{"id":2325,"date":"2020-05-28T17:46:35","date_gmt":"2020-05-28T08:46:35","guid":{"rendered":"https:\/\/math-travel.com\/?p=2325"},"modified":"2026-02-11T16:06:30","modified_gmt":"2026-02-11T07:06:30","slug":"cosine-theorem","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-1\/cosine-theorem\/","title":{"rendered":"\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u3068\u899a\u3048\u65b9\u3092\u56f3\u89e3\uff01\u300c\u3044\u3064\u4f7f\u3046\u304b\u300d\u306e\u5224\u65ad\u57fa\u6e96\u3068\u8a3c\u660e\u3092\u4e01\u5be7\u306b\u89e3\u8aac"},"content":{"rendered":"\n

\u6570\u5b66\u2160\u4e09\u89d2\u6bd4\u306e\u306a\u304b\u3067\u591a\u304f\u306e\u9ad8\u6821\u751f\u3092\u56f0\u3089\u305b\u308b\u306e\u304c\u300c\u4f59\u5f26\u5b9a\u7406\u300d<\/span>\u3067\u3059\u306d\u3002<\/span><\/p>\n\n\n\n

\u4eca\u56de\u89e3\u6c7a\u3059\u308b\u60a9\u307f<\/span><\/div>
\n

\u300c\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u304c\u77e5\u308a\u305f\u3044\u300d <\/p>\n\n\n\n

\u300c\u4f59\u5f26\u5b9a\u7406\u306e\u4f7f\u3044\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u300d<\/p>\n<\/div><\/div>\n\n\n\n

\u4eca\u56de\u306f\u4e09\u89d2\u6bd4\u306e\u4e2d\u304b\u3089”\u4f59\u5f26\u5b9a\u7406”\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u4f59\u5f26\u5b9a\u7406\u304c\u826f\u304f\u5206\u304b\u3089\u306a\u304f\u3066\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u3055\u3063\u305d\u304f\u3067\u3059\u304c\u3001\u8fbaBC\u306e\u9577\u3055\u304c\u3044\u304f\u3064\u306b\u306a\u308b\u304b\u5206\u304b\u308a\u307e\u3059\u304b\uff1f<\/p>\n\n\n\n

\u300c\u3053\u308c\u3060\u3051\u306e\u60c5\u5831\u3067\u306f\u6c42\u3081\u3089\u308c\u306a\u3044\u3067\u3059\u3088\u300d<\/p>\n\n\n\n

\u305d\u3046\u601d\u3063\u305f\u65b9\u306f\u4f59\u5f26\u5b9a\u7406\u304c\u4f7f\u3048\u3066\u3044\u306a\u3044\u306e\u3067\u5371\u967a\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u5b9f\u306f\u300c\u4f59\u5f26\u5b9a\u7406\u300d\u3092\u4f7f\u3048\u3070\u8fbaBC\u306e\u9577\u3055\u3092\u7c21\u5358\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff01<\/span><\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u3084\u8a3c\u660e\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f<\/h2>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306f\u4e09\u89d2\u5f62\u306e\u8fba\u3084\u89d2\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u3089\u308c\u308b\u5b9a\u7406\u3067\u3059\u3002<\/p>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306f\u8fba\u306e\u9577\u3055\u3068\u4e09\u89d2\u6bd4\u3092\u7528\u3044\u305f\u91cd\u8981\u5b9a\u7406\u306e\uff11\u3064\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u21d3\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3053\u3068\u3067\u89d2\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\\begin{eqnarray}
\\displaystyle \\cos A&=&\\frac{b^{2}+c^{2}-a^{2}}{2bc}\\\\
\\displaystyle \\cos B&=&\\frac{a^{2}+c^{2}-b^{2}}{2ac}\\\\
\\displaystyle \\cos C&=&\\frac{a^{2}+b^{2}-c^{2}}{2ab}
\\end{eqnarray}<\/p>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306e\u4f7f\u3044\u65b9<\/h2>\n\n\n\n

\u5192\u982d\u3067\u767b\u5834\u3057\u305f\u4f59\u5f26\u5b9a\u7406\u306e\u4f8b\u984c\u3092\u4e00\u7dd2\u306b\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

2\u8fba\u3068\u305d\u306e\u9593\u306e\u89d2\u306e\u5927\u304d\u3055\u304c\u5206\u304b\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u5411\u304b\u3044\u5408\u3046\u8fbaBC\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u8fbaAB\u306e\u5927\u304d\u3055\u3092\\(a\\)\u3068\u3059\u308b\u3068\u6b63\u5f26\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
a^{2}&=&b^{2}+c^{2}-2bc \\cos A\\\\
a^{2}&=&(3\\sqrt{3})^{2}+4^{2}-2 \\times (3\\sqrt{3}) \\times 4 \\times \\cos 30^\\circ\\\\
a^{2}&=&27+16-36\\\\
a^{2}&=&7
\\end{eqnarray}<\/p>\n\n\n\n

\\(a>0\\)\u306a\u306e\u3067<\/p>\n\n\n\n

\\[a=\\sqrt{7}\\]<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001BC\u306e\u5927\u304d\u3055\u306f\\(\\sqrt{7}\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3053\u306e\u3088\u3046\u306b\u4f59\u5f26\u5b9a\u7406\u306f\u8fba\u306e\u9577\u3055\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u8fba\u3084\u89d2\u306e\u5927\u304d\u3055\u3092\u4ee3\u5165\u3059\u308b\u3060\u3051\u3067\u3059\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u3000\u8a3c\u660e<\/h2>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306e\u4f7f\u3044\u65b9\u3092\u78ba\u8a8d\u3057\u305f\u3068\u3053\u308d\u3067\u3001\u306a\u305c\u4f59\u5f26\u5b9a\u7406\u304c\u6210\u308a\u7acb\u3064\u306e\u304b\u8a3c\u660e\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\\([1]\u30000^\\circ < A < 90^\\circ\\)\u306e\u3068\u304d<\/h3>\n\n\n
\n
\"\u4f59\u5f26\u5b9a\u7406\u3000\u8a3c\u660e1\"<\/figure>\n<\/div>\n\n\n

\u4e0a\u306e\u56f3\u306e\u3088\u3046\u306b\u70b9A,B,C\u3092\u3068\u308b\u3002<\/p>\n\n\n\n

\\(A(0 , 0)\u3001B(c , 0)\\)\u3068\u3059\u308b\u3068\u3001C\u306f\\((b \\cos A , b \\sin A)\\)\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\u9802\u70b9C\u304b\u3089X\u8ef8\u3078\u5782\u7dda\u3092\u4e0b\u3057\u3066\u3001\u305d\u306e\u4ea4\u70b9\u3092H\u3068\u304a\u304f\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u5f62\\(CHB\\)\u306b\u6ce8\u76ee\u3057\u3066\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3092\u7528\u3044\u308b\u3068\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
a^{2} &=& |c \u2013 b \\cos A|^{2} + (b \\sin A)^{2}\\\\
&=& c^{2} \u2013 2bc\u30fb\\cos A + b^{2} (\\cos^{2}A + \\sin^{2}A)
\\end{eqnarray}<\/p>\n\n\n\n

\u3059\u306a\u308f\u3061<\/p>\n\n\n\n

\\(a^{2} = b^{2} + c^{2} \u2013 2bc\u30fb\\cos A\\) \u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\\([2]\u3000A = 90^\\circ\\)\u306e\u3068\u304d<\/h3>\n\n\n
\n
\"\u4f59\u5f26\u5b9a\u7406\u3000\u8a3c\u660e2\"<\/figure>\n<\/div>\n\n\n

\u89d2A\u304c\u76f4\u89d2\u306e\u5834\u5408\u3001\u25b3ABC\u306f\u76f4\u89d2\u4e09\u89d2\u5f62\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3088\u308a<\/p>\n\n\n\n

\\(a^{2}=b^{2}+c^{2}\\)<\/p>\n\n\n\n

\u3068\u306a\u308b\u3002<\/p>\n\n\n\n

\\(\\cos A = \\cos 90 = 0\\)\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
a^{2}&=&b^{2}+c^{2}-2bc \\cos A\\\\
a^{2}&=&b^{2}+c^{2}
\\end{eqnarray}<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\u89d2A\u304c\u76f4\u89d2\u306e\u5834\u5408\u3082<\/p>\n\n\n\n

\\(a^{2} = b^{2} + c^{2} \u2013 2bc\u30fb\\cos A\\) \u304c\u6210\u7acb\u3059\u308b\u3002<\/p>\n\n\n\n

\\([3]\u300090^\\circ < A < 180^\\circ\\)\u306e\u3068\u304d<\/h3>\n\n\n
\n
\"\u4f59\u5f26\u5b9a\u7406\u3000\u8a3c\u660e3\"<\/figure>\n<\/div>\n\n\n

\u4e0a\u306e\u56f3\u3088\u308a\u3001BH\u3068CH\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n

\\begin{eqnarray}
BH&=&c+AH\\\\
&=&c+b \\cos (180^circ-A)\\\\
&=&c-b \\cos A
\\end{eqnarray}<\/p>\n\n\n\n

\\begin{eqnarray}
CH&=&b \\sin (180^circ-A)\\\\
&=&b \\sin A
\\end{eqnarray}<\/p>\n\n\n\n

\\(BCH\\)\u306b\u7f6e\u3044\u3066\u3001\u4e09\u5e73\u65b9\u306e\u5b9a\u7406\u3088\u308a<\/p>\n\n\n\n

\\(BC^{2}=BH^{2}+CH^{2}\\)<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066<\/p>\n\n\n\n

\\begin{eqnarray}
a^{2}&=&(c-b \\cos A)^{2} + b^{2} \\sin^{2}A\\\\
&=&(c^{2}-2bc \\cos A+b^{2} \\cos^{2}A)+b^{2} \\sin^{2}A\\\\
&=&c^{2}-2bc\u30fbcosA+b^{2}
\\end{eqnarray}<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

\\(a^{2}=b^{2}+c^{2}-2bc\u30fbcosA\\)\u3000\u306f\u6210\u7acb\u3059\u308b<\/p>\n\n\n\n

\\([1][2][3]\\)\u3088\u308a\u4f59\u5f26\u5b9a\u7406\u306e\u8a3c\u660e\u7d42\u4e86\u3002<\/p>\n\n\n\n

\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\uff1c\u7df4\u7fd2\u554f\u984c\uff1e<\/h2>\n\n\n
\n
\"\u4f59\u5f26\u5b9a\u7406\uff1c\u7df4\u7fd2\u554f\u984c\uff1e\"<\/figure>\n<\/div>\n\n\n

\u4eca\u56de\u5b66\u3093\u3060\u300c\u4f59\u5f26\u5b9a\u7406\u300d\u3092\u7528\u3044\u3066\u3001\u7df4\u7fd2\u554f\u984c\u306b\u6311\u6226\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u4eca\u56de\u306f\uff12\u3064\u306e\u30d1\u30bf\u30fc\u30f3\u306e\u7df4\u7fd2\u554f\u984c\u3092\u7528\u610f\u3057\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n

    \n
  • \u8fba\u306e\u9577\u3055\u3092\u6c42\u3081\u308b\u554f\u984c<\/li>\n\n\n\n
  • \u89d2\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u308b\u554f\u984c<\/li>\n<\/ul>\n\n\n\n

    \u89e3\u8aac<\/span><\/p>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    a^{2}&=&b^{2}+c^{2}-2bc \\cos A\\\\
    a^{2}&=&4^{2}+6^{2}-2 \\times 4 \\times 6 \\times \\cos 60^\\circ \\\\
    a^{2}&=&16+36-24\\\\
    a^{2}&=&28
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(a>0\\)\u306a\u306e\u3067<\/p>\n\n\n\n

    \\(a=2\\sqrt{7}\\)<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n

    \u8fbaBC\u306e\u9577\u3055\u306f\u3001\\(2\\sqrt{7}\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

    \u6b21\u306f\u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\u89d2\u5ea6\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u308b\u7df4\u7fd2\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406<\/p>\n\n\n\n

    \\(a^{2}=b^{2}+c^{2}-2bc \\cos A\\)\u3092\u5909\u5f62\u3057\u3066<\/p>\n\n\n\n

    \\[\\displaystyle \\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2bc} \\]<\/p>\n\n\n\n

    \u5909\u5f62\u3057\u305f\u5f0f\u306b\u5206\u304b\u3063\u3066\u3044\u308b\u5024\u3092\u4ee3\u5165\u3057\u3066\u3044\u304f<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\displaystyle \\cos A&=&\\frac{b^{2}+c^{2}-a^{2}}{2bc} \\\\
    \\displaystyle &=&\\frac{6^{2}+(2\\sqrt{2})^{2}-(2\\sqrt{5})^{2}}{2\\times 6 \\times 2\\sqrt{2}} \\\\
    \\displaystyle &=&\\frac{36+8-20}{24\\sqrt{2}} \\\\
    \\displaystyle &=&\\frac{24}{24\\sqrt{2}} \\\\
    \\displaystyle &=&\\frac{1}{\\sqrt{2}}
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(0 < \\angle A < 180^\\circ\\)<\/p>\n\n\n\n

    \u306a\u306e\u3067<\/p>\n\n\n\n

    \\(A=45^\\circ\\)<\/p>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

    \u4eca\u56de\u306f\u6570\u5b66\u2160\u306e\u4e09\u89d2\u6bd4\u304b\u3089\u300c\u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f\u300d\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406\u306e\u516c\u5f0f<\/span><\/div>
    \n

    \\(\u25b3ABC\\)\u306b\u304a\u3044\u3066\u5404\u8fba\u3092\\(a,b,c\\)\u3068\u3059\u308b\u3068\u304d\u3001\u4ee5\u4e0b\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    a^{2}&=&b^{2}+c^{2}-2bc \\cos A\\\\
    b^{2}&=&a^{2}+c^{2}-2ac \\cos B\\\\
    c^{2}&=&a^{2}+b^{2}-2ab \\cos C\\\\
    \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u3053\u3068\u3067\u3001\u8fba\u306e\u9577\u3055\u3084\u89d2\u306e\u5927\u304d\u3055\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \u6b63\u5f26\u5b9a\u7406\u3068\u4f59\u5f26\u5b9a\u7406<\/span><\/div>
    \n

    \u6b63\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u6642<\/p>\n\n\n\n

    \u24601\u8fba\u30682\u3064\u306e\u89d2\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u5834\u5408
    \u24611\u7d44\u306e\u5411\u304b\u3044\u5408\u3046\u8fba\u3068\u89d2\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u5834\u5408
    \u2462\u5916\u63a5\u5186\u306e\u534a\u5f84\u3092\u6c42\u3081\u305f\u3044\u5834\u5408<\/p>\n\n\n\n

    \u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u6642<\/p>\n\n\n\n

    \u24602\u8fba\u3068\u305d\u306e\u9593\u306e\u89d2\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u5834\u5408
    \u24613\u8fba\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u304d<\/p>\n<\/div><\/div>\n\n\n\n

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