{"id":6737,"date":"2025-12-24T17:19:15","date_gmt":"2025-12-24T08:19:15","guid":{"rendered":"https:\/\/math-travel.com\/?p=6737"},"modified":"2026-02-11T16:30:19","modified_gmt":"2026-02-11T07:30:19","slug":"half-angle","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-2\/half-angle\/","title":{"rendered":"\u534a\u89d2\u306e\u516c\u5f0f\u306e\u899a\u3048\u65b9\u3068\u4f7f\u3044\u65b9\uff1a\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u5c0e\u304f\u624b\u9806\u3068\u8a08\u7b97\u306e\u6ce8\u610f\u70b9\u3092\u5fb9\u5e95\u89e3\u8aac"},"content":{"rendered":"\n

\u300c\u534a\u89d2\u306e\u516c\u5f0f\u3063\u3066\u306a\u3093\u3060\u3063\u3051\u300d<\/span>
\u300c\u534a\u89d2\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u304c\u77e5\u308a\u305f\u3044\u300d<\/span>
\u4eca\u56de\u306f\u534a\u89d2\u306e\u516c\u5f0f\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

\u534a\u89d2\u306e\u516c\u5f0f\u3092\u3059\u3050\u306b\u5fd8\u308c\u3066\u3057\u307e\u3044\u307e\u3059<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u534a\u89d2\u306e\u516c\u5f0f\u306f\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u306a\u516c\u5f0f\u306e1\u3064\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u534a\u89d2\u306e\u516c\u5f0f<\/span><\/div>
\n

\\begin{eqnarray}
\\displaystyle \\sin^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{2}\\\\
\\displaystyle \\cos ^{2} \\frac{\\theta}{2}&=&\\frac{1+\\cos \\theta}{2}\\\\
\\displaystyle \\tan ^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{1+\\cos \\theta}
\\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

\u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3053\u3068\u3067\u3001\\(\\sin 15^\\circ\\)\u306a\u3069\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u305f\u3060\u3001\u534a\u89d2\u306e\u516c\u5f0f\u306f\u898b\u305f\u76ee\u3082\u8907\u96d1\u3067\u3059\u3057\u3001\u4f7f\u3044\u65b9\u304c\u5206\u304b\u308a\u3065\u3089\u3044\u516c\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n

\u304d\u3063\u3068\u534a\u89d2\u306e\u516c\u5f0f\u30842\u500d\u89d2\u306e\u516c\u5f0f\u304c\u30cb\u30ac\u30c6\u306a\u65b9\u3082\u591a\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u534a\u89d2\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u306a\u3069\u3092\u5fb9\u5e95\u89e3\u8aac<\/span>\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u95a2\u6570\u304c\u82e6\u624b\u306a\u65b9\u306b\u3068\u3063\u3066\u53c2\u8003\u306b\u306a\u308b\u3053\u3068\u3082\u591a\u3044\u306e\u3067\u3001\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u6c17\u306b\u306a\u308b\u898b\u51fa\u3057\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\u3001
\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u534a\u89d2\u306e\u516c\u5f0f<\/h2>\n\n\n\n

\u4e09\u89d2\u95a2\u6570\u306b\u306f\u91cd\u8981\u306a\u516c\u5f0f<\/span>\u304c\u3044\u304f\u3064\u304b\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u516c\u5f0f<\/span><\/div>
\n
    \n
  • \\(\\sin,\\cos,\\tan\\)\u306e\u57fa\u672c\u516c\u5f0f<\/li>\n\n\n\n
  • \u6b63\u5f26\u5b9a\u7406<\/li>\n\n\n\n
  • \u4f59\u5f26\u5b9a\u7406<\/li>\n\n\n\n
  • \u52a0\u6cd5\u5b9a\u7406<\/li>\n\n\n\n
  • 2\u500d\u89d2\u306e\u516c\u5f0f\u3000\u306a\u3069<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

    \u305d\u3057\u3066\u534a\u89d2\u306e\u516c\u5f0f\u3082\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u516c\u5f0f\u306e1\u3064\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f<\/span><\/div>
    \n

    \\begin{eqnarray}
    \\displaystyle \\sin^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{2}\\\\
    \\displaystyle \\cos ^{2} \\frac{\\theta}{2}&=&\\frac{1+\\cos \\theta}{2}\\\\
    \\displaystyle \\tan ^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{1+\\cos \\theta}
    \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u306f\u3053\u3093\u306a\u5f0f\u306e\u5f62\u3092\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u3000\u8a3c\u660e<\/h2>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u306f\\(\\cos\\)\u306e2\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5909\u5f62\u3057\u3066\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    2\u500d\u89d2\u306e\u516c\u5f0f<\/span><\/div>
    \n

    \\begin{eqnarray}
    \\cos 2\\theta&=&\\cos^{2} \\theta -\\sin^{2} \\theta\\\\
    &=&1-2\\sin^{2}\\theta\\\\
    &=&2\\cos^{2} \\theta-1
    \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

    \\(\\cos\\)\u306e2\u500d\u89d2\u306e\u516c\u5f0f<\/p>\n\n\n\n

    \\[\\cos 2\\theta=1-2\\sin^{2}\\theta\\]<\/p>\n\n\n\n

    \\(\\theta\\)\u3092\\(\\displaystyle \\frac{\\theta}{2}\\)\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001<\/p>\n\n\n\n

    \\[\\displaystyle \\cos 2\\cdot \\frac{\\theta}{2}=1-2\\sin^{2} \\frac{\\theta}{2}\\]<\/p>\n\n\n\n

    \u3086\u3048\u306b<\/p>\n\n\n\n

    \\[\\displaystyle \\cos \\theta=1-2\\sin^{2} \\frac{\\theta}{2}\\]<\/p>\n\n\n\n

    \u3068\u306a\u308a\u3001\u5f0f\u3092\u6574\u7406\u3059\u308b\u3068<\/p>\n\n\n\n

    \\[\\displaystyle \\sin^{2} \\frac{\\theta}{2}=\\frac{1-\\cos \\theta}{2}\\]<\/p>\n\n\n\n

    \u3053\u308c\u3067\\(\\sin\\)\u306e\u534a\u89d2\u306e\u516c\u5f0f\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

    \\(\\displaystyle \\cos^{2} \\frac{\\theta}{2}\\)\u3082\u540c\u69d8\u306b\u30012\u500d\u89d2\u306e\u516c\u5f0f\u304b\u3089\u8a3c\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u3000\u4f7f\u3044\u65b9<\/h2>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u306f\\(\\theta\\)\u3092\u4f7f\u3063\u3066\u3001\\(\\displaystyle \\frac{\\theta}{2}\\)\u306e\u4e09\u89d2\u6bd4\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n

    \u516c\u5f0f\u304b\u3089\u5206\u304b\u308b\u3088\u3046\u306b\u3001\\(\\cos \\theta\\)\u3055\u3048\u5206\u304b\u308c\u3070\u534a\u89d2\u306e\u516c\u5f0f\u304c\u4f7f\u3048\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\u534a\u89d2\u306e\u516c\u5f0f\"<\/figure>\n<\/div>\n\n\n

    \u4f8b\u3068\u3057\u3066\u3001\\(\\displaystyle \\sin \\frac{\\pi}{12}\\)\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \\(\\displaystyle \\sin^{2} \\frac{\\pi}{12}\\)\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\\(\\displaystyle \\cos \\frac{\\pi}{6}\\)\u306f\u5fc5\u8981\u3067\u3059\u3002<\/p>\n\n\n\n

    \\[\\displaystyle \\cos \\frac{\\pi}{6}=\\frac{\\sqrt{3}}{2} \\cdots \u2460\\]<\/p>\n\n\n\n

    \u2460\u3068\u534a\u89d2\u306e\u516c\u5f0f\u304b\u3089\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\displaystyle \\sin^{2} \\frac{\\pi}{12}&=&\\frac{1-\\cos \\frac{\\pi}{6}}{2}\\\\
    \\displaystyle &=&\\frac{1-\\frac{\\sqrt{3}}{2}}{2}\\\\
    \\displaystyle &=&\\frac{2-\\sqrt{3}}{4}
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(\\displaystyle \\sin \\frac{\\pi}{12}>0\\)\u3088\u308a\u3001<\/p>\n\n\n\n

    \\[\\displaystyle \\sin \\frac{\\pi}{12}=\\frac{\\sqrt{2-\\sqrt{3}}}{2}\\]<\/span><\/p>\n\n\n\n

    \u307e\u305a\u306f\\(\\cos \\theta\\)\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u610f\u8b58\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

    \"\"\u30b7\u30fc\u30bf<\/span><\/div>
    \n

    \u516c\u5f0f\u3092\u899a\u3048\u305f\u3089\u3001\u4f7f\u3044\u3053\u306a\u305b\u308b\u3088\u3046\u306b\u306a\u308d\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u6642\u306e\u6ce8\u610f\u70b9<\/h2>\n\n\n\n

    \u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u6642\u306e\u6ce8\u610f\u70b9\u304c2\u3064\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

    \u6ce8\u610f\u30dd\u30a4\u30f3\u30c8<\/span><\/div>
    \n
      \n
    • 2\u4e57\u3067\u3042\u308b\u3053\u3068\u5fd8\u308c\u306a\u3044<\/li>\n\n\n\n
    • \\(\\theta\\)\u306e\u7bc4\u56f2\u306b\u6c17\u3092\u4ed8\u3051\u3088\u3046<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u3067\u6c17\u3092\u4ed8\u3051\u305f\u30442\u70b9\u3092\u89e3\u8aac\u3057\u307e\u3059\u3002<\/p>\n\n\n

      \"\"\u30b7\u30fc\u30bf<\/span><\/div>
      \n

      \u6cb9\u65ad\u3059\u308b\u3068\u30df\u30b9\u3059\u308b\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      2\u4e57\u3067\u3042\u308b\u3053\u3068\u5fd8\u308c\u306a\u3044<\/h3>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u306f\u5de6\u8fba\u304c2\u4e57\u3067\u3042\u308b\u3053\u3068\u306b\u6c17\u3092\u4ed8\u3051\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

      \n
      \"\u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u6642\u306e\u6ce8\u610f\u70b9\"<\/figure>\n<\/div>\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u3092\u5fd8\u308c\u305f\u3068\u304d\u306e\u305f\u3081\u306b\u30012\u500d\u89d2\u306e\u516c\u5f0f\u304b\u3089\u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f5c\u308c\u308b\u3088\u3046\u306b\u3057\u3066\u304a\u304f\u3068\u826f\u3044\u3067\u3059\u3002<\/p>\n\n\n\n

      \\(theta\\)\u306e\u7bc4\u56f2\u306b\u6c17\u3092\u4ed8\u3051\u3088\u3046<\/h3>\n\n\n\n

      \u3082\u30461\u3064\u6ce8\u610f\u3059\u3079\u304d\u306a\u306e\u304c\\(\\theta\\)\u306e\u7bc4\u56f2\u3067\u3059\u3002<\/p>\n\n\n\n

      \\(\\displaystyle \\sin^{2} \\frac{\\theta}{2}\\)\u304b\u3089\\(\\displaystyle \\sin \\frac{\\theta}{2}\\)\u3092\u6c42\u3081\u308b\u3068\u304d\u3001<\/p>\n\n\n\n

      \n

      \\(\\displaystyle \\frac{\\theta}{2}>0\\)\u306a\u3089\u3070\u3001\\(\\displaystyle \\sin \\frac{\\theta}{2}>0\\)<\/p>\n\n\n\n

      \\(\\displaystyle \\frac{\\theta}{2}<0\\)\u306a\u3089\u3070\u3001\\(\\displaystyle \\sin \\frac{\\theta}{2}<0\\)<\/p>\n<\/div><\/div>\n\n\n\n

      \u96e3\u3057\u3044\u3053\u3068\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u3053\u3053\u3067\u30df\u30b9\u3059\u308b\u3068\u3082\u3063\u305f\u3044\u306a\u3044\u3067\u3059\u3002<\/p>\n\n\n\n

      \\(\\sin\\)\u306b\u9650\u3089\u305a\u3001\\(\\cos\\)\u3084\\(\\tan\\)\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u3082\u7bc4\u56f2\u3092\u610f\u8b58\u3059\u308b\u3088\u3046\u306b\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

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

      2\u4e57\u3082\u7b26\u53f7\u3082\u6cb9\u65ad\u3057\u306a\u3044\u3088\u3046\u306b\u6c17\u3092\u4ed8\u3051\u307e\u3059<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u306e\u899a\u3048\u65b9<\/h2>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u306e\u899a\u3048\u65b9\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

      \n

      \\begin{eqnarray}
      \\displaystyle \\sin^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{2}\\\\
      \\displaystyle \\cos ^{2} \\frac{\\theta}{2}&=&\\frac{1+\\cos \\theta}{2}
      \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n

      \n
      \"\u534a\u89d2\u306e\u516c\u5f0f\u899a\u3048\u65b9\"\/<\/figure>\n<\/div>\n\n\n

      \u3069\u3061\u3089\u3082\u3053\u3053\u307e\u3067\u306f\u540c\u3058\u5f62\u3092\u3057\u3066\u3044\u307e\u3059\u3002
      <\/span>
      \u3053\u3053\u3067\u53f3\u8fba\u306e\u7b26\u53f7<\/span>\u306b\u6ce8\u76ee\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

      \u30dd\u30a4\u30f3\u30c8<\/span><\/div>
      \n

      \u53f3\u8fba\u306b\u306f\\(\\cos \\theta\\)\u304c\u3042\u308a\u307e\u3059\u3002
      \\(\\displaystyle \\sin^{2}\\frac{\\theta}{2}\\)\u21d2\\(\\cos\\)\u3068\u7570\u306a\u308b\u306e\u3067\u30de\u30a4\u30ca\u30b9
      \\(\\displaystyle \\cos^{2}\\frac{\\theta}{2}\\)\u21d2\\(\\cos\\)\u3068\u540c\u3058\u306a\u306e\u3067\u30d7\u30e9\u30b9<\/p>\n<\/div><\/div>\n\n\n\n

      \\(\\tan\\)\u306e\u516c\u5f0f\u306f\u4e38\u6697\u8a18\u3067\u306f\u306a\u304f\u3001<\/p>\n\n\n\n

      \\[\\displaystyle \\tan \\theta=\\frac{\\sin \\theta}{\\cos \\theta}\\]<\/p>\n\n\n\n

      \u3092\u5229\u7528\u3057\u3066\u4f5c\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

      \\begin{eqnarray}
      \\displaystyle \\tan^{2}\\frac{\\theta}{2}&=&\\frac{\\sin^{2} \\frac{\\theta}{2}}{\\cos^{2} \\frac{\\theta}{2}}\\\\
      \\displaystyle &=&\\frac{1-\\cos \\theta}{1+\\cos \\theta}
      \\end{eqnarray}<\/p>\n\n\n\n

      \\(\\sin\\)\u3068\\(\\cos\\)\u306e\u534a\u89d2\u306e\u516c\u5f0f\u3055\u3048\u899a\u3048\u3066\u304a\u3051\u3070\u3001\\(\\tan\\)\u306f\u7c21\u5358\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n

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

      \u4e21\u8fba\u304ccos\u306e\u3068\u304d\u304c\uff0b\u306b\u306a\u308b\u3093\u3067\u3059\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

      \"\"\u30b7\u30fc\u30bf<\/span><\/div>
      \n

      \u305d\u3046\u3060\u3088\uff01\u81ea\u5206\u306e\u899a\u3048\u3084\u3059\u3044\u3088\u3046\u306b\u899a\u3048\u3088\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      2\u500d\u89d2\u306e\u516c\u5f0f<\/h2>\n\n\n\n

      2\u500d\u89d2\u306e\u516c\u5f0f\u3082\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u306a\u516c\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n

      2\u500d\u89d2\u306e\u516c\u5f0f<\/span><\/div>
      \n

      \\begin{eqnarray}
      \\sin 2 \\alpha&=&2 \\sin \\alpha \\cos \\alpha\\\\
      \\cos 2 \\alpha&=&\\cos^{2} \\alpha – \\sin^{2} \\alpha\\\\
      &=&1-2 \\sin^{2} \\alpha\\\\
      &=&2 \\cos^{2}-1\\\\
      \\displaystyle \\tan 2\\alpha&=&\\frac{2 \\tan \\alpha}{1-\\tan^{2}\\alpha}
      \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n

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

      2\u500d\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u534a\u89d2\u306e\u516c\u5f0f\u3092\u8a3c\u660e\u3057\u307e\u3057\u305f\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      \u305d\u306e2\u500d\u89d2\u306e\u516c\u5f0f\u306f“\u52a0\u6cd5\u5b9a\u7406”<\/span>\u3092\u6d3b\u7528\u3057\u3066\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

      \u52a0\u6cd5\u5b9a\u7406<\/span><\/div>
      \n

      \\begin{eqnarray}
      \\sin(\\alpha+\\beta)&=&\\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta\\\\
      \\sin(\\alpha-\\beta)&=&\\sin \\alpha \\cos \\beta-\\cos \\alpha \\sin \\beta\\\\
      \\cos(\\alpha+\\beta)&=&\\cos \\alpha \\cos \\beta-\\sin \\alpha \\sin \\beta\\\\
      \\cos(\\alpha-\\beta)&=&\\cos \\alpha \\cos \\beta+\\sin \\alpha \\sin \\beta\\\\
      \\displaystyle \\tan(\\alpha+\\beta)&=&\\frac{\\tan \\alpha+\\tan \\beta}{1-\\tan \\alpha \\tan \\beta}\\\\
      \\displaystyle \\tan(\\alpha-\\beta)&=&\\frac{\\tan \\alpha -\\tan \\beta}{1+\\tan \\alpha \\tan \\beta}
      \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u305f\u7df4\u7fd2\u554f\u984c\u306b\u30c1\u30e3\u30ec\u30f3\u30b8\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

      \u7df4\u7fd2\u554f\u984c<\/span><\/div>
      \n

      \\(0\uff1c\\theta\uff1c\\pi\\)\u3067\\(\\displaystyle \\cos \\theta=-\\frac{2}{3}\\)\u306e\u3068\u304d\u3001<\/p>\n\n\n\n

      \\(\\displaystyle \\sin \\frac{\\theta}{2},\\cos \\frac{\\theta}{2},\\tan \\frac{\\theta}{2}\\)\u3092\u6c42\u3081\u3088\u3046\u3002<\/p>\n<\/div><\/div>\n\n\n

      \"\"\u30b7\u30fc\u30bf<\/span><\/div>
      \n

      \u516c\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u3044\u3053\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      \\(\\displaystyle \\sin \\frac{\\theta}{2}\\)\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

      \\(\\displaystyle \\cos \\theta=-\\frac{2}{3}\\)\u3092\u534a\u89d2\u306e\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u3001<\/p>\n\n\n\n

      \\begin{eqnarray}
      \\displaystyle \\sin^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{2}\\\\
      \\displaystyle &=&\\frac{1+\\frac{2}{3}}{2}\\\\
      \\displaystyle &=&\\frac{5}{6}
      \\end{eqnarray}<\/p>\n\n\n\n

      \u3053\u3053\u3067\\(0<\\frac{\\theta}{2}<\\frac{\\pi}{2}\\)\u306a\u306e\u3067\u3001\\(\\sin \\frac{\\theta}{2}>0\\)<\/p>\n\n\n\n

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

      \\[\\displaystyle \\sin \\frac{\\theta}{2}=\\frac{\\sqrt{30}}{6}\\]<\/span><\/p>\n\n\n\n

      \\(\\displaystyle \\cos \\frac{\\theta}{2}\\)\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

      \\(\\sin\\)\u3068\u540c\u69d8\u306b\u3001\u516c\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u8003\u3048\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

      \\begin{eqnarray}
      \\displaystyle \\cos^{2} \\frac{\\theta}{2}&=&\\frac{1+\\cos \\theta}{2}\\\\
      \\displaystyle &=&\\frac{1-\\frac{2}{3}}{2}\\\\
      \\displaystyle &=&\\frac{1}{6}
      \\end{eqnarray}<\/p>\n\n\n\n

      \u3053\u3053\u3067\\(0<\\frac{\\theta}{2}<\\frac{\\pi}{2}\\)\u306a\u306e\u3067\u3001\\(\\cos \\frac{\\theta}{2}>0\\)<\/p>\n\n\n\n

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

      \\[\\displaystyle \\cos \\frac{\\theta}{2}=\\frac{\\sqrt{6}}{6}\\]<\/span><\/p>\n\n\n\n

      \\(\\displaystyle \\tan \\frac{\\theta}{2}\\)\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

      \\(\\displaystyle \\sin \\frac{\\theta}{2}=\\frac{\\sqrt{30}}{6},\\cos \\frac{\\theta}{2}=\\frac{\\sqrt{6}}{6}\\)<\/p>\n\n\n\n

      \u3088\u3063\u3066\u3001<\/p>\n\n\n\n

      \\begin{eqnarray}
      \\displaystyle \\tan\\frac{\\theta}{2}&=&\\frac{\\sin \\frac{\\theta}{2}}{\\cos \\frac{\\theta}{2}}\\\\
      \\displaystyle &=&\\frac{\\frac{\\sqrt{30}}{6}}{\\frac{\\sqrt{6}}{6}}\\\\
      &=&\\sqrt{5}
      \\end{eqnarray}<\/p>\n\n\n\n

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

      \\[\\displaystyle \\tan \\frac{\\theta}{2}=\\sqrt{5}\\]<\/span><\/p>\n\n\n

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

      \u516c\u5f0f\u3092\u899a\u3048\u305f\u306e\u3067\u89e3\u3051\u307e\u3057\u305f\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

      \"\"\u30b7\u30fc\u30bf<\/span><\/div>
      \n

      \u3059\u3070\u3089\u3057\u3044\uff01\uff01\u3053\u308c\u3067\u534a\u89d2\u306e\u516c\u5f0f\u3082\u30d0\u30c3\u30c1\u30ea\u3060\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

      \u4eca\u56de\u306f\u534a\u89d2\u306e\u516c\u5f0f\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n

      \u534a\u89d2\u306e\u516c\u5f0f<\/span><\/div>
      \n

      \\begin{eqnarray}
      \\displaystyle \\sin^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{2}\\\\
      \\displaystyle \\cos ^{2} \\frac{\\theta}{2}&=&\\frac{1+\\cos \\theta}{2}\\\\
      \\displaystyle \\tan ^{2} \\frac{\\theta}{2}&=&\\frac{1-\\cos \\theta}{1+\\cos \\theta}
      \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

      \u4eca\u56de\u306f\u534a\u89d2\u306e\u516c\u5f0f\u306b\u7126\u70b9\u3092\u3042\u3066\u3066\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

      \u4e09\u89d2\u6bd4\u3084\u4e09\u89d2\u95a2\u6570\u306b\u95a2\u3059\u308b\u8a18\u4e8b\u3092\u30d4\u30c3\u30af\u30a2\u30c3\u30d7\u3057\u305f\u306e\u3067\u3001\u305c\u3072\u53c2\u8003\u306b\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

      \u307f\u3093\u306a\u306e\u52aa\u529b\u304c\u5831\u308f\u308c\u307e\u3059\u3088\u3046\u306b\uff01<\/p>\n","protected":false},"excerpt":{"rendered":"

      \u300c\u534a\u89d2\u306e\u516c\u5f0f\u3063\u3066\u306a\u3093\u3060\u3063\u3051\u300d\u300c\u534a\u89d2\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u304c\u77e5\u308a\u305f\u3044\u300d\u4eca\u56de\u306f\u534a\u89d2\u306e\u516c\u5f0f\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 \u534a\u89d2\u306e\u516c\u5f0f\u306f\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u306a\u516c\u5f0f\u306e1\u3064\u3067\u3059\u3002 \u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3053\u3068\u3067\u3001\\(\\sin 15^\\circ\\)\u306a\u3069\u3092 […]<\/p>\n","protected":false},"author":1,"featured_media":6764,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","footnotes":""},"categories":[35,224],"tags":[36,14,11],"class_list":["post-6737","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sincos","category-math-2","tag-36","tag-b","tag-11"],"yoast_head":"\n\u534a\u89d2\u306e\u516c\u5f0f\u306e\u899a\u3048\u65b9\u3068\u4f7f\u3044\u65b9\uff1a\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u5c0e\u304f\u624b\u9806\u3068\u8a08\u7b97\u306e\u6ce8\u610f\u70b9\u3092\u5fb9\u5e95\u89e3\u8aac<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/math-travel.jp\/math-2\/half-angle\/\" \/>\n<meta property=\"og:locale\" content=\"ja_JP\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u534a\u89d2\u306e\u516c\u5f0f\u306e\u899a\u3048\u65b9\u3068\u4f7f\u3044\u65b9\uff1a\u52a0\u6cd5\u5b9a\u7406\u304b\u3089\u5c0e\u304f\u624b\u9806\u3068\u8a08\u7b97\u306e\u6ce8\u610f\u70b9\u3092\u5fb9\u5e95\u89e3\u8aac\" \/>\n<meta property=\"og:description\" content=\"\u300c\u534a\u89d2\u306e\u516c\u5f0f\u3063\u3066\u306a\u3093\u3060\u3063\u3051\u300d\u300c\u534a\u89d2\u306e\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u304c\u77e5\u308a\u305f\u3044\u300d\u4eca\u56de\u306f\u534a\u89d2\u306e\u516c\u5f0f\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002 \u534a\u89d2\u306e\u516c\u5f0f\u306f\u4e09\u89d2\u95a2\u6570\u306e\u91cd\u8981\u306a\u516c\u5f0f\u306e1\u3064\u3067\u3059\u3002 \u534a\u89d2\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3053\u3068\u3067\u3001(sin 15^circ)\u306a\u3069\u3092 […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/math-travel.jp\/math-2\/half-angle\/\" \/>\n<meta property=\"og:site_name\" content=\"\u30de\u30b9\u30c8\u30e9\u9ad8\u6821\u6570\u5b66\u307e\u3068\u3081\u30b5\u30a4\u30c8\" \/>\n<meta property=\"article:published_time\" content=\"2025-12-24T08:19:15+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-02-11T07:30:19+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/math-travel.jp\/wp-content\/uploads\/2021\/08\/\u534a\u89d2\u306e\u516c\u5f0f\u307e\u3068\u3081\uff01\u516c\u5f0f\u306e\u899a\u3048\u65b9\u3084\u4f7f\u3044\u65b9\u3092\u5fb9\u5e95\u89e3\u8aac\uff01.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1200\" \/>\n\t<meta property=\"og:image:height\" content=\"630\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"\u3086\u3046\u3084\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@https:\/\/twitter.com\/mathtora\" \/>\n<meta name=\"twitter:label1\" content=\"\u57f7\u7b46\u8005\" \/>\n\t<meta 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