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三角関数 - 10倍角の公式

イメージ 1

三角関数の10倍角の公式です。
倍角の公式、5倍角の公式から導出できます。

準備

$\sin^{2} \alpha + \cos^{2} \alpha = 1 \Leftrightarrow \cos^{2} \alpha = 1 - \sin^{2} \alpha$

$\displaystyle \sin 2\alpha = 2 \sin \alpha \cos \alpha$

$\displaystyle \sin 5 \alpha = \sin \alpha \left( 16 \sin ^ {4} \alpha - 20 \sin ^ {2} \alpha + 5 \right)$

$\displaystyle \cos 2\alpha = 2 \cos ^ {2} \alpha - 1$

$\displaystyle \cos 5 \alpha = \cos \alpha \left( 16 \cos ^ {4} \alpha - 20 \cos ^ {2} \alpha + 5 \right)$

$\displaystyle \tan 2 \alpha = \frac{ 2 \tan \alpha } { 1 - \tan ^ {2} \alpha }$

$\displaystyle \tan 5 \alpha = \frac{ \tan \alpha \left( 5 - 10 \tan ^ {2} \alpha + \tan ^ {4} \alpha \right) }{ 1 - 10 \tan ^ {2} \alpha + 5 \tan ^ {4} \alpha }$

導出

$\displaystyle \sin 10 \alpha = \sin 5 ( 2 \alpha ) = \sin 2 \alpha \left( 16 \sin^{4} 2 \alpha - 20 \sin^{2} 2 \alpha + 5 \right) \\ = 2 \sin \alpha \cos \alpha \left\{ 16 \left( \sin \alpha \cos \alpha \right)^{4} - 20 \left( \sin \alpha \cos \alpha \right)^{2} + 5 \right\} \\ = 2 \sin \alpha \cos \alpha \left( 256 \sin^{4} \alpha \cos^{4} \alpha - 80 \sin^{2} \alpha \cos^{2} \alpha + 5 \right) \\ = 2 \sin \alpha \cos \alpha \left\{ 256 \sin^{4} \alpha \left( 1 - \sin^{2} \alpha \right)^{2} - 80 \sin^{2} \alpha \left( 1 - \sin^{2} \alpha \right) + 5 \right\} \\ = 2 \sin \alpha \cos \alpha \left\{ 256 \sin^{4} \alpha \left( 1 - 2 \sin^{2} + \sin^{4} \alpha \right) - 80 \sin^{2} \alpha + 80 \sin^{4} \alpha + 5 \right\} \\ = 2 \sin \alpha \cos \alpha \left( 256 \sin^{4} \alpha - 512 \sin^{6} \alpha + 256 \sin^{8} \alpha - 80 \sin^{2} \alpha + 80 \sin^{4} \alpha + 5 \right) \\ = 2 \sin \alpha \cos \alpha \left( 256 \sin^{8} \alpha - 512 \sin^{6} \alpha + 336 \sin^{4} \alpha - 80 \sin^{2} \alpha + 5 \right) \\ = \cos \alpha \left( 512 \sin ^ 9 \alpha - 1024 \sin ^ 7 \alpha + 672 \sin ^ 5 \alpha - 160 \sin ^ 3 \alpha + 10 \sin \alpha \right)$

$\displaystyle \cos 10 \alpha = \cos 5 ( 2 \alpha ) = \cos 2 \alpha \left( 16 \cos^{4} 2 \alpha - 20 \cos^{2} 2 \alpha + 5 \right) = \cos 2 \alpha \left\{ 4 \cos^{2} 2 \alpha \left( 4 \cos^{2} 2 \alpha - 5 \right) + 5 \right\} \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left[ 4 \left( 2 \cos^{2} \alpha - 1 \right)^{2} \left\{ 4 \left( 2 \cos^{2} \alpha - 1 \right)^{2} - 5 \right\} + 5 \right] \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left[ 4 \left( 4 \cos^{4} \alpha - 4 \cos^{2} \alpha + 1 \right) \left\{ 4 \left( 4 \cos^{4} \alpha - 4 \cos^{2} \alpha + 1 \right) - 5 \right\} + 5 \right] \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left\{ 4 \left( 4 \cos^{4} \alpha - 4 \cos^{2} \alpha + 1 \right) \left( 16 \cos^{4} \alpha - 16 \cos^{2} \alpha + 4 - 5 \right) + 5 \right\} \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left\{ 4 \left( 4 \cos^{4} \alpha - 4 \cos^{2} \alpha + 1 \right) \left( 16 \cos^{4} \alpha - 16 \cos^{2} \alpha -1 \right) + 5 \right\} \\ \small = \left( 2 \cos^{2} \alpha - 1 \right) \left\{ 4 \left( 64 \cos^{8} \alpha - 64 \cos^{6} \alpha + 16 \cos^{4} \alpha - 64 \cos^{6} \alpha + 64 \cos^{4} \alpha - 16 \cos^{2} \alpha - 4 \cos^{4} \alpha + 4 \cos^{2} \alpha - 1 \right) + 5 \right\} \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left\{ 4 \left( 64 \cos^{8} \alpha - 128 \cos^{6} \alpha + 76 \cos^{4} \alpha - 12 \cos^{2} \alpha - 1 \right) + 5 \right\} \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left( 256 \cos^{8} \alpha - 512 \cos^{6} \alpha + 304 \cos^{4} \alpha - 48 \cos^{2} \alpha - 4 + 5 \right) \\ = \left( 2 \cos^{2} \alpha - 1 \right) \left( 256 \cos^{8} \alpha - 512 \cos^{6} \alpha + 304 \cos^{4} \alpha - 48 \cos^{2} \alpha + 1 \right) \\ \small = 512 \cos ^ {10} \alpha - 1024 \cos ^ {8} \alpha + 608 \cos ^ {6} \alpha - 96 \cos ^ {4} \alpha + 2 \cos ^ {2} \alpha - 256 \cos ^ {8} \alpha + 512 \cos ^ {6} \alpha - 304 \cos ^ {4} \alpha + 48 \cos ^ {2} \alpha - 1 \\ = 512 \cos ^ {10} \alpha - 1280 \cos ^ 8 \alpha + 1120 \cos ^ 6 \alpha - 400 \cos ^ 4 \alpha + 50 \cos ^ 2 \alpha - 1$

$\displaystyle \tan 10 \alpha = \tan 5 ( 2 \alpha ) = \frac{ \tan 2 \alpha \left( \tan^{4} 2 \alpha - 10 \tan^{2} 2 \alpha + 5 \right) }{ 5 \tan^{4} 2 \alpha - 10 \tan^{2} 2 \alpha + 1 } \\ \displaystyle = \dfrac{ \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \left\{ \left( \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \right)^{4} - 10 \left( \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \right)^{2} + 5 \right\} }{ 5 \left( \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \right)^{4} - 10 \left( \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \right)^{2} + 1 } \\ \displaystyle = \dfrac{ \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \left\{ \dfrac{ ( 2 \tan \alpha )^{4} - 10 ( 2 \tan \alpha )^{2} \left( \tan^{2} \alpha - 1 \right)^{2} + 5 \left( \tan^{2} \alpha - 1 \right)^{4} }{ \left( \tan^{2} \alpha - 1 \right)^{4} } \right\} }{ \dfrac{ 5 ( 2 \tan \alpha )^{4} - 10 ( 2 \tan \alpha )^{2} \left( \tan^{2} \alpha - 1 \right)^{2} + \left( \tan^{2} \alpha - 1 \right)^{4} }{ \left( \tan^{2} \alpha - 1 \right)^{4} } } \\ \displaystyle = \dfrac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \dfrac{ ( 2 \tan \alpha )^{4} - 10 ( 2 \tan \alpha )^{2} \left( \tan^{2} \alpha - 1 \right)^{2} + 5 \left( \tan^{2} \alpha - 1 \right)^{4} }{ 5 ( 2 \tan \alpha )^{4} - 10 ( 2 \tan \alpha )^{2} \left( \tan^{2} \alpha - 1 \right)^{2} + \left( \tan^{2} \alpha - 1 \right)^{4} } \\ = \frac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \frac{ 16 \tan^{4} \alpha - 10 \left( 4 \tan^{2} \alpha \right) \left( \tan^{4} \alpha - 2 \tan^{2} \alpha + 1 \right) + 5 \left( \tan^{8} \alpha - 4 \tan^{6} \alpha + 6 \tan^{4} \alpha - 4 \tan^{2} \alpha + 1 \right) }{ 5 \cdot 16 \tan^{4} \alpha - 10 \left( 4 \tan^{2} \alpha \right) \left( \tan^{4} \alpha - 2 \tan^{2} \alpha + 1 \right) + \left( \tan^{8} \alpha - 4 \tan^{6} \alpha + 6 \tan^{4} \alpha - 4 \tan^{2} \alpha + 1 \right) } \\ = \frac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \frac{ 16 \tan^{4} \alpha - 10 \cdot 4 \left( \tan^{6} \alpha - 2 \tan^{4} \alpha + \tan^{2} \alpha \right) + 5 \left( \tan^{8} \alpha - 4 \tan^{6} \alpha + 6 \tan^{4} \alpha - 4 \tan^{2} \alpha + 1 \right) }{ 5 \cdot 16 \tan^{4} \alpha - 10 \cdot 4 \left( \tan^{6} \alpha - 2 \tan^{4} \alpha + \tan^{2} \alpha \right) + \left( \tan^{8} \alpha - 4 \tan^{6} \alpha + 6 \tan^{4} \alpha - 4 \tan^{2} \alpha + 1 \right) } \\ \displaystyle = \frac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \frac{ 5 \tan^{8} \alpha - \left( 10 \cdot 4 + 5 \cdot 4 \right) \tan^{6} \alpha + \left( 16 + 10 \cdot 4 \cdot 2 + 5 \cdot 6 \right) \tan^{4} \alpha - \left( 10 \cdot 4 + 5 \cdot 4 \right) \tan^{2} \alpha + 5 }{ \tan^{8} \alpha - \left( 10 \cdot 4 + 4 \right) \tan^{6} \alpha + \left( 5 \cdot 16 + 10 \cdot 4 \cdot 2 + 6 \right) \tan^{4} \alpha - \left( 10 \cdot 4 + 4 \right) \tan^{2} \alpha + 1 } \\ \displaystyle = \frac{ 2 \tan \alpha }{ \tan^{2} \alpha - 1 } \cdot \frac{ 5 \tan^{8} \alpha - 60 \tan^{6} \alpha + 126 \tan^{4} \alpha - 60 \tan^{2} \alpha + 5 }{ \tan^{8} \alpha - 44 \tan^{6} \alpha + 166 \tan^{4} \alpha - 44 \tan^{2} \alpha + 1 } \\ \displaystyle = \frac{ 10 \tan^{9} \alpha - 120 \tan^{7} \alpha + 252 \tan^{5} \alpha - 120 \tan^{3} \alpha + 10 \tan \alpha }{ \left( \tan^{2} \alpha - 1 \right) \left( \tan^{8} \alpha - 44 \tan^{6} \alpha + 166 \tan^{4} \alpha - 44 \tan^{2} \alpha + 1 \right) } \\ \displaystyle = \frac{ 10 \tan^{9} \alpha - 120 \tan^{7} \alpha + 252 \tan^{5} \alpha - 120 \tan^{3} \alpha + 10 \tan \alpha }{ \tan^{10} \alpha - \left( 1 + 44 \right) \tan^{8} \alpha + \left( 44 + 166 \right) \tan^{6} \alpha - \left( 166 + 44 \right) \tan^{4} \alpha + \left( 44 + 1 \right) \tan^{2} \alpha - 1 } \\ \displaystyle = \frac{ 10 \tan^{9} \alpha - 120 \tan^{7} \alpha + 252 \tan^{5} \alpha - 120 \tan^{3} \alpha + 10 \tan \alpha }{ \tan^{10} \alpha - 45 \tan^{8} \alpha + 210 \tan^{6} \alpha - 210 \tan^{4} \alpha + 45 \tan^{2} \alpha - 1 }$

LibreOffice 数式(Math) のソース：

alignl sin 10 %alpha = sin 5 ( 2 %alpha ) = sin 2 %alpha ( 16 sin^4 2 %alpha - 20 sin^2 2 %alpha + 5 )
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alignl phantom { y } = 2 sin %alpha cos %alpha lbrace 16 ( 2 sin %alpha cos %alpha ) ^4 - 20 ( 2 sin %alpha cos %alpha ) ^2 + 5 rbrace
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alignl phantom { y } = 2 sin %alpha cos %alpha ( 256 sin ^4 %alpha cos ^4 %alpha - 80 sin ^2 %alpha cos ^2 %alpha + 5 )
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alignl phantom { y } = 2 sin %alpha cos %alpha lbrace 256 sin ^4 %alpha ( 1 - sin ^2 %alpha) ^2 - 80 sin ^2 %alpha ( 1 - sin ^2 %alpha) ^2 + 5 rbrace
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alignl phantom { y } = 2 sin %alpha cos %alpha ( 256 sin ^8 %alpha - 512 sin ^6 %alpha + 336 sin ^4 %alpha - 80 sin ^2 %alpha + 5 )
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alignl phantom { y } = cos %alpha ( 512 sin ^9 %alpha - 1024 sin ^7 %alpha + 672 sin ^5 %alpha - 160 sin ^3 %alpha + 10 sin %alpha )

alignl cos 10 %alpha = cos 5 ( 2 %alpha ) = cos 2 %alpha ( 16 cos ^4 2 %alpha - 20 cos ^2 2 %alpha + 5 ) = cos 2 %alpha lbrace 4 cos ^2 2 %alpha ( 4 cos ^2 2 %alpha - 5 ) + 5 rbrace
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alignl phantom { y } = ( 2 cos ^2 %alpha - 1 ) [ 4 ( 2 cos ^2 %alpha - 1 ) ^2 lbrace 4 ( 2 cos ^2 %alpha - 1 ) ^2 - 5 rbrace + 5 ]
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alignl phantom { y } = ( 2 cos ^2 %alpha - 1 ) lbrace 4 ( 4 cos ^4 2 %alpha - 4 cos ^2 2 %alpha + 1 ) ( 16 cos ^4 2 %alpha - 16 cos ^2 2 %alpha -1 ) + 5 rbrace
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alignl phantom { y } = ( 2 cos ^2 %alpha - 1 ) ( 256 cos ^8 %alpha - 512 cos ^6 %alpha + 304 cos ^4 %alpha - 48 cos ^2 %alpha + 1)
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alignl phantom { y } = 512 cos ^10 %alpha - 1280 cos ^8 %alpha + 1120 cos ^6 %alpha - 400 cos ^4 %alpha + 50 cos ^2 %alpha - 1

alignl tan 10 %alpha = tan 5 ( 2 %alpha ) = { alignc { tan 2 %alpha ( tan ^4 2 %alpha - 10 tan ^2 2 %alpha + 5 ) } over { 5 tan ^4 2 %alpha - 10 tan ^2 2 %alpha + 1 } }
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alignl phantom { y } = { alignc { { { 2 tan %alpha } over { 1 - tan ^2 %alpha } } cdot left lbrace left ( { 2 tan %alpha } over { 1 - tan ^2 %alpha } right ) ^4 - 10 cdot left ( { 2 tan %alpha } over { 1 - tan ^2 %alpha } right ) ^2 + 5 right rbrace } over { 5 cdot left ( { 2 tan %alpha } over { 1 - tan ^2 %alpha } right ) ^4 - 10 cdot left ( { 2 tan %alpha } over { 1 - tan ^2 %alpha } right ) ^2 + 1 } }
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alignl phantom { y } = { alignc { { { 2 tan %alpha } over { 1 - tan ^2 %alpha } } cdot left lbrace { ( 2 tan %alpha ) ^4 - 10 ( 2 tan %alpha ) ^2 ( 1 - tan ^2 %alpha ) ^2 + 5 ( 1 - tan ^2 %alpha ) ^4 } over { ( 1 - tan ^2 %alpha ) ^4 } right rbrace } over { { 5 ( 2 tan %alpha ) ^4 - 10 ( 2 tan %alpha ) ^2 ( 1 - tan ^2 %alpha ) ^2 + ( 1 - tan ^2 %alpha ) ^4 } over { ( 1 - tan ^2 %alpha ) ^4 } } }
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alignl phantom { y } = { alignc { 2 tan %alpha lbrace 16 tan ^4 %alpha - 40 tan ^2 %alpha ( 1 - tan ^2 %alpha ) ^2 + 5 ( 1 - tan ^2 %alpha ) ^4 rbrace } over { ( 1 - tan ^2 %alpha ) lbrace 80 tan ^4 %alpha - 40 tan ^2 %alpha ( 1 - tan ^2 %alpha ) ^2 + ( 1 - tan ^2 %alpha ) ^4 rbrace } }
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alignl phantom { y } = { alignc { 2 tan %alpha ( 5 - 60 tan ^2 %alpha + 126 tan ^4 %alpha - 60 tan ^6 %alpha + 5 tan ^8 %alpha ) } over { ( 1 - tan ^2 %alpha ) ( 1 - 44 tan ^2 %alpha + 166 tan ^4 %alpha - 44 tan ^6 %alpha + tan ^8 %alpha ) } }
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alignl phantom { y } = { alignc { 10 tan %alpha - 120 tan ^3 %alpha + 252 tan ^5 %alpha - 120 tan ^7 %alpha + 10 tan ^9 %alpha } over { 1 - 45 tan ^2 %alpha + 210 tan ^4 %alpha - 210 tan ^6 %alpha + 45 tan ^8 %alpha - tan ^10 %alpha } }