{"id":3005,"date":"2025-09-03T02:45:55","date_gmt":"2025-09-02T17:45:55","guid":{"rendered":"https:\/\/math-friend.com\/?p=3005"},"modified":"2025-09-03T02:50:14","modified_gmt":"2025-09-02T17:50:14","slug":"%e3%80%90%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e3%81%a8%e5%80%8d%e8%a7%92%e5%85%ac%e5%bc%8f%e3%81%ae%e6%b4%bb%e7%94%a82","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3005","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u3011\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u3068\u500d\u89d2\u516c\u5f0f\u306e\u6d3b\u7528(2008)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066, \\(a_n=(\\cos{2^n})(\\cos{2^{n-1}})\\cdots(\\cos{2})(\\cos{1})\\)\u3068\u3059\u308b. \u305f\u3060\u3057, \u89d2\u306e\u5927\u304d\u3055\u3092\u8868\u3059\u306e\u306b\u5f27\u5ea6\u6cd5\u3092\u7528\u3044\u3066\u3044\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.

(1) \\(\\displaystyle a_1=\\frac{\\sin{4}}{4\\sin{1}}\\)\u3092\u793a\u305b.

(2) \\(\\displaystyle a_n=\\frac{\\sin{2^{n+1}}}{2^{n+1}\\sin{1}}\\)\u3092\u793a\u305b.

(3) \\(\\displaystyle a_n<\\frac{\\sqrt{2}}{2^{n+1}}\\)\u3092\u793a\u305b.
(2008 \u4e5d\u5dde\u5927\u5b66 \u6587\u7cfb [1])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(a_1=(\\cos{2})(\\cos{1})\\)\u306e\u4e21\u8fba\u306b\\(\\sin{1}\\)\u3092\u304b\u3051\u308b\u3068,$$
a_1\\sin{1}=(\\cos{2})(\\cos{1})(\\sin{1})
$$\u3068\u306a\u308b\u304c, \\(sin\\)\u306e\u500d\u89d2\u306e\u516c\u5f0f, \\(\\sin{2\\theta}=2\\sin{\\theta}\\cos{\\theta}\\)\u304b\u3089, \\( \\displaystyle \\sin{\\theta}\\cos{\\theta}=\\frac{1}{2}\\sin{2\\theta}\\)\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066,$$
\\begin{align}
a_1\\sin{1}&=(\\cos{2})(\\cos{1})(\\sin{1})\\\\[1.5ex]
&=\\frac{1}{2}(\\cos{2})(\\sin{2})\\\\[1.5ex]
&=\\frac{1}{4}\\sin{4}
\\end{align}
$$\u3068\u306a\u308b. \u3053\u3053\u3067, \\(0<1<\\pi\\)\u3067\u3042\u308b\u304b\u3089, \\(\\sin{1}\\neq 0\\)\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \u4e21\u8fba\u3092\\(\\sin{1}\\)\u3067\u5272\u308b\u3068,$$
a_1=\\frac{\\sin{4}}{4\\sin{1}}
$$\u3068\u306a\u3063\u3066\u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) \u81ea\u7136\u6570\\(n\\)\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u793a\u3059.
\u2460 \\(n=1\\)\u306e\u3068\u304d
$$
\\begin{align}
(\u5de6\u8fba)&=a_1\\\\[1.5ex]
(\u53f3\u8fba)&=\\frac{\\sin{2^{1+1}}}{2^{1+1}\\sin{1}}=\\frac{\\sin{4}}{4\\sin{1}}
\\end{align}
$$\u3068\u306a\u308a, (1)\u3088\u308a\u3053\u306e\u53f3\u8fba\u306f\\(a_1\\)\u3068\u7b49\u3057\u3044. \u3088\u3063\u3066, \\(n=1\\)\u306e\u3068\u304d\u306b\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.

\u2461 \\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066, \\(n=k+1\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059.
\\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066\u3044\u308b\u306e\u3067,$$
a_k=\\frac{\\sin{2^{k+1}}}{2^{k+1}\\sin{1}}
$$\u3067\u3042\u308b. \u3053\u306e\u3068\u304d, $$
\\begin{align}
a_{k+1}&=(\\cos{2^{k+1}})(\\cos{2^k})\\cdots(\\cos{2})(\\cos{1})\\\\[1.5ex]
&=(\\cos{2^{k+1}})a_k\\\\[1.5ex]
&=\\cos{2^{k+1}}\\times \\frac{\\sin{2^{k+1}}}{2^{k+1}\\sin{1}}\\\\[1.5ex]
&=\\frac{\\cos{2^{k+1}}\\sin{2^{k+1}}}{2^{k+1}\\sin{1}}
\\end{align}
$$\u3068\u306a\u308b. \u3053\u3053\u3067, \\(\\sin\\)\u306e\u500d\u89d2\u306e\u516c\u5f0f\u304b\u3089,$$
\\cos{2^{k+1}}\\sin{2^{k+1}}=\\frac{1}{2}\\sin{2\\times 2^{k+1}}=\\frac{1}{2}\\sin{2^{k+2}}
$$\u3067\u3042\u308b\u304b\u3089, $$
a_{k+1}=\\frac{1}{2}\\sin{2^{k+2}}\\times\\frac{1}{2^{k+1}\\sin{1}}=\\frac{\\sin{2^{k+2}}}{2^{k+2}\\sin{1}}
$$\u3068\u306a\u308a, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b.

\u2460, \u2461 \u304b\u3089\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066, $$
a_n=\\frac{\\sin{2^{n+1}}}{2^{n+1}\\sin{1}}
$$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(3) \\(\\displaystyle 0<\\frac{\\pi}{4}<1<\\frac{\\pi}{2}\\)\u3067\u3042\u308a, \\(\\sin{x}\\)\u306f\\(\\displaystyle 0\\leq x\\leq \\frac{\\pi}{2}\\)\u3067\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, $$
\\frac{1}{\\sqrt{2}}=\\sin{\\frac{\\pi}{4}}<\\sin{1}
$$\u304c\u308f\u304b\u308b. \u4e21\u8fba\u306b\\(\\displaystyle \\frac{\\sqrt{2}}{\\sin{1}}>0\\)\u3092\u304b\u3051\u308b\u3068,$$
\\frac{1}{\\sin{1}}<\\sqrt{2}
$$\u3068\u306a\u308b.

\u307e\u305f, $$
\\sin{2^{n+1}}\\leq 1
$$\u3067\u3042\u308a, \u3053\u306e\u4e21\u8fba\u3092\\(\\sin{1}>0\\)\u3067\u5272\u308b\u3068, $$
\\frac{\\sin{2^{n+1}}}{\\sin{1}}\\leq\\frac{1}{\\sin{1}}
$$\u3067\u3042\u308a, \\(\\displaystyle \\frac{1}{\\sin{1}}<\\sqrt{2}\\)\u3060\u3063\u305f\u306e\u3067, $$
\\frac{\\sin{2^{n+1}}}{\\sin{1}}\\leq\\frac{1}{\\sin{1}}<\\sqrt{2}
$$\u3068\u306a\u308b. \u3055\u3089\u306b\u3053\u306e\u4e21\u8fba\u3092, \\(2^{n+1}>0\\)\u3067\u5272\u308b\u3068,$$
\\frac{\\sin{2^{n+1}}}{2^{n+1}\\sin{1}}<\\frac{\\sqrt{2}}{2^{n+1}}
$$\u3068\u306a\u308b. (2)\u304b\u3089\u3053\u306e\u5de6\u8fba\u306f, \\(a_n\\)\u306a\u306e\u3067,$$
a_n<\\frac{\\sqrt{2}}{2^{n+1}}
$$\u304c\u793a\u3055\u308c\u305f.<\/p>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n