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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