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三角関数 - 7倍角の公式(2)

イメージ 1

三角関数の7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 $n = 7$ を代入し展開

$\cos 7 \theta + i \sin 7 \theta = ( \cos \theta + i \sin \theta )^{7} \\ = \cos^{7} \theta + 7 \cos^{6} \theta \left( i \sin \theta \right) + 21 \cos^{5} \theta \left( i \sin \theta \right)^{2} + 35 \cos^{4} \theta \left( i \sin \theta \right)^{3} + 35 \cos^{3} \theta \left( i \sin \theta \right)^{4} + 21 \cos^{2} \theta \left( i \sin \theta \right)^{5} \\ + 7 \cos \theta \left( i \sin \theta \right)^{6} + \left( i \sin \theta \right)^{7} \\ = \cos^{7} \theta + 7 i \cos^{6} \theta \sin \theta - 21 \cos^{5} \theta \sin^{2} \theta + 35 i \cos^{4} \theta \sin^{3} \theta + 35 \cos^{3} \theta \sin^{4} \theta + 21 i \cos^{2} \theta \sin^{5} \theta \\ - 7 \cos \theta \sin^{6} \theta - i \sin^{7} \theta \\ = \cos^{7} \theta - 21 \cos^{5} \theta \sin^{2} \theta + 35 \cos^{3} \theta \sin^{4} \theta - 7 \cos \theta \sin^{6} \theta \\ - i \left( 7 \cos^{6} \theta \sin \theta - 35 \cos^{4} \theta \sin^{3} \theta + 21 \cos^{2} \theta \sin^{5} \theta - \sin^{7} \theta \right) \\ = \cos \theta \left( \cos^{6} \theta - 21 \cos^{4} \theta \sin^{2} \theta + 35 \cos^{2} \theta \sin^{4} \theta - 7 \sin^{6} \theta \right) \\ + i \sin \theta \left( 7 \cos^{6} \theta - 35 \cos^{4} \theta \sin^{2} \theta + 21 \cos^{2} \theta \sin^{4} \theta - \sin^{6} \theta \right)$

ここで実部と虚部に分けて、展開していきます。

実部

$\cos \theta \left( \cos^{6} \theta - 21 \cos^{4} \theta \sin^{2} \theta + 35 \cos^{2} \theta \sin^{4} \theta - 7 \sin^{6} \theta \right) \\ = \cos \theta \left\{ \cos^{6} \theta - 21 \cos^{4} \theta \left( 1 - \cos^{2} \theta \right) + 35 \cos^{2} \theta \left( 1 - \cos^{2} \theta \right)^{2} - 7 \left( 1 - \cos^{2} \theta \right)^{3} \right\} \\ \small = \cos \theta \left\{ \cos^{6} \theta - 21 \cos^{4} \theta \left( 1 - \cos^{2} \theta \right) + 35 \cos^{2} \theta \left( 1 - 2 \cos^{2} \theta + \cos^{4} \theta \right) - 7 \left( 1 - 3 \cos^{2} \theta + 3 \cos^{4} \theta - \cos^{6} \theta \right) \right\} \\ = \cos \theta \left\{ ( 1 + 21 + 35 + 7 ) \cos^{6} \theta - ( 21 + 35 \cdot 2 + 7 \cdot 3 ) \cos^{4} \theta + ( 35 + 7 \cdot 3 ) \cos^{2} \theta - 7 \right\} \\ = \cos \theta \left( 64 \cos^{6} \theta - 112 \cos^{4} \theta + 56 \cos^{2} \theta - 7 \right) \\ = 64 \cos^{7} \theta - 112 \cos^{5} \theta + 56 \cos^{3} \theta - 7 \cos \theta$

∴ $\cos 7 \theta = 64 \cos^{7} \theta - 112 \cos^{5} \theta + 56 \cos^{3} \theta - 7 \cos \theta$

虚部

$\sin \theta \left( 7 \cos^{6} \theta - 35 \cos^{4} \theta \sin^{2} \theta + 21 \cos^{2} \theta \sin^{4} \theta - \sin^{6} \theta \right) \\ = \sin \theta \left\{ 7 \left( 1 - \sin^{2} \theta \right)^{3} - 35 \left( 1 - \sin^{2} \theta \right)^{2} \sin^{2} \theta + 21 \left( 1 - \sin^{2} \theta \right) \sin^{4} \theta - \sin^{6} \theta \right\} \\ \small = \sin \theta \left\{ 7 \left( 1 - 3 \sin^{2} \theta + 3 \sin^{4} \theta - \sin^{6} \theta \right) - 35 \left( 1 - 2 \sin^{2} \theta + \sin^{4} \theta \right) \sin^{2} \theta + 21 \left( 1 - \sin^{2} \theta \right) \sin^{4} \theta - \sin^{6} \theta \right\} \\ = \sin \theta \left\{ 7 - ( 7 \cdot 3 + 35 ) \sin^{2} \theta + ( 7 \cdot 3 + 35 \cdot 2 + 21 ) \sin^{4} \theta - ( 7 + 35 + 21 + 1 ) \sin^{6} \theta \right\} \\ = \sin \theta \left( 7 - 56 \sin^{2} \theta + 112 \sin^{4} \theta - 64 \sin^{6} \theta \right) \\ = 7 \sin \theta - 56 \sin^{3} \theta + 112 \sin^{5} \theta - 64 \sin^{7} \theta$

∴ $\sin 7 \theta = 7 \sin \theta - 56 \sin^{3} \theta + 112 \sin^{5} \theta - 64 \sin^{7} \theta$

$\tan 7 \theta$ の導出

これら $\sin 7 \theta$ ， $\cos 7 \theta$ から、

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