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bunkei-2-3.tex
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bunkei-2-3.tex
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\documentclass[a4j, 11pt]{jarticle}
% \usepackage{amsmath}
% \usepackage{amssymb}
% \usepackage{amsthm}
\begin{document}
\subsection*{(3)}
(2) より P$_n$, \ P$_{n + 1}$ の座標はそれぞれ $(\cos n \alpha, \sin n \alpha), \ (\cos (n + 1) \alpha, \sin (n + 1) \alpha)$ であるので,
\begin{align*}
\triangle \textrm{P}_n \textrm{O} \textrm{P}_{n + 1} &= |\textrm{O} \textrm{P}_n| |\textrm{O} \textrm{P}_{n + 1}| \frac{1}{2} \sin \angle \textrm{P}_n \textrm{O} \textrm{P}_{n + 1} \\
&= \frac{1}{2} \left| \sin (n + 1) \alpha \cos n \alpha - \sin n \alpha \cos (n + 1) \alpha \right| \\
&= \frac{1}{2} \left| \sin \alpha \right| \\
&= \frac{1}{2} \sin \alpha \\
\end{align*}
となる. ここで $\displaystyle \tan \left( \frac{\alpha}{2} \right) = k$ より, $\sin \alpha = \displaystyle \frac{2k}{1 + k^2}$ なので,
\begin{align*}
\triangle P_n O P_{n + 1} &= \frac{1}{2} \sin \alpha = \frac{k}{1 + k^2}
\end{align*}
となる.
\end{document}