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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逆3角関数と逆双曲線関数の微分
\[
\frac{d}{dx}\sin^{\bullet}x=\frac{1}{\sqrt{1-x^{2}}}
\]
三角関数と双曲線関数のn乗積分
\[
\int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\}
\]
三角関数・双曲線関数の実部と虚部
\[
\sin z=\sin\left(\Re z\right)\cosh\left(\Im z\right)+i\cos\left(\Re z\right)\sinh\left(\Im z\right)
\]
偏角の3角関数
\[
\sin\Arg z=\frac{\Im z}{\left|z\right|}
\]