sinh7倍角までの公式

\begin{alignat}{2}
&(1)  \sinh 2x=2 \sinh x\cosh x\\
&(2)  \sinh 3x=3 \sinh x+ 4 \sinh^3 x\\
&(3)  \sinh 4x=\cosh x(4 \sinh x +8 \sinh^3 x)\\
&(4)  \sinh 5x=5 \sinh x+20 \sinh^3 x+16 \sinh^5 x\\
&(5)  \sinh 6x=\cosh x(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)\\
&(6)  \sinh 7x=7\sinh x-56 \sinh^3 x+112 \sinh^5 x+64 \sinh^7 x\\
\end{alignat}










\((3)\) から \((7)\) のみ証明します。

また、途中で用いている \(\cosh\) の倍角の公式の詳細はこちらです。

<証明>

\begin{alignat}{2}
(3)  \sinh 4x&=2 \sinh 2x \cosh 2x\\
&=2 \cdot 2 \sinh x\cosh x(1+ 2 \sinh^2 x)\\
&=4 \sinh x(1+ 2 \sinh^2 x)\cosh x\\
&=\cosh x(4 \sinh x +8 \sinh^3 x)
\end{alignat}以上より$$\sinh 4x=\cosh x(4 \sinh x +8 \sinh^3 x)$$







\begin{alignat}{2}
(4)  \sinh 5x&=\sinh (4x+x)\\
&=\sinh 4x \cosh x+\cosh 4x \sinh x\\
&=\cosh^2 x(4 \sinh x+8\sinh^3 x)+(8\cosh^4 x-8 \cosh^2 x+1)\sinh x\\
&=(4 \cosh^2 x+8 \sinh^2 x \cosh^2 x+8 \cosh^4 x-8\cosh^2 x+1)\sinh x\\
&=(8 \cosh^4 x+8\sinh^2 x \cosh^2 x-4 \cosh^2 x+1)\sinh x\\
&=\{8 (1+\sinh^2 x)^2+8 \sinh^2 x (1+\sinh^2 x) -4(1+\sinh^2 x)+1\}\sinh x\\
&=\{8 (1+2\sinh^2 x+\sinh^4 x)+8 \sinh^2 x (1+\sinh^2 x) -4(1+\sinh^2 x)+1\}\sinh x\\
&=(8+16 \sinh^2 x+8 \sinh^4 x+8 \sinh^2 x+8 \sinh^4 x-4-4 \sinh^2 x+1)\sinh x\\
&=(5+20 \sinh^2 x+16 \sinh^4 x)\sinh x\\
&=5 \sinh x+20 \sinh^3 x+16 \sinh^5 x
\end{alignat}以上より$$\sinh 5x=5 \sinh x+20 \sinh^3 x+16 \sinh^5 x$$







\begin{alignat}{2}
(5)  \sinh 6x&=2 \sinh 3x \cosh 3x\\
&=2(3 \sinh x+4 \sinh^3 x)(4 \cosh^3 x-3 \cosh x)\\
&=2(3 \sinh x+4 \sinh^3 x)(4 \cos^2 x-3)\cosh x\\
&=2(3 \sinh x+4 \sinh^3 x)(1+4 \sinh^2 x)\cosh x\\
&=2(3 \sinh x+12 \sinh^3 x+4 \sinh^3 x+16 \sinh^5 x)\cosh x\\
&=2(3 \sinh x+16 \sinh^3 x+16 \sinh^5 x)\cosh x\\
&=\cosh x(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)
\end{alignat}以上より$$\sinh 6x=\cosh x(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)$$







\begin{alignat}{2}
(6)  \sinh 7x&=\sinh (6x+x)\\
&=\sinh 6x \cosh x+\cosh 6x \sinh x\\
&=\cosh^2 x(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)+(32 \cosh^6 x-48 \cosh^4 x+18 \cosh^2 x-1) \sinh x\\
&=(1+\sinh^2 x)(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)+\{32 (1+\sinh^2 x)^3-48 (1+\sinh^2 x)^2+18 (1+\sinh^2 x)-1\} \sin x\\
&=(1+\sinh^2 x)(6 \sinh x-32 \sinh^3 x+32 \sinh^5 x)+\{32 (1+3\sinh^2 x+3 \sinh^4 x+\sinh^6 x)-48 (1+2 \sinh^2 x+\sinh^4 x)+18 (1+\sinh^2 x)-1\} \sinh x\\
&=(1+\sinh^2 x)(6 \sinh x+32 \sinh^3 x+32 \sinh^5 x)+(32+96 \sinh^2 x+96 \sinh^4 x+32 \sinh^6 x-48-96 \sinh^2 x-48 \sinh^4 x+18+18 \sinh^2 x-1)\sinh x\\
&=6 \sinh x+32 \sinh^3 x+32 \sinh^5 x+6\sinh^3 x+32 \sinh^5 x+32 \sinh^7 x+32 \sinh x+96 \sinh^3 x+96 \sinh^5 x+32 \sinh^7 x-48 \sinh x-96 \sinh^3 x-48 \sinh^5 x+18 \sinh x+18 \sinh^3 x-\sinh x\\
&=7\sinh x+56 \sinh^3 x+112 \sinh^5 x+64 \sinh^7 x
\end{alignat} 以上より$$\sinh 7x=7\sinh x+56 \sinh^3 x+112 \sinh^5 x+64 \sinh^7 x$$

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