{"id":6675,"date":"2025-12-24T17:19:15","date_gmt":"2025-12-24T08:19:15","guid":{"rendered":"https:\/\/math-travel.com\/?p=6675"},"modified":"2026-02-11T16:29:35","modified_gmt":"2026-02-11T07:29:35","slug":"addition-theorem-proof","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-2\/addition-theorem-proof\/","title":{"rendered":"\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\u3092\u5206\u304b\u308a\u3084\u3059\u304f\u56f3\u89e3\uff012\u70b9\u9593\u306e\u8ddd\u96e2\u3068\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3063\u305f\u5c0e\u304d\u65b9"},"content":{"rendered":"\n

\u300c\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\u304c\u77e5\u308a\u305f\u3044\u300d<\/span>
\u300c\u3069\u3046\u3057\u3066\u52a0\u6cd5\u5b9a\u7406\u306f\u6210\u308a\u7acb\u3064\u306e\uff1f\u300d<\/span>
\u4eca\u56de\u306f\u52a0\u6cd5\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

\u52a0\u6cd5\u5b9a\u7406\u304c\u306a\u305c\u6210\u308a\u7acb\u3064\u306e\u304b\u77e5\u308a\u305f\u3044\u3067\u3059<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u4e09\u89d2\u95a2\u6570\u306e\u306a\u304b\u3067\u3082\u52a0\u6cd5\u5b9a\u7406\u306f\u91cd\u8981\u306a\u516c\u5f0f<\/span>\u306e1\u3064\u3067\u3059\u3002<\/p>\n\n\n\n

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


\\begin{eqnarray}
\\sin(\u03b1+\u03b2)&=&\\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta\\\\
\\sin(\u03b1-\u03b2)&=&\\sin \u03b1 \\cos\u03b2-\\cos\u03b1 \\sin\u03b2\\\\
\\cos(\u03b1+\u03b2)&=&\\cos \u03b1 \\cos\u03b2-\\sin\u03b1 \\sin\u03b2\\\\
\\cos(\u03b1-\u03b2)&=&\\cos \u03b1 \\cos \u03b2+\\sin\u03b1 \\sin\u03b2\\\\
\\displaystyle \\tan(\u03b1+\u03b2)&=&\\frac{\\tan\u03b1+\\tan\u03b2}{1-\\tan\u03b1 \\tan\u03b2}\\\\
\\displaystyle \\tan(\u03b1-\u03b2)&=&\\frac{\\tan\u03b1-\\tan\u03b2}{1+\\tan\u03b1 \\tan\u03b2}
\\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

\u898b\u305f\u76ee\u304c\u8907\u96d1\u306a\u5f62\u3092\u3057\u3066\u3044\u308b\u306e\u3067\u3001<\/p>\n\n\n\n

\u300c\u52a0\u6cd5\u5b9a\u7406\u306f\u3069\u3046\u3057\u3066\u305d\u3093\u306a\u516c\u5f0f\u306b\u306a\u308b\u306e\uff1f\u300d<\/span><\/p>\n\n\n\n

\u305d\u3046\u601d\u3046\u65b9\u3082\u3044\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e<\/span>\u3092\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u5b9f\u306f\u305d\u3053\u307e\u3067\u96e3\u3057\u3044\u8a3c\u660e\u3067\u306f\u306a\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

\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e<\/h2>\n\n\n\n

\u52a0\u6cd5\u5b9a\u7406\u3092\u5168\u3066\u8a3c\u660e\u3059\u308b\u306b\u306f\u3001<\/p>\n\n\n\n

\\[\\cos(\u03b1+\u03b2)=\\cos \u03b1 \\cos\u03b2-\\sin\u03b1 \\sin\u03b2\\]<\/p>\n\n\n\n

\u3092\u307e\u305a\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\\(\\cos(\u03b1+\u03b2)\\)\u3092\u8a3c\u660e\u3059\u308b\u306b\u306f\u3044\u304f\u3064\u304b\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u4eca\u56de\u306f\u5b9a\u756a\u306e2\u3064\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\n
    \n
  • \u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\u8a3c\u660e<\/li>\n\n\n\n
  • \u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u306a\u3044\u8a3c\u660e<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

    \u3055\u3063\u305d\u304f\u52a0\u6cd5\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\uff01<\/p>\n\n\n

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

    \u8a3c\u660e\u65b9\u6cd5\u306f\u3044\u308d\u3044\u308d\u77e5\u3063\u3066\u304a\u304f\u3068\u826f\u3044\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

    \u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\uff1a\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046<\/h3>\n\n\n\n

    \u307e\u305a\u306f\u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\u52a0\u6cd5\u5b9a\u7406\u3092\u8a3c\u660e<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u70b9\\(P,Q\\)\u306e\u5ea7\u6a19\u3092\\(P(\\cos \\alpha ,\\sin \\alpha),Q(\\cos \\beta ,\\sin \\beta)\\)\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\uff1a\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\"<\/figure>\n<\/div>\n\n\n

    \u307e\u305a\u306f\u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\\(PQ^{2}\\)\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u70b9\\(P,Q\\)\u306f\u5358\u4f4d\u5186\u4e0a\u306e\u70b9\u306a\u306e\u3067\u3001\\(OP=OQ=1\\)\u3067\u3059\u3002<\/p>\n\n\n\n

    \\begin{eqnarray}
    PQ^{2}&=&OP^{2}+OQ^{2}-2OP \\cdot OQ \\cos (\\beta – \\alpha)\\\\
    &=&1+1-2\\cos(\\beta – \\alpha)\\\\
    &=&2-2 \\cos(\\alpha – \\beta) \\cdots \u2460
    \\end{eqnarray}<\/p>\n\n\n\n

    \u6b21\u306b\u70b9\\(P\\)\u3068\u70b9\\(Q\\)\u306e2\u70b9\u9593\u306e\u8ddd\u96e2\u3092\u3082\u3068\u3081\u3066\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    PQ^{2}&=&(\\cos \\beta -\\cos \\alpha)^{2}+(\\sin \\beta -\\sin \\alpha)^{2}\\\\
    &=&2-2\\cos \\alpha \\cos \\beta – 2 \\sin \\alpha \\sin \\beta\\\\
    &=&2-2(\\cos \\alpha \\cos \\beta +  \\sin \\alpha \\sin \\beta) \\cdots \u2461
    \\end{eqnarray}<\/p>\n\n\n\n

    \u2460,\u2461\u3088\u308a\u3001<\/p>\n\n\n\n

    \\[2-2 \\cos(\\alpha – \\beta)=2-2(\\cos \\alpha \\cos \\beta + \\sin \\alpha \\sin \\beta)\\]<\/p>\n\n\n\n

    \u3086\u3048\u306b\u3001<\/p>\n\n\n\n

    \\[2 \\cos(\\alpha – \\beta)=2(\\cos \\alpha \\cos \\beta + \\sin \\alpha \\sin \\beta)\\]<\/p>\n\n\n\n

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

    \\[\\cos(\\alpha – \\beta)=\\cos \\alpha \\cos \\beta +  \\sin \\alpha \\sin \\beta\\]<\/span><\/p>\n\n\n\n

    \u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\uff1a\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u308f\u306a\u3044<\/h3>\n\n\n\n

    \u6b21\u306b\u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u308f\u306a\u3044\u3067\u8a3c\u660e<\/span>\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\"<\/figure>\n<\/div>\n\n\n

    \u4e0a\u306e\u56f3\u306b\u304a\u3044\u3066\u3001\\(AP\\)\u9593\u306e\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u3068<\/p>\n\n\n\n

    \\begin{eqnarray}
    AP^{2}&=&\\{\\cos(\\alpha + \\beta)-1\\}^{2}+\\sin^{2}(\\alpha + \\beta)\\\\
    &=&2-2\\cos (\\alpha + \\beta)
    \\end{eqnarray}<\/p>\n\n\n\n

    \u6b21\u306b\u3001\uff12\u70b9\\(P,A\\)\u3092\u539f\u70b9\u3092\u4e2d\u5fc3\u306b\\(- \\alpha\\)\u3060\u3051\u56de\u8ee2\u3057\u305f\u4f4d\u7f6e\u306b\u3042\u308b\u70b9\u3092\u3001\u305d\u308c\u305e\u308c\\(Q,R\\)\u3068\u3059\u308b\u3002<\/p>\n\n\n

    \n
    \"\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\"<\/figure>\n<\/div>\n\n\n

    \\(Q,R\\)\u306e\u5ea7\u6a19\u306f<\/p>\n\n\n\n

    \\(Q(\\cos \\beta , \\sin \\beta),R(\\cos \\alpha ,- \\sin \\alpha)\\)\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

    \\(QR\\)\u9593\u306e\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u3068\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    RQ^{2}&=&(\\cos \\beta – \\cos \\alpha)^{2}+(\\sin \\beta + \\sin \\alpha)^{2}\\\\
    &=&2-2(\\cos \\alpha \\cos \\beta – \\sin \\alpha \\sin \\beta)
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(\\angle AOP =\\angle ROQ\\)\u3088\u308a\u3001\\(AP^{2}=RQ^{2}\\)\u306a\u306e\u3067<\/p>\n\n\n\n

    \\[2-2\\cos (\\alpha + \\beta)=2-2(\\cos \\alpha \\cos \\beta –  \\sin \\alpha \\sin \\beta)\\]<\/p>\n\n\n\n

    \u3086\u3048\u306b\u3001<\/p>\n\n\n\n

    \\[2\\cos (\\alpha + \\beta)=2(\\cos \\alpha \\cos \\beta –  \\sin \\alpha \\sin \\beta)\\]<\/p>\n\n\n\n

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

    \\[\\cos(\\alpha + \\beta)=\\cos \\alpha \\cos \\beta –  \\sin \\alpha \\sin \\beta\\]<\/span><\/p>\n\n\n

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

    \u3069\u3061\u3089\u306e\u8a3c\u660e\u3082\u77e5\u3063\u3066\u304a\u3044\u3066\u640d\u306f\u306a\u3044\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

    \u305d\u306e\u4ed6\u306e\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e<\/h3>\n\n\n\n

    \\(\\cos (\\alpha + \\beta)\\)\u3084\\(\\cos (\\alpha – \\beta)\\)\u306e\u52a0\u6cd5\u5b9a\u7406\u3092\u793a\u305b\u3070\u3001\u305d\u306e\u4ed6\u306e\u52a0\u6cd5\u5b9a\u7406\u3082\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

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

    \\begin{eqnarray}
    \\sin (\\alpha+\\beta)&=&\\cos \\{90-(\\alpha+\\beta)\\}\\\\
    &=&\\cos \\{(90-\\alpha)-\\beta\\} \\\\
    &=&\\cos (90-\\alpha) \\cos \\beta+\\sin (90-\\alpha) \\sin \\beta \\\\
    &=&\\sin \\alpha \\cos \\beta+\\cos \\alpha \\sin \\beta
    \\end{eqnarray}<\/p>\n\n\n\n

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

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

    \u5206\u5b50, \u5206\u6bcd\u3092 \\(\\cos \\alpha \\cos \\beta\\)\u3067\u5272\u3063\u3066\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\displaystyle \u3000&=&\\frac{\\tan \\alpha+\\tan \\beta}{1-\\tan \\alpha \\tan \\beta}
    \\end{eqnarray}
    \\begin{eqnarray}
    \\displaystyle \\tan (\\alpha-\\beta)&=&\\frac{\\sin (\\alpha-\\beta)}{\\cos (\\alpha-\\beta)} \\\\
    \\displaystyle &=&\\frac{\\sin \\alpha \\cos \\beta-\\cos \\alpha \\sin \\beta}{\\cos \\alpha \\cos \\beta+\\sin \\alpha \\sin \\beta}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u5206\u5b50, \u5206\u6bcd\u3092\\(\\cos \\alpha \\cos \\beta \\)\u3067\u5272\u3063\u3066\u3001<\/p>\n\n\n\n

    \\begin{eqnarray}
    \\displaystyle \u3000&=&\\frac{\\tan \\alpha-\\tan \\beta}{1+\\tan \\alpha \\tan \\beta}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3053\u308c\u3067\u3059\u3079\u3066\u306e\u52a0\u6cd5\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

    \u307e\u305a\u306f\\(\\cos\\)\u306e\u52a0\u6cd5\u5b9a\u7406\u3092\u793a\u3059\u3053\u3068\u3092\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n\n\n

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

    \u4f59\u5f26\u5b9a\u7406\u3092\u4f7f\u3046\u8a3c\u660e\u304c\u5206\u304b\u308a\u3084\u3059\u304b\u3063\u305f\u3067\u3059\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

    \u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

    \u4eca\u56de\u306f\u52a0\u6cd5\u5b9a\u7406\u306e\u8a3c\u660e\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n

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

    \\begin{eqnarray}
    \\sin(\u03b1+\u03b2)&=&\\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta\\\\
    \\sin(\u03b1-\u03b2)&=&\\sin \u03b1 \\cos\u03b2-\\cos\u03b1 \\sin\u03b2\\\\
    \\cos(\u03b1+\u03b2)&=&\\cos \u03b1 \\cos\u03b2-\\sin\u03b1 \\sin\u03b2\\\\
    \\cos(\u03b1-\u03b2)&=&\\cos \u03b1 \\cos \u03b2+\\sin\u03b1 \\sin\u03b2\\\\
    \\displaystyle \\tan(\u03b1+\u03b2)&=&\\frac{\\tan\u03b1+\\tan\u03b2}{1-\\tan\u03b1 \\tan\u03b2}\\\\
    \\displaystyle \\tan(\u03b1-\u03b2)&=&\\frac{\\tan\u03b1-\\tan\u03b2}{1+\\tan\u03b1 \\tan\u03b2}
    \\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

    \u5168\u3066\u306e\u516c\u5f0f\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b<\/p>\n\n\n\n

    \\[\\cos(\u03b1+\u03b2)=\\cos \u03b1 \\cos\u03b2-\\sin\u03b1 \\sin\u03b2\\]<\/p>\n\n\n\n

    \u3092\u307e\u305a\u8a3c\u660e\u3059\u308b\u3002<\/p>\n\n\n\n

    \\(P(\\cos \\alpha,\\sin \\alpha),Q(\\cos \\beta,\\sin \\beta)\\)\u3068\u3057\u3066<\/p>\n\n\n\n

    \u2460\\(PQ^{2}=\\)\u4f59\u5f26\u5b9a\u7406<\/p>\n\n\n\n

    \u2461\\(PQ^{2}=\\)\u5358\u4f4d\u5186\u4e0a\u306e2\u70b9\u9593\u306e\u8ddd\u96e2<\/p>\n\n\n\n

    \u8a3c\u660e\u306e\u65b9\u91dd<\/span><\/div>
    \n

    \u4f59\u5f26\u5b9a\u7406\uff1d\u5358\u4f4d\u5186\u4e0a\u306e2\u70b9\u9593\u306e\u8ddd\u96e2<\/p>\n<\/div><\/div>\n\n\n\n

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