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三角関数 - 9倍角の公式(2)

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

三角関数 - 9倍角の公式(2)

ド・モアブルの公式から導出する9倍角の公式

三角関数の9倍角の公式をド・モアブルの定理を用いて導出します。

展開時の係数は、パスカルの三角形を用います。

準備

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

 \displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}

 \displaystyle \tan^{2} \theta + 1 = \frac{1}{\cos^{2} \theta}

  \cos n \theta + i \sin n \theta = ( \cos \theta + i \sin \theta )^{n}

導出

ド・モアブルの定理で  n = 9 を代入し展開

 \cos 9 \theta + i \sin 9 \theta  = ( \cos \theta  + i \sin \theta )^{9} \\ = \cos^{9} \theta + 9 \cos^{8} \theta ( i \sin \theta ) + 36 \cos^{7} \theta ( i \sin \theta )^{2} + 84 \cos^{6} \theta ( i \sin \theta )^{3} + 126 \cos^{5} \theta ( i \sin \theta )^{4} \\ + 126 \cos^{4} \theta ( i \sin \theta )^{5} + 84 \cos^{3} \theta ( i \sin \theta )^{6} + 36 \cos^{2} \theta ( i \sin \theta )^{7} + 9 \cos \theta ( i \sin \theta )^{8} + ( i \sin \theta )^{9} \\ = \cos^{9} \theta + 9 i \cos^{8} \theta \sin \theta - 36 \cos^{7} \theta \sin^{2} \theta - 84 i \cos^{6} \theta \sin^{3} \theta + 126 \cos^{5} \theta \sin^{4} \theta + 126 i \cos^{4} \theta \sin^{5} \theta \\ - 84 \cos^{3} \theta \sin^{6} \theta - 36 i \cos^{2} \theta \sin^{7} \theta + 9 \cos \theta \sin^{8} \theta \\ = \cos^{9} \theta - 36 \cos^{7} \theta \sin^{2} \theta + 126 \cos^{5} \theta \sin^{4} \theta - 84 \cos^{3} \theta \sin^{6} \theta + 9 \cos \theta \sin^{8} \theta \\ + i \left( 9 \cos^{8} \theta \sin \theta - 84 \cos^{6} \theta \sin^{3} \theta + 126 \cos^{4} \theta \sin^{5} \theta - 36 \cos^{2} \theta \sin^{7} \theta + \sin^{9} \theta \right) \\ = \cos \theta \left( \cos^{8} \theta - 36 \cos^{6} \theta \sin^{2} \theta + 126 \cos^{4} \theta \sin^{4} \theta - 84 \cos^{2} \theta \sin^{6} \theta + 9 \sin^{8} \theta \right) \\ + i \sin \theta \left( 9 \cos^{8} \theta - 84 \cos^{6} \theta \sin^{2} \theta + 126 \cos^{4} \theta \sin^{4} \theta - 36 \cos^{2} \theta \sin^{6} \theta + \sin^{8} \theta \right)  

ここで実部と虚部に分けて、展開していきます。

実部

  \cos \theta \left( \cos^{8} \theta - 36 \cos^{6} \theta \sin^{2} \theta + 126 \cos^{4} \theta \sin^{4} \theta - 84 \cos^{2} \theta \sin^{6} \theta + 9 \sin^{8} \theta \right) \\ = \cos \theta \left\{ \cos^{8} \theta - 36 \cos^{6} \theta \left( 1 - \cos^{2} \theta \right) + 126 \cos^{4} \theta \left( 1 - \cos^{2} \theta \right)^{2} - 84 \cos^{2} \theta \left( 1 - \cos^{2} \theta \right)^{3} + 9 \left( 1 - \cos^{2} \theta \right)^{4} \right\} \\ \tiny = \cos \theta \left\{ \cos^{8} \theta - 36 \cos^{6} \theta \left( 1 - \cos^{2} \theta \right) + 126 \cos^{4} \theta \left( 1 - 2 \cos^{2} \theta + \cos^{4} \theta \right) - 84 \cos^{2} \theta \left( 1 - 3 \cos^{2} \theta + 3 \cos^{4} \theta - \cos^{6} \theta   \right) + 9 \left( 1 - 4 \cos^{2} \theta + 6 \cos^{4} \theta - 4 \cos^{6} \theta + 4 \cos^{8} \theta \right) \right\} \\ \small = \cos \theta \left\{ ( 1 + 36 + 126 + 84 + 9 ) \cos^{8} \theta - ( 36 + 126 \cdot 2 + 84 \cdot 3 + 9 \cdot 4 ) \cos^{6} \theta + ( 126 + 84 \cdot 3 + 9 \cdot 6 ) \cos^{4} \theta - ( 84 + 9 \cdot 4 ) \cos^{2} \theta + 9 \right\} \\ = \cos \theta \left( 256 \cos^{8} \theta - 576 \cos^{6} \theta + 432 \cos^{4} \theta - 120 \cos^{2} \theta + 9 \right) \\ = 256 \cos^{9} \theta - 576 \cos^{7} \theta + 432 \cos^{5} \theta - 120 \cos^{3} \theta + 9 \cos \theta

 \cos 9 \theta = 256 \cos^{9} \theta - 576 \cos^{7} \theta + 432 \cos^{5} \theta - 120 \cos^{3} \theta + 9 \cos \theta

虚部

 \sin \theta \left( 9 \cos^{8} \theta - 84 \cos^{6} \theta \sin^{2} \theta + 126 \cos^{4} \theta \sin^{4} \theta - 36 \cos^{2} \theta \sin^{6} \theta + \sin^{8} \theta \right) \\ = \sin \theta \left\{ 9 \left( 1 - \sin^{2} \theta \right)^{4} - 84 \left( 1 - \sin^{2} \theta \right)^{3} \sin^{2} \theta + 126 \left( 1 - \sin^{2} \theta \right)^{2} \sin^{4} \theta - 36 \left( 1 - \sin^{2} \theta \right) \sin^{6} \theta + \sin^{8} \theta \right\} \\ \tiny = \sin \theta \left\{ 9 \left( 1 - 4 \sin^{2} \theta + 6 \sin^{4} \theta - 4 \sin^{6} \theta + \sin^{8} \theta \right) - 84 \left( 1 - 3 \sin^{2} \theta + 3 \sin^{4} \theta - \sin^{6} \theta \right) \sin^{2} \theta + 126 \left( 1 - 2 \sin^{2} \theta + \sin^{4} \theta \right) \sin^{4} \theta - 36 \left( 1 - \sin^{2} \theta \right) \sin^{6} \theta + \sin^{8} \theta \right\} \\ \small = \sin \theta \left\{ ( 9 + 84 + 126 + 36 + 1 ) \sin^{8} \theta - ( 9 \cdot 4 + 84 \cdot 3 + 126 \cdot 2 + 36 ) \sin^{6} \theta + ( 9 \cdot 6 + 84 \cdot 3 + 126 ) \sin^{4} \theta - ( 9 \cdot 4 + 84 ) \sin^{2} \theta + 9  \right\} \\ = \sin \theta \left( 256 \sin^{8} \theta - 576 \sin^{6} \theta + 432 \sin^{4} \theta - 120 \sin^{2} \theta + 9  \right) \\ = 256 \sin^{9} \theta - 576 \sin^{7} \theta + 432 \sin^{5} \theta - 120 \sin^{3} \theta + 9 \sin \theta

 \sin 9 \theta = 256 \sin^{9} \theta - 576 \sin^{7} \theta + 432 \sin^{5} \theta - 120 \sin^{3} \theta + 9 \sin \theta

 \tan 9 \theta の導出

 これら \sin 9 \theta \cos 9 \theta から、

 \displaystyle \tan 9 \theta = \frac{\sin 9 \theta}{\cos 9 \theta} = \frac{ \sin \theta \left( 256 \sin^{8} \theta - 576 \sin^{6} \theta + 432 \sin^{4} \theta - 120 \sin^{2} \theta + 9  \right) }{ \cos \theta \left( 256 \cos^{8} \theta - 576 \cos^{6} \theta + 432 \cos^{4} \theta - 120 \cos^{2} \theta + 9 \right)  } \\ \displaystyle = \frac{ \sin \theta }{ \cos \theta } \cdot \frac{ \dfrac{ 256 \sin^{8} \theta }{ \cos^{8} \theta } - \dfrac{ 576 \sin^{6} \theta }{ \cos^{6} \theta } \cdot \dfrac{ 1 }{ \cos^{2} \theta } + \dfrac{ 432 \sin^{4} \theta }{ \cos^{4} \theta } \cdot \dfrac{ 1 }{ \cos^{4} \theta } - \dfrac{ 120 \sin^{2} \theta }{ \cos^{2} \theta } \cdot \dfrac{ 1 }{ \cos^{6} \theta } + \dfrac{ 9 }{ \cos^{8} \theta } }{ 256 - \dfrac{ 576 }{ \cos^{2} \theta } + \dfrac{ 432 }{ \cos^{4} \theta } - \dfrac{ 120 }{ \cos^{6} \theta } + \dfrac{ 9 }{ \cos^{8} \theta } } \\ \displaystyle = \tan \theta \cdot \frac{ 256 \tan^{8} \theta - 576 \tan^{6} \theta \left( \tan^{2} \theta + 1 \right) + 432 \tan^{4} \theta \left( \tan^{2} \theta + 1 \right)^{2} - 120 \tan^{2} \theta \left( \tan^{2} \theta + 1 \right)^{3} + 9 \left( \tan^{2} \theta + 1 \right)^{4} }{ 256 - 576 \left( \tan^{2} \theta + 1 \right) + 432 \left( \tan^{2} \theta + 1 \right)^{2} - 120 \left( \tan^{2} \theta + 1 \right)^{3} + 9 \left( \tan^{2} \theta + 1 \right)^{4} } \\ \small = \tan \theta \cdot \frac{ 256 \tan^{8} \theta - 576 \tan^{6} \theta \left( \tan^{2} \theta + 1 \right) + 432 \tan^{4} \theta \left( \tan^{4} \theta + 2 \tan^{2} \theta + 1 \right) - 120 \tan^{2} \theta \left( \tan^{6} \theta + 3 \tan^{4} \theta + 3 \tan^{2} \theta + 1 \right) + 9 \left( \tan^{8} \theta + 4 \tan^{6 } \theta + 6 \tan^{4} \theta + 4 \tan^{2} \theta + 1 \right) }{ 256 - 576 \left( \tan^{2} \theta + 1 \right) + 432 \left( \tan^{4} \theta + 2 \tan^{2} \theta + 1 \right) - 120 \left( \tan^{6} \theta + 3 \tan^{4} \theta + 3 \tan^{2} \theta + 1 \right) + 9 \left( \tan^{8} \theta + 4 \tan^{6 } \theta + 6 \tan^{4} \theta + 4 \tan^{2} \theta + 1 \right) } \\ = \tan \theta \cdot \frac{ ( 256 - 576 + 432 - 120 + 9 ) \tan^{8} \theta - ( 576 - 432 \cdot 2 + 120 \cdot 3 - 9 \cdot 4 ) \tan^{6} \theta + ( 432 - 120 \cdot 3 + 9 \cdot 6 ) \tan^{4} \theta - ( 120 - 9 \cdot 4 ) \tan^{2} \theta + 9 }{ 9 \tan^{8} \theta - ( 120 - 9 \cdot 4 ) \tan^{6} \theta + ( 432 - 120 \cdot 3 + 9 \cdot 6 ) \tan^{4} \theta - ( 576 - 432 \cdot 2 + 120 \cdot 3 - 9 \cdot 4 )\tan^{2} \theta + ( 256 - 576 + 432 - 120 + 9 ) } \\ \displaystyle = \tan \theta \cdot \frac{ \tan^{8} \theta - 36 \tan^{6} \theta + 126 \tan^{4} \theta - 84 \tan^{2} \theta + 9 }{ 9 \tan^{8} \theta - 84 \tan^{6} \theta + 126 \tan^{4} \theta - 36 \tan^{2} \theta + 1 } \\ \displaystyle = \frac{ \tan^{9} \theta - 36 \tan^{7} \theta + 126 \tan^{5} \theta - 84 \tan^{3} \theta + 9  \tan \theta }{ 9 \tan^{8} \theta - 84 \tan^{6} \theta + 126 \tan^{4} \theta - 36 \tan^{2} \theta + 1 }