今週の問題　問127　答え｜数学の偏差値を上げて合格を目指す

当サイトは、PRを含む場合があります。

上野竜生です。問127の答えを発表します。

問127

7つの三角関数の積
\(\displaystyle \cos{\left(\frac{\pi}{127}\right)}\cos{\left(\frac{2\pi}{127}\right)}\cos{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \)
の値を求めよ。

答え

求めるものをXとする。両辺を\(\displaystyle \sin{\frac{\pi}{127}} \)倍すると
\(\displaystyle \left( \sin{\frac{\pi}{127}} \right) X = \\ \displaystyle \sin{\left(\frac{\pi}{127}\right)}\cos{\left(\frac{\pi}{127}\right)}\cos{\left(\frac{2\pi}{127}\right)}\cos{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{2} \sin{\left(\frac{2\pi}{127}\right)}\cos{\left(\frac{2\pi}{127}\right)}\cos{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{4} \sin{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{8} \sin{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{16} \sin{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{32} \sin{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{64} \sin{\left(\frac{64\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)} \\ = \displaystyle \frac{1}{128} \sin{\left(\frac{128\pi}{127}\right)}\)
∵2倍角の公式
つまり
\(\displaystyle \left( \sin{\frac{\pi}{127}} \right) X =\frac{1}{128} \sin{\left(\frac{128\pi}{127}\right)} = -\frac{1}{128} \sin{\left(\frac{\pi}{127}\right)} \)
（∵sin(θ+π)=-sinθ)
\(\displaystyle \sin{\frac{\pi}{127}} \neq 0 \)なので両辺を\(\displaystyle \sin{\frac{\pi}{127}} \)で割ると
\( \displaystyle X=-\frac{1}{128} \)
よって
\(\displaystyle \cos{\left(\frac{\pi}{127}\right)}\cos{\left(\frac{2\pi}{127}\right)}\cos{\left(\frac{4\pi}{127}\right)}\cos{\left(\frac{8\pi}{127}\right)}\cos{\left(\frac{16\pi}{127}\right)}\cos{\left(\frac{32\pi}{127}\right)}\cos{\left(\frac{64\pi}{127}\right)}=-\frac{1}{128} \)

解説を読んで数学がわかった「つもり」になりましたか？数学は読んでいるうちはわかったつもりになりますが演習をこなさないと実力になりません。そのためには問題集で問題を解く練習も必要です。オススメの参考書を厳選しました

<高校数学>