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三角関数 - 7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7倍角の公式です。
加法定理、3倍角の公式、4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 3\alpha = \sin \alpha \left( 3 - 4 \sin ^ {2} \alpha \right) = \sin \alpha \left( 4 \cos ^ 2 \alpha - 1 \right)
 \displaystyle \sin 4 \alpha = 4 \sin \alpha \cos \alpha \left( 1 - 2 \sin ^ {2} \alpha \right) = 4 \sin \alpha \cos \alpha \left( 2 \cos ^ {2} \alpha - 1 \right)
 \displaystyle \cos 3\alpha = \cos \alpha \left( 4 \cos ^ {2} \alpha - 3 \right) = \cos \alpha \left( 1 - 4 \sin ^ 2 \alpha \right)
 \displaystyle \cos 4 \alpha = 8 \cos ^ {4} \alpha - 8 \cos ^ {2} \alpha + 1 = 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1
 \displaystyle \tan 3\alpha = \frac{ \tan \alpha \left( 3 - \tan ^ {2} \alpha \right) }{ 1 - 3 \tan ^ {2} \alpha }
 \displaystyle \tan 4 \alpha = \frac{ 4 \tan \alpha \left( 1 - \tan ^ {2} \alpha \right) }{ 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha }

導出

 \displaystyle \sin 7 \alpha = \sin \left( 4 \alpha + 3 \alpha \right) = \sin 4 \alpha \cos 3 \alpha + \cos 4 \alpha \sin 3 \alpha \\ \displaystyle = 4 \sin \alpha \cos \alpha \left( 1 - 2 \sin ^ {2} \alpha \right) \cos \alpha \left( 1 - 4 \sin ^ 2 \alpha \right) + \left( 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1 \right) \sin \alpha \left( 3 - 4 \sin ^ {2} \alpha \right) \\ \displaystyle = \sin \alpha \left\{ 4 \cos ^ {2} \alpha \left( 1 - 2 \sin ^ {2} \alpha \right) \left( 1 - 4 \sin ^ 2 \alpha \right) + \left( 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1 \right) \left( 3 - 4 \sin ^ {2} \alpha \right) \right\} \\ \displaystyle = \sin \alpha \left\{ 4 \left( 1 - \sin ^ {2} \alpha \right) \left( 1 - 2 \sin ^ {2} \alpha \right) \left( 1 - 4 \sin ^ 2 \alpha \right) + \left( 8 \sin ^ {4} \alpha - 8 \sin ^ {2} \alpha + 1 \right) \left( 3 - 4 \sin ^ {2} \alpha \right) \right\} \\ \displaystyle = 7 \sin \alpha - 56 \sin ^ {3} \alpha + 112 \sin ^ {5} \alpha - 64 \sin ^ {7} \alpha
 \displaystyle \cos 7 \alpha = \cos \left( 4 \alpha + 3 \alpha \right) = \cos 4 \alpha \cos 3 \alpha - \sin 4 \alpha \sin 3 \alpha \\ \displaystyle = \left( 8 \cos ^ {4} \alpha - 8 \cos ^ {2} \alpha + 1 \right) \cos \alpha \left( 4 \cos ^ {2} \alpha - 3 \right) - 4 \sin \alpha \cos \alpha \left( 2 \cos ^ {2} \alpha - 1 \right) \sin \alpha \left( 4 \cos ^ {2} \alpha - 1 \right) \\ \displaystyle = 64 \cos ^ {7} \alpha - 112 \cos ^ {5} \alpha + 56 \cos ^ {3} \alpha - 7 \cos \alpha
 \displaystyle \tan 7 \alpha = \tan \left( 4 \alpha + 3 \alpha \right) = \frac {\tan 4 \alpha + \tan 3 \alpha}{ 1 - {\tan 4 \alpha} {\tan 3 \alpha} } \\ \displaystyle = \frac {\frac{ 4 \tan \alpha \left( 1 - \tan ^ {2} \alpha \right) }{ 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha } + \frac{ \tan \alpha \left( 3 - \tan ^ {2} \alpha \right) }{ 1 - 3 \tan ^ {2} \alpha }}{ 1 - {\frac{ 4 \tan \alpha \left( 1 - \tan ^ {2} \alpha \right) }{ 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha }} \cdot {\frac{ \tan \alpha \left( 3 - \tan ^ {2} \alpha \right) }{ 1 - 3 \tan ^ {2} \alpha }} } \\ \displaystyle = \frac{ 4 \tan \alpha \left( 1 - \tan ^ {2} \alpha \right) \left( 1 - 3 \tan ^ {2} \alpha \right) + \tan \alpha \left( 3 - \tan ^ {2} \alpha \right) \left( 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha \right) }{ \left( 1 - 6 \tan ^ {2} \alpha + \tan ^ {4} \alpha \right) \left( 1 - 3 \tan ^ {2} \alpha \right) - 4 \tan \alpha \left( 1 - \tan ^ {2} \alpha \right) \tan \alpha \left( 3 - \tan ^ {2} \alpha \right) } \\ \displaystyle = \frac{ 7 \tan \alpha - 35 \tan ^ {3} \alpha + 21 \tan ^ {5} \alpha - \tan ^ {7} \alpha }{ 1 - 21 \tan ^ {2} \alpha + 35 \tan ^ {4} \alpha - 7 \tan ^ {6} \alpha }

 


 

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LibreOffice 数式(Math) のソース:

 

alignl sin 7 %alpha = sin ( 4 %alpha + 3 %alpha ) = sin 4 %alpha cos 3 %alpha + cos 4 %alpha sin 3 %alpha
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alignl phantom { y } = lbrace 4 sin %alpha cos %alpha ( 1 - 2 sin ^2 %alpha ) rbrace lbrace cos %alpha ( 1 - 4 sin ^2 %alpha ) rbrace + ( 8 sin ^4 %alpha - 8 sin ^2 %alpha + 1 ) lbrace sin %alpha ( 3 - 4 sin ^2 %alpha ) rbrace
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alignl phantom { y } = sin %alpha lbrace cos ^2 %alpha ( 1 - 2 sin ^2 %alpha ) ( 1 - 4 sin ^2 %alpha ) + ( 8 sin ^4 %alpha - 8 sin ^2 %alpha + 1 ) ( 3 - 4 sin ^2 %alpha ) rbrace
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alignl phantom { y } = sin %alpha lbrace ( 1 - sin ^2 %alpha ) ( 1 - 2 sin ^2 %alpha ) ( 1 - 4 sin ^2 %alpha ) + ( 8 sin ^4 %alpha - 8 sin ^2 %alpha + 1 ) ( 3 - 4 sin ^2 %alpha ) rbrace
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alignl phantom { y } = sin %alpha ( 7 - 56 sin ^2 %alpha + 112 sin ^4 %alpha - 64 sin ^6 %alpha )
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alignl phantom { y } = 7 sin %alpha - 56 sin ^3 %alpha + 112 sin ^5 %alpha - 64 sin ^7 %alpha

 

alignl cos 7 %alpha = cos ( 4 %alpha + 3 %alpha ) = cos 4 %alpha cos 3 %alpha - sin 4 %alpha sin 3 %alpha
newline
alignl phantom { y } = ( 8 cos ^4 %alpha - 8 cos^2 %alpha + 1 ) lbrace cos %alpha ( 4 cos ^2 %alpha - 3 ) rbrace - lbrace 4 sin %alpha cos %alpha ( 2 cos ^2 %alpha - 1) rbrace lbrace sin %alpha ( 4 cos ^2 %alpha - 1) rbrace
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alignl phantom { y } = cos %alpha lbrace ( 8 cos ^4 %alpha - 8 cos^2 %alpha + 1 ) ( 4 cos ^2 %alpha - 3 ) - 4 sin ^2 %alpha ( 2 cos ^2 %alpha - 1) ( 4 cos ^2 %alpha - 1) rbrace
newline
alignl phantom { y } = cos %alpha lbrace ( 8 cos ^4 %alpha - 8 cos^2 %alpha + 1 ) ( 4 cos ^2 %alpha - 3 ) + 4 ( cos ^2 %alpha - 1 ) ( 2 cos ^2 %alpha - 1) ( 4 cos ^2 %alpha - 1) rbrace
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alignl phantom { y } = cos %alpha ( 64 cos ^6 %alpha - 112 cos ^4 %alpha + 56 cos ^2 %alpha - 7 )
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alignl phantom { y } = 64 cos ^7 %alpha - 112 cos ^5 %alpha + 56 cos ^3 %alpha - 7 cos %alpha

 

alignl tan 7 %alpha = tan ( 4 %alpha + 3 %alpha ) = { alignc { tan 4 %alpha + tan 3 %alpha } over { 1 - tan 4 %alpha tan 3 %alpha } } = { alignc { { 4 tan %alpha ( 1 - tan ^2 %alpha ) } over { 1 - 6 tan ^2 %alpha + tan ^4 %alpha } + { tan %alpha ( 3 - tan ^2 %alpha ) } over { 1 - 3 tan ^2 %alpha } } over { 1 - { { 4 tan %alpha ( 1 - tan ^2 %alpha ) } over { 1 - 6 tan ^2 %alpha + tan ^4 %alpha } } cdot { { tan %alpha ( 3 - tan ^2 %alpha ) } over { 1 - 3 tan ^2 %alpha } } } }
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alignl phantom { y } = { alignc { tan %alpha lbrace 4 ( 1 - tan ^2 %alpha ) ( 1 - 3 tan ^2 %alpha ) + ( 3 - tan ^2 %alpha ) ( 1 - 6 tan ^2 %alpha + tan ^4 %alpha ) rbrace } over { ( 1 - 6 tan ^2 %alpha + tan ^4 %alpha ) ( 1 - 3 tan ^2 %alpha ) - 4 tan ^2 %alpha ( 1 - tan ^2 %alpha ) ( 3 - tan ^2 %alpha ) } }
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alignl phantom { y } = { alignc { tan %alpha ( 7 - 35 tan ^2 %alpha + 21 tan ^4 %alpha - tan ^6 %alpha ) } over { 1 - 21 tan ^2 %alpha + 35 tan ^4 %alpha - 7 tan ^6 %alpha } }
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alignl phantom { y } = { alignc { 7 tan %alpha - 35 tan ^3 %alpha + 21 tan ^5 %alpha - tan ^7 %alpha } over { 1 - 21 tan ^2 %alpha + 35 tan ^4 %alpha - 7 tan ^6 %alpha } }