{"id":1716,"date":"2025-07-24T17:52:54","date_gmt":"2025-07-24T08:52:54","guid":{"rendered":"https:\/\/math-friend.com\/?p=1716"},"modified":"2025-07-29T11:09:49","modified_gmt":"2025-07-29T02:09:49","slug":"%e3%80%90-%e5%8c%97%e6%b5%b7%e9%81%93%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e6%95%b4%e6%95%b0%e3%81%ae%e8%be%ba%e3%81%a7%e4%bd%9c%e3%82%8b%e4%b8%89%e8%a7%92%e5%bd%a2%e3%81%a8cos%e8%a8%88","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1716","title":{"rendered":"\u3010 \u5317\u6d77\u9053\u5927\u5b66\u5165\u8a66\u3011\u6574\u6570\u306e\u8fba\u3067\u4f5c\u308b\u4e09\u89d2\u5f62\u3068cos\u8a08\u7b97(2025)"},"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

\u6574\u6570\\(a\\), \\(b\\), \\(c\\)\u306f \\(2\\leq a<b<c\\leq 6\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(a+b>c\\) \u3092\u6e80\u305f\u3059\\(a\\), \\(b\\), \\(c\\) \u306e\u7d44\\((a, b, c)\\) \u3092\u5168\u3066\u6c42\u3081\u3088.
(2) \\(a^2+b^2\\geq c^2\\) \u3092\u6e80\u305f\u3059\\(a\\), \\(b\\), \\(c\\)\u306e\u7d44\\((a, b, c)\\) \u3092\u5168\u3066\u6c42\u3081\u3088.
(3) (2)\u3067\u6c42\u3081\u305f\u5404\\((a, b, c)\\) \u306b\u3064\u3044\u3066, \u9802\u70b9\\(\\mathrm{A}\\), \\(\\mathrm{B} \\), \\(\\mathrm{C}\\)\u306b\u5411\u304b\u3044\u5408\u3046\u8fba\u306e\u9577\u3055\u304c\\(a\\), \\(b\\), \\(c\\) \u3068\u306a\u308b\u4e09\u89d2\u5f62\\(\\mathrm{ABC}\\)\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \\(\\cos{ \\angle{\\mathrm{ACB}} }\\) \u3092\u6c42\u3081\u3088.
(2025 \u5317\u6d77\u9053\u5927\u5b66 [2])<\/span><\/p>\n\n\n\n

\u3053\u3061\u3089, (3)\u3092\u9664\u3044\u3066\u5358\u306a\u308b\u6574\u6570\u554f\u984c\u306b\u898b\u3048\u307e\u3059\u304c, \u5b9f\u306f(1)\u3067\u306f\u8fba\u306e\u9577\u3055\u304c\\(a\\), \\(b\\), \\(c\\) \u3068\u306a\u308b\u4e09\u89d2\u5f62\u304c\u6210\u7acb\u3059\u308b\u6761\u4ef6\u3092\u8003\u3048\u3066\u304a\u308a, (2)\u3067\u306f\u305d\u306e\u4e09\u89d2\u5f62\u304c\u76f4\u89d2\u4e09\u89d2\u5f62, \u3082\u3057\u304f\u306f\u92ed\u89d2\u4e09\u89d2\u5f62\u306b\u306a\u308b\u6761\u4ef6(=\u920d\u89d2\u4e09\u89d2\u5f62\u3067\u306a\u3044\u6761\u4ef6)\u3092\u8003\u3048\u3066\u3044\u307e\u3059. \u3053\u306e\u3053\u3068\u306f\u89e3\u7b54\u5f8c\u306b\u5c11\u3057\u88dc\u8db3\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059.

\u4eca\u56de, \u6700\u521d\u306b\u4e0e\u3048\u3089\u308c\u305f\\(2\\leq a<b<c\\leq 6\\)\u306e\u6761\u4ef6\u304b\u3089, \\((a, b, c)\\)\u306e\u8003\u3048\u3089\u308c\u308b\u7d44\u306f, \\(2\\)\u304b\u3089\\(6\\)\u306e\\(5\\)\u3064\u306e\u6574\u6570\u304b\u3089\\(3\\)\u500b\u306e\u6574\u6570\u3092\u9078\u3073\u51fa\u3059\u7d44\u306e\u7dcf\u6570\u3060\u3051\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u5b9f\u969b, \u9078\u3073\u51fa\u3055\u308c\u305f\\(3\\)\u500b\u306e\u6574\u6570\u306b\u5bfe\u3057\u3066, \u5c0f\u3055\u3044\u65b9\u304b\u3089\\(a\\), \\(b\\), \\(c\\)\u3068\u3057\u3066\u3044\u3051\u3070\u826f\u3044\u3067\u3059. \u3088\u3063\u3066, \\((a, b, c)\\)\u306e\u8003\u3048\u3089\u308c\u308b\u7d44\u306f\\({}_5C_3=10\\)\u901a\u308a\u3057\u304b\u306a\u3044\u306e\u3067, (1), (2)\u306f\u5168\u3066\u5217\u6319\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u306b\u3057\u307e\u3059.

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

\n

(1) \\((a, b, c)\\)\u306e\u7d44\u306e\u7dcf\u6570\u306f, \\(2\\)\u304b\u3089\\(6\\)\u306e\\(5\\)\u3064\u306e\u6574\u6570\u304b\u3089\\(3\\)\u500b\u306e\u6574\u6570\u3092\u9078\u3073\u51fa\u3059\u7d44\u307f\u5408\u308f\u305b\u306e\u7dcf\u6570\\({}_5C_3=10\\)\u901a\u308a\u3067\u3042\u308a, \u3053\u308c\u3089\u3092\u5217\u6319\u3057\u3066\u78ba\u8a8d\u3059\u308b.<\/p>\n\n\n

\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059 <\/i><\/span><\/div>\n
\\(a\\)<\/th>\\(b\\)<\/th>\\(c\\)<\/th>\\(a+b\\)<\/th>\\(a+b>c\\)<\/th><\/tr><\/thead>
 <\/span>\\(2\\)<\/td> <\/span>\\(3\\)<\/td> <\/span>\\(4\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(2\\)<\/td> <\/span>\\(3\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
 <\/span>\\(2\\)<\/td> <\/span>\\(3\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
 <\/span>\\(2\\)<\/td> <\/span>\\(4\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(2\\)<\/td> <\/span>\\(4\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
 <\/span>\\(2\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(7\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(3\\)<\/td> <\/span>\\(4\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(7\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(3\\)<\/td> <\/span>\\(4\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(7\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(3\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(8\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr>
 <\/span>\\(4\\)<\/td> <\/span>\\(5\\)<\/td> <\/span>\\(6\\)<\/td> <\/span>\\(9\\)<\/td> <\/span>\u6e80\u305f\u3059<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u3088\u3063\u3066\u3053\u308c\u304b\u3089, \u6761\u4ef6\u3092\u6e80\u305f\u3059\\((a, b, c)\\)\u306e\u7d44\u306f, \\((2, 3, 4)\\), \\((2, 4, 5)\\), \\((2, 5, 6)\\), \\((3, 4, 5)\\), \\((3, 4, 6)\\), \\((3, 5, 6)\\), \\((4, 5, 6)\\)\u306e7\u7d44\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

(2) (1)\u3068\u540c\u69d8\u306b\u5217\u6319\u3057\u3066\u78ba\u8a8d\u3059\u308b.<\/p>\n\n\n

\u30b9\u30af\u30ed\u30fc\u30eb\u3067\u304d\u307e\u3059 <\/i><\/span><\/div>\n
\\(a\\)<\/th>\\(b\\)<\/th>\\(c\\)<\/th>\\(a^2+b^2\\)<\/th>\\(c^2\\)<\/th>\\(a^2+b^2>c^2\\)<\/th><\/tr><\/thead>
\\(2\\)<\/td>\\(3\\)<\/td>\\(4\\)<\/td>\\(13\\)<\/td>\\(16\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(2\\)<\/td>\\(3\\)<\/td>\\(5\\)<\/td>\\(13\\)<\/td>\\(25\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(2\\)<\/td>\\(3\\)<\/td>\\(6\\)<\/td>\\(13\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(2\\)<\/td>\\(4\\)<\/td>\\(5\\)<\/td>\\(20\\)<\/td>\\(25\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(2\\)<\/td>\\(4\\)<\/td>\\(6\\)<\/td>\\(20\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(2\\)<\/td>\\(5\\)<\/td>\\(6\\)<\/td>\\(29\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(3\\)<\/td>\\(4\\)<\/td>\\(5\\)<\/td>\\(25\\)<\/td>\\(25\\)<\/td>\u6e80\u305f\u3059<\/td><\/tr>
\\(3\\)<\/td>\\(4\\)<\/td>\\(6\\)<\/td>\\(25\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(3\\)<\/td>\\(5\\)<\/td>\\(6\\)<\/td>\\(34\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3055\u306a\u3044<\/td><\/tr>
\\(4\\)<\/td>\\(5\\)<\/td>\\(6\\)<\/td>\\(41\\)<\/td>\\(36\\)<\/td>\u6e80\u305f\u3059<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u3088\u3063\u3066\u3053\u308c\u304b\u3089, \u6761\u4ef6\u3092\u6e80\u305f\u3059\\((a, b, c)\\)\u306e\u7d44\u306f, \\((3, 4, 5)\\), \\((4, 5, 6)\\)\u306e2\u7d44\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

(3) \\((a, b, c)=(3, 4, 5)\\)\u306e\u3068\u304d, \u4f59\u5f26\u5b9a\u7406\u304b\u3089
$$
\\cos{\\angle{\\mathrm{ACB}}}=\\frac{a^2+b^2-c^2}{2ab}=\\frac{3^2+4^2-5^2}{2\\cdot 3\\cdot 4}=0
$$\u3068\u306a\u308a, \\((a, b, c)=(4, 5, 6)\\)\u306e\u3068\u304d\u540c\u69d8\u306b,
$$
\\cos{\\angle{\\mathrm{ACB}}}=\\frac{a^2+b^2-c^2}{2ab}=\\frac{4^2+5^2-6^2}{2\\cdot 4\\cdot 5}=\\frac{1}{8}
$$\u3068\u306a\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u305d\u308c\u3067\u306f\u4eca\u56de\u306e\u554f\u984c\u306e\u80cc\u666f\u306e\u88dc\u8db3\u3092\u3057\u307e\u3059. \u307e\u305a\u4e09\u89d2\u5f62\u306e\u6210\u7acb\u6761\u4ef6\u3092\u7d39\u4ecb\u3057\u307e\u3059.<\/p>\n\n\n\n

\u4e09\u89d2\u5f62\u306e\u6210\u7acb\u6761\u4ef6<\/strong>
\u6b63\u306e\u5b9f\u6570\\(a\\), \\(b\\), \\(c\\)\u3092\u8fba\u306e\u9577\u3055\u3068\u3059\u308b\u4e09\u89d2\u5f62\u304c\u5b58\u5728\u3059\u308b\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f,
$$
a+b>c\\,\\,\u304b\u3064\\,\\,b+c>a\\,\\,\u304b\u3064\\,\\,c+a>b
$$\u3067\u3042\u308b.<\/p>\n\n\n\n

\u4eca\u56de\u306e\u554f\u984c\u3067\u306f\\(a<b<c\\)\u304c\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u306e\u3067, \\(b+c>a\\)\u3068\\(c+a>b\\)\u306f\u81ea\u52d5\u3067\u6210\u308a\u7acb\u3061\u307e\u3059. \u3088\u3063\u3066(1)\u3067\u78ba\u8a8d\u3057\u3066\u3044\u308b\u4e0d\u7b49\u5f0f\\(a+b>c\\)\u304c\u6e80\u305f\u3055\u308c\u308c\u3070, 3\u8fba\u306e\u9577\u3055\u3092\\(a\\), \\(b\\), \\(c\\)\u3068\u3059\u308b\u4e09\u89d2\u5f62\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059.<\/p>\n\n\n\n

\u6b21\u306b(2)\u306b\u95a2\u3057\u3066\u3067\u3059\u304c, \u307e\u305a, \u6b63\u306e\u5b9f\u6570\\(a\\), \\(b\\), \\(c\\)\u306b\u5bfe\u3057\u3066, \\(a^2+b^2\\geq c^2\\)\u3067\u3042\u308c\u3070, (1)\u306e\u4e0d\u7b49\u5f0f\\(a+b>c\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c,
$$
(a+b)^2=a^2+2ab+b^2>a^2+b^2\\geq c^2
$$\u304b\u3089\u7c21\u5358\u306b\u308f\u304b\u308a\u307e\u3059. \u305d\u306e\u4e0a\u3067, \u8fba\u306e\u9577\u3055\u304c\\(a\\), \\(b\\), \\(c\\)\u3067\u3042\u308b\u4e09\u89d2\u5f62\u304c, \\(a^2+b^2\\geq c^2\\)\u3092\u6e80\u305f\u3059\u306e\u306f, \u4e09\u89d2\u5f62\u304c\u76f4\u89d2\u4e09\u89d2\u5f62, \u307e\u305f\u306f\u92ed\u89d2\u4e09\u89d2\u5f62\u306e\u3068\u304d\u306b\u9650\u308a\u307e\u3059. \u306a\u305c\u306a\u3089, \\(a<b<c\\)\u3088\u308a, \u4e09\u89d2\u5f62\\(\\mathrm{ABC}\\)\u306e\u6700\u5927\u89d2\u306f\\(\\angle{\\mathrm{C}}\\)\u306b\u306a\u308a\u307e\u3059(\u3053\u308c\u306f\u6b63\u5f26\u5b9a\u7406\u304b\u3089\u308f\u304b\u308b\u6709\u540d\u4e8b\u5b9f\u3067\u3059). \u4f59\u5f26\u5b9a\u7406\u304b\u3089
$$
\\cos{\\mathrm{C}}=\\frac{a^2+b^2-c^2}{2ab}
$$\u3067\u3042\u308a, \\(0<a<b<c\\)\u3092\u524d\u63d0\u3068\u3057\u3066,
$$
a^2+b^2-c^2\\geq 0 \\iff \\cos{\\mathrm{C}}\\geq 0 \\iff 0< \\mathrm{C}\\leq \\frac{\\pi}{2}\\,\\,(0^\\circ < \\mathrm{C} \\leq 90^\\circ)
$$\u3068\u306a\u308b\u304b\u3089\u3067\u3059. <\/p>\n\n\n\n

\u88dc\u8db3\u3067\u3059\u304c, \\(a<b<c\\)\u3092\u524d\u63d0\u3068\u3057\u306a\u3044\u3068\u4e0a\u306e\u540c\u5024\u95a2\u4fc2\u306f\u5d29\u308c\u307e\u3059. \u4f8b\u3048\u3070, \\(a=5\\), \\(b=3\\), \\(c=2\\)\u3068\u3059\u308b\u3068, \\(\\cos{\\mathrm{C}}=1\\)\u3068\u306a\u308a, \\(C=0\\)\u3068\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059.<\/p>\n\n\n\n

\u4ee5\u4e0a\u304b\u3089, (1)\u306e\u4e0d\u7b49\u5f0f\u306f\u4e09\u89d2\u5f62\u306e\u6210\u7acb\u6761\u4ef6, (2)\u306e\u4e0d\u7b49\u5f0f\u306f\u305d\u306e\u4e09\u89d2\u5f62\u304c\u76f4\u89d2\u4e09\u89d2\u5f62, \u3082\u3057\u304f\u306f\u92ed\u89d2\u4e09\u89d2\u5f62\u306b\u306a\u308b\u6761\u4ef6\u3092\u8003\u3048\u3066\u3044\u305f\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3057\u305f.<\/p>\n\n\n\n

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