三角関数の合成
三角関数の合成
(1)
\begin{align*} a\sin\theta+b\cos\theta & =\sqrt{a^{2}+b^{2}}\sin(\theta+\alpha) \end{align*} \begin{align*} \alpha & =\arcsin\frac{b}{\sqrt{a^{2}+b^{2}}}\\ & =\arccos\frac{a}{\sqrt{a^{2}+b^{2}}} \end{align*}(2)
\begin{align*} a\sin\theta+b\cos\theta & =\sqrt{a^{2}+b^{2}}\cos(\theta-\beta) \end{align*} \begin{align*} \beta & =\arcsin\frac{a}{\sqrt{a^{2}+b^{2}}}\\ & =\arccos\frac{b}{\sqrt{a^{2}+b^{2}}} \end{align*}(1)
\begin{align*} \alpha & =\arcsin\frac{b}{\sqrt{a^{2}+b^{2}}}\\ & =\arccos\frac{a}{\sqrt{a^{2}+b^{2}}} \end{align*} とおくと、\begin{align*} a\sin\theta+b\cos\theta & =\sqrt{a^{2}+b^{2}}\left(\sin\theta\frac{a}{\sqrt{a^{2}+b^{2}}}+\cos\theta\frac{b}{\sqrt{a^{2}+b^{2}}}\right)\\ & =\sqrt{a^{2}+b^{2}}\left(\sin\theta\cos\alpha+\cos\theta\sin\alpha\right)\\ & =\sqrt{a^{2}+b^{2}}\sin(\theta+\alpha) \end{align*}
(2)
(1)より、\begin{align*} a\sin\theta+b\cos\theta & =\sqrt{a^{2}+b^{2}}\sin(\theta+\alpha)\\ & =\sqrt{a^{2}+b^{2}}\cos(\theta+\alpha-\frac{\pi}{2})\\ & =\sqrt{a^{2}+b^{2}}\cos(\theta-\beta)\qquad,\qquad\beta=\frac{\pi}{2}-\alpha \end{align*} \begin{align*} \beta & =\frac{\pi}{2}-\alpha\\ & =\frac{\pi}{2}-\arcsin\frac{b}{\sqrt{a^{2}+b^{2}}}\\ & =\arccos\frac{b}{\sqrt{a^{2}+b^{2}}} \end{align*} 同様に、
\begin{align*} \beta & =\frac{\pi}{2}-\alpha\\ & =\frac{\pi}{2}-\arccos\frac{a}{\sqrt{a^{2}+b^{2}}}\\ & =\arcsin\frac{a}{\sqrt{a^{2}+b^{2}}} \end{align*} これより、
\begin{align*} \beta & =\arccos\frac{b}{\sqrt{a^{2}+b^{2}}}\\ & =\arcsin\frac{a}{\sqrt{a^{2}+b^{2}}} \end{align*}
(2)別解
\begin{align*} \beta & =\arccos\frac{b}{\sqrt{a^{2}+b^{2}}}\\ & =\arcsin\frac{a}{\sqrt{a^{2}+b^{2}}} \end{align*} とおくと、\begin{align*} a\sin\theta+b\cos\theta & =\sqrt{a^{2}+b^{2}}\left(\cos\theta\frac{b}{\sqrt{a^{2}+b^{2}}}+\sin\theta\frac{a}{\sqrt{a^{2}+b^{2}}}\right)\\ & =\sqrt{a^{2}+b^{2}}\left(\cos\theta\cos\beta+\sin\theta\sin\beta\right)\\ & =\sqrt{a^{2}+b^{2}}\sin(\theta-\beta) \end{align*}
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三角関数を正接の半角、双曲線関数を双曲線正接の半角で表す。
\[
\sin z=\frac{2\tan\frac{z}{2}}{1+\tan^{2}\frac{z}{2}}
\]
3角関数・双曲線関数の総和
\[
\sum_{k=m_{1}}^{m_{2}}\sin\left(ak+b\right)=\sin^{-1}\left(\frac{a}{2}\right)\sin\left(\left(m_{1}+m_{2}\right)\frac{a}{2}+b\right)\sin\left(\left(1+m_{2}-m_{1}\right)\frac{a}{2}\right)
\]
三角関数・双曲線関数の微分
\[
\left(\sin x\right)'=\cos x
\]
逆三角関数と逆双曲線関数の冪乗積分漸化式
\[
\int\sin^{\bullet,n}xdx=x\sin^{\bullet,n}x+n\sqrt{1-x^{2}}\sin^{\bullet,n-1}x-n(n-1)\int\sin^{\bullet,n-2}xdx
\]