ 1

三角関数の9倍角の公式です。
加法定理、4倍角の公式、5倍角の公式から導出できます。

準備

 \displaystyle \sin ( \alpha + \beta ) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
 \displaystyle \cos ( \alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
 \displaystyle \sin 4 \alpha = 4 \cos \alpha \sin \alpha \left( 1 - 2 \sin ^ {2} \alpha \right) = 4 \cos \alpha \sin \alpha \left( 2 \cos ^ {2} \alpha - 1 \right)
 \displaystyle \sin 5 \alpha = \sin \alpha \left( 16 \sin ^ {4} \alpha - 20 \sin ^ {2} \alpha + 5 \right) = \sin \alpha \left( 16 \cos ^ {4} \alpha - 12 \cos ^ {2} \alpha + 1 \right)
 \displaystyle \cos 4 \alpha = 8 \cos ^ {4} \alpha - 8 \cos ^ {2} \alpha + 1 = 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1
 \displaystyle \cos 5 \alpha = \cos \alpha \left( 16 \cos ^ {4} \alpha - 20 \cos ^ {2} \alpha + 5 \right) = \cos \alpha \left( 16 \sin^ {4} \alpha - 12 \sin ^ {2} \alpha + 1 \right)
 \displaystyle \tan 4 \alpha = \frac{ 4 \tan \alpha \left( 1 - \tan ^{2} \alpha \right) }{ 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha }
 \displaystyle \tan 5 \alpha = \frac{ \tan \alpha \left( 5 - 10 \tan ^ {2} \alpha + \tan ^ {4} \alpha \right) }{ 1 - 10 \tan ^ {2} \alpha + 5 \tan ^ {4} \alpha }

導出

 \displaystyle \sin 9 \alpha = \sin \left( 5 \alpha + 4 \alpha \right) = \sin 5 \alpha \cos 4 \alpha + \cos 5 \alpha \sin 4 \alpha \\ \small = \sin \alpha \left( 16 \sin ^ {4} \alpha - 20 \sin ^ {2} \alpha + 5 \right) \left( 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1 \right) + \cos \alpha \left( 16 \sin^ {4} \alpha - 12 \sin ^ {2} \alpha + 1 \right) \cdot  4 \cos \alpha \sin \alpha \left( 1 - 2 \sin ^ {2} \alpha \right) \\ \small = \sin \alpha \left\{ \left( 16 \sin ^ {4} \alpha - 20 \sin ^ {2} \alpha + 5 \right) \left( 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1 \right) + 4 \cos^{2} \alpha \left( 16 \sin^ {4} \alpha - 12 \sin ^ {2} \alpha + 1 \right) \left( 1 - 2 \sin ^ {2} \alpha \right) \right\} \\ \tiny = \sin \alpha \left\{ 16 \cdot 8 \sin ^ {8} \alpha - 20 \cdot 8 \sin ^ {6} \alpha   + 5 \cdot 8 \sin ^ {4} \alpha - 16 \cdot 8 \sin ^ {6} \alpha + 20 \cdot 8 \sin ^ {4} \alpha - 5 \cdot 8 \sin ^ {2} \alpha + 16 \sin ^ {4} \alpha - 20 \sin ^ {2} \alpha + 5 + 4 \left( 16 \sin^ {4} \alpha - 12 \sin ^ {2} \alpha + 1 \right) \left( 1 - 2 \sin ^ {2} \alpha \right) \left( 1 - \sin ^ {2} \alpha \right) \right\} \\ \tiny = \sin \alpha \left\{ 16 \cdot 8 \sin ^ {8} \alpha - \left( 20 \cdot 8 + 16 \cdot 8 \right) \sin ^ {6} \alpha   + \left( 5 \cdot 8 + 20 \cdot 8 + 16 \right) \sin ^ {4} \alpha  - \left( 5 \cdot 8 + 20 \right) \sin ^ {2} \alpha + 5 + 4 \left( 16 \sin^ {4} \alpha - 12 \sin ^ {2} \alpha + 1 \right) \left( 2 \sin ^ {4} \alpha - 3 \sin ^ {2} \alpha + 1 \right) \right\} \\ \tiny = \sin \alpha \left\{ 128 \sin ^ {8} \alpha - 288 \sin ^ {6} \alpha   + 216 \sin ^ {4} \alpha  - 60 \sin ^ {2} \alpha + 5 + 4 \left( 16 \cdot 2 \sin^ {8} \alpha - 12 \cdot 2 \sin^ {6} \alpha + 2 \sin^ {4}  \alpha - 16 \cdot 3 \sin^ {6} \alpha + 12 \cdot 3 \sin^ {4} \alpha - 3 \sin^ {2} \alpha + 16 \sin^ {4} \alpha - 12 \sin^ {2} \alpha + 1 \right) \right\} \\ \tiny = \sin \alpha \left[ 128 \sin ^ {8} \alpha - 288 \sin ^ {6} \alpha   + 216 \sin ^ {4} \alpha  - 60 \sin ^ {2} \alpha + 5 + 4 \left\{ 16 \cdot 2 \sin^ {8} \alpha - \left( 12 \cdot 2 + 16 \cdot 3 \right) \sin^ {6} \alpha + \left( 2 + 12 \cdot 3 + 16 \right) \sin^ {4}  \alpha - \left( 3 + 12 \right) \sin^ {2}  \alpha + 1 \right\} \right] \\ \small = \sin \alpha \left\{ 128 \sin ^ {8} \alpha - 288 \sin ^ {6} \alpha   + 216 \sin ^ {4} \alpha  - 60 \sin ^ {2} \alpha + 5 + 4 \left( 32 \sin^ {8} \alpha - 72 \sin^ {6} \alpha + 54 \sin^ {4}  \alpha - 15 \sin^ {2}  \alpha + 1 \right) \right\} \\ \small = \sin \alpha \left\{ \left( 128 + 4 \cdot 32 \right) \sin ^ {8} \alpha - \left( 288 + 4 \cdot 72 \right) \sin ^ {6} \alpha + \left( 216 + 4 \cdot 54 \right) \sin ^ {4} \alpha - \left( 60 + 4 \cdot 15 \right) \sin ^ {2} \alpha + 5 + 4 \sin \alpha \right\} \\ = \sin \alpha \left( 256 \sin ^ {8} \alpha - 576 \sin ^ {6} \alpha + 432 \sin ^ {4} \alpha - 120 \sin ^ {2} \alpha + 9 \sin \alpha \right) \\ = 256 \sin ^ {9} \alpha - 576 \sin ^ {7} \alpha +432 \sin ^ {5} \alpha - 120 \sin ^ {3} \alpha + 9 \sin \alpha
 
 \displaystyle \cos 9 \alpha = 256 \cos ^ {9} \alpha - 576 \cos ^ {7} \alpha +432 \cos ^ {5} \alpha - 120 \cos ^ {3} \alpha + 9 \cos \alpha
 \displaystyle \tan 9 \alpha = \frac{ \tan ^ {9} \alpha - 36 \tan ^ {7} \alpha +126 \tan ^ {5} \alpha - 84 \tan ^ {3} \alpha + 9 \tan \alpha }{ 9 \tan ^ {8} \alpha - 84 \tan ^ {6} \alpha + 126 \tan ^ {4} \alpha - 36 \tan ^ {2} \alpha + 1 }

 



LibreOffice 数式(Math) のソース:

 

alignl sin 9 %alpha = sin ( 5 %alpha + 4 %alpha ) = sin 5 %alpha cos 4 %alpha + cos 5 %alpha sin 4 %alpha
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alignl size 10 { phantom { y } = lbrace sin %alpha ( 16 sin ^4 %alpha - 20 sin ^2 %alpha + 5 ) rbrace ( 8 sin ^4 %alpha - 8 sin ^2 %alpha + 1 ) + lbrace cos %alpha ( 16 sin ^4 %alpha - 12 sin ^2 %alpha + 1 ) rbrace lbrace 4 sin %alpha cos %alpha ( 1 - 2 sin ^2 %alpha ) rbrace }
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alignl size 10 { phantom { y } = sin %alpha lbrace ( 16 sin ^4 %alpha - 20 sin ^2 %alpha + 5 ) ( 8 sin ^4 %alpha -8 sin ^2 %alpha + 1) + 4 cos ^2 %alpha ( 16 sin ^4 %alpha - 12 sin ^2 %alpha + 1 ) ( 1 - 2 sin ^2 %alpha ) rbrace }
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alignl size 10 { phantom { y } = sin %alpha lbrace ( 16 sin ^4 %alpha - 20 sin ^2 %alpha + 5 ) ( 8 sin ^4 %alpha -8 sin ^2 %alpha + 1) + 4 ( sin ^2 %alpha - 1 ) ( 16 sin ^4 %alpha - 12 sin ^2 %alpha + 1 ) ( 2 sin ^2 %alpha - 1 ) rbrace }
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alignl phantom { y } = sin %alpha ( 256 sin ^8 %alpha - 576 sin ^6 %alpha + 432 sin ^4 %alpha - 120 sin ^2 %alpha + 9)
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alignl phantom { y } = 256 sin ^9 %alpha - 576 sin ^7 %alpha + 432 sin ^5 %alpha - 120 sin ^3 %alpha + 9 sin %alpha

 

alignl cos 9 %alpha = cos ( 5 %alpha + 4 %alpha ) = cos 5 %alpha cos 4 %alpha - sin 5 %alpha sin 4 %alpha
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alignl size 10 { phantom { y } = lbrace cos %alpha ( 16 cos ^4 %alpha - 20 cos ^2 %alpha + 5 ) rbrace ( 8 cos ^4 %alpha - 8 cos ^2 %alpha + 1 ) + lbrace sin %alpha ( 16 cos ^4 %alpha - 12 cos ^2 %alpha + 1 ) rbrace lbrace 4 sin %alpha cos %alpha ( 2 cos ^2 %alpha - 1) rbrace }
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alignl size 10 { phantom { y } = cos %alpha lbrace ( 16 cos ^4 %alpha - 20 cos ^2 %alpha + 5 ) ( 8 cos ^4 %alpha -8 cos ^2 %alpha + 1) - 4 sin ^2 %alpha ( 16 cos ^4 %alpha - 12 cos ^2 %alpha + 1 ) ( 2 cos ^2 %alpha - 1) rbrace }
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alignl size 10 { phantom { y } = cos %alpha lbrace ( 16 cos ^4 %alpha - 20 cos ^2 %alpha + 5 ) ( 8 cos ^4 %alpha -8 cos ^2 %alpha + 1) + 4 ( cos ^2 %alpha - 1) ( 16 cos ^4 %alpha - 12 cos ^2 %alpha + 1 ) ( 2 cos ^2 %alpha - 1) rbrace }
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alignl phantom { y } = cos %alpha ( 256 cos ^8 %alpha - 576 cos ^6 %alpha + 432 cos ^4 %alpha - 120 cos ^2 %alpha + 9)
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alignl phantom { y } = 256 cos ^9 %alpha - 576 cos ^7 %alpha + 432 cos ^5 %alpha - 120 cos ^3 %alpha + 9 cos %alpha

 

alignl tan 9 %alpha = tan ( 5 %alpha + 4 %alpha ) = { alignc { tan 5 %alpha + tan 4 %alpha } over { 1 - tan 5 %alpha tan 4 %alpha } } = { alignc { { tan %alpha ( tan ^4 %alpha - 10 tan ^2 %alpha + 5 ) } over { 5 tan ^4 %alpha - 10 tan ^2 %alpha + 1 } + { 4 tan %alpha ( 1 - tan ^2 %alpha ) } over { tan ^4 %alpha - 6 tan ^2 %alpha + 1 } } over { 1 - { { tan %alpha ( tan ^4 %alpha - 10 tan ^2 %alpha + 5 ) } over { 5 tan ^4 %alpha - 10 tan ^2 %alpha + 1 } } cdot { { 4 tan %alpha ( 1 - tan ^2 %alpha) } over { tan ^4 %alpha - 6 tan ^2 %alpha + 1 } } } }
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alignl phantom { y } = { alignc { tan %alpha lbrace ( tan ^4 %alpha - 10 tan ^2 %alpha + 5 ) ( tan ^4 %alpha - 6 tan ^2 %alpha + 1 ) + 4 ( 1 - tan ^2 %alpha ) ( 5 tan ^4 %alpha - 10 tan ^2 %alpha + 1 ) rbrace } over { ( 5 tan ^4 %alpha - 10 tan ^2 %alpha + 1 ) ( tan ^4 %alpha - 6 tan ^2 %alpha + 1 ) - 4 tan ^2 %alpha ( tan ^4 %alpha - 10 tan ^2 %alpha + 5 ) ( 1 - tan ^2 %alpha ) } }
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alignl phantom { y } = { alignc { tan %alpha ( tan ^8 %alpha - 36 tan ^6 %alpha) + 126 tan ^4 %alpha -84 tan ^2 %alpha + 9 } over { 9 tan ^8 %alpha - 84 tan ^6 %alpha + 126 tan ^4 %alpha - 36 tan ^2 %alpha + 1 } }
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alignl phantom { y } = { alignc { tan ^9 %alpha - 36 tan ^7 %alpha + 126 tan ^5 %alpha -84 tan ^3 %alpha + 9 tan %alpha } over { 9 tan ^8 %alpha - 84 tan ^6 %alpha + 126 tan ^4 %alpha - 36 tan ^2 %alpha + 1 } }