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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正接・双曲線正接の総和展開
\[
\tan\pi z=\frac{2z}{\pi}\sum_{k=1}^{\infty}\frac{1}{\left(k-\frac{1}{2}\right)^{2}-z^{2}}
\]
3角関数(双曲線関数)の逆3角関数(逆双曲線関数)が恒等写像になる条件
\[
\sin^{\bullet}\sin z=?z
\]
三角関数と双曲線関数の対数の積分
\[
\int\Log\sin^{\alpha}zdz=z\Log\sin^{\alpha}x+\frac{i\alpha}{2}z^{2}+\alpha z\Li_{1}\left(e^{2iz}\right)+\frac{i\alpha}{2}\Li_{2}\left(e^{2iz}\right)+\C{}
\]
三角関数・双曲線関数の一次結合の逆数の積分
\[
\int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz=-\frac{2}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\bullet}\frac{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}+C
\]