\u3064\u307e\u308a\\((cos\u03b8+isin\u03b8)\\)\u3092\\(n\\)\u4e57\u3059\u308b\u5834\u5408\u3001\\(n\\)\u3068\u89d2\u5ea6\\(\u03b8\\)\u3092\u304b\u3051\u305f\u5f0f\u306b\u5909\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u308c\u304c\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3067\u3059\u3002<\/p>\n
\u305f\u3060\u3001\u3053\u306e\u516c\u5f0f\u3092\u899a\u3048\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u8907\u7d20\u6570\u5e73\u9762\u3092\u5b66\u3093\u3067\u3044\u308b\u5834\u5408\u3001\u3053\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u306e\u306f\u5f53\u7136\u3060\u304b\u3089\u3067\u3059\u3002\\(cos\u03b8+isin\u03b8\\)\u306b\u3064\u3044\u3066\u3001\u8907\u7d20\u6570\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3059\u3068\u9577\u3055\uff08\u7d76\u5bfe\u5024\uff09\u306f1\u3067\u3042\u308b\u305f\u3081\u3001\u304b\u3051\u3066\u3082\u9577\u3055\u306b\u5909\u5316\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n
\u4e00\u65b9\u3067\\(cos\u03b8+isin\u03b8\\)\u3092\u304b\u3051\u308b\u3068\u3001\u504f\u89d2\\(\u03b8\\)\u306e\u5206\u3060\u3051\u89d2\u5ea6\u304c\u5897\u3048\u307e\u3059\u3002\u305d\u306e\u305f\u3081\\(z=cos\u03b8+isin\u03b8\\)\u306e\u5834\u5408\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
- \n
- \\(z^2=cos2\u03b8+isin2\u03b8\\)<\/li>\n
- \\(z^3=cos3\u03b8+isin3\u03b8\\)<\/li>\n
- \\(z^4=cos4\u03b8+isin4\u03b8\\)<\/li>\n
- …<\/li>\n
- \\(z^n=cos\\ n\u03b8+isin\\ n\u03b8\\)<\/li>\n<\/ul>\n
\u3053\u3046\u3057\u3066\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u304c\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3067\u304d\u307e\u3057\u305f\u3002\u3088\u308a\u6b63\u78ba\u306b\u8a3c\u660e\u3057\u305f\u3044\u5834\u5408\u306f\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u307e\u3059\u304c\u3001\u6b63\u78ba\u306b\u8a3c\u660e\u3092\u3057\u306a\u304f\u3066\u3082\u3001\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u5bb9\u6613\u306b\u7406\u89e3\u3067\u304d\u307e\u3059\u3002\u307e\u305f\u6975\u5f62\u5f0f\u306e\u6027\u8cea\u3092\u7406\u89e3\u3057\u3066\u3044\u308c\u3070\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u899a\u3048\u308b\u5fc5\u8981\u306f\u306a\u3044\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n
\u8907\u7d20\u6570\u306e\\(n\\)\u4e57\u6839\u306e\u8a08\u7b97\u3092\u884c\u3046<\/h3>\n
\u305d\u308c\u3067\u306f\u3001\u3069\u306e\u3088\u3046\u306a\u3068\u304d\u306b\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u304c\u6709\u52b9\u306a\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u8907\u7d20\u6570\u306b\u3064\u3044\u3066\\(n\\)\u4e57\u6839\u306e\u8a08\u7b97\u3092\u884c\u3046\u3068\u304d\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3057\u307e\u3057\u3087\u3046\u3002\u4f8b\u3048\u3070\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u306e\u7b54\u3048\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n
- \n
- \\((1+\\sqrt{3}i)^6\\)<\/li>\n<\/ul>\n
\u5f0f\u306e\u5c55\u958b\u306b\u3088\u3063\u3066\u8a08\u7b97\u3059\u308b\u5834\u5408\u3001\u8a08\u7b97\u304c\u7169\u96d1\u3067\u3059\u3002\u307e\u305f\u4e8c\u9805\u5b9a\u7406\u3092\u5229\u7528\u3059\u308b\u306b\u3057\u3066\u3082\u8a08\u7b97\u304c\u5927\u5909\u306a\u306e\u306f\u540c\u3058\u3067\u3059\u3002\u305d\u3053\u3067\u8907\u7d20\u6570\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3057\u305f\u5f8c\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
\\((1+\\sqrt{3}i)^6\\)<\/p>\n
\\(=\\left\\{2\\left(cos\\displaystyle\\frac{\u03c0}{3}+isin\\displaystyle\\frac{\u03c0}{3}\\right)\\right\\}^6\\)<\/p>\n
\\(=2^6\\left(cos\\displaystyle\\frac{6\u03c0}{3}+isin\\displaystyle\\frac{6\u03c0}{3}\\right)\\)<\/p>\n
\\(=64(cos2\u03c0+isin2\u03c0)\\)<\/p>\n
\\(=64\\)<\/p>\n
\u3053\u3046\u3057\u3066\u3001\u7b54\u3048\u306f64\u3068\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u5f0f\u306b\u865a\u6570\u304c\u5b58\u5728\u3057\u3001\u3055\u3089\u306b\u306f\\(n\\)\u4e57\u6839\u306e\u8a08\u7b97\u3092\u884c\u3046\u5834\u5408\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u8907\u7d20\u6570\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3057\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308b<\/h3>\n
\u306a\u304a\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308b\u3068\u304d\u3001\\(z=cos\u03b8+isin\u03b8\\)\u306b\u3064\u3044\u3066\u3001\\(z^0=0\\)\u3068\u8003\u3048\u308b\u3068\u3001\\(n=0\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n
\u307e\u305f\\(n\\)\u304c\u8ca0\u306e\u6574\u6570\u3067\u3042\u3063\u3066\u3082\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3067\u304d\u307e\u3059\u3002\\(z^{-n}=\\displaystyle\\frac{1}{z^n}\\)\u3067\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8003\u3048\u307e\u3059\u3002<\/p>\n
- \n
- \\((cos\u03b8+isin\u03b8)^{-n}\\)\\(=\\displaystyle\\frac{1}{(cos\u03b8+isin\u03b8)^{n}}\\)\\(=\\displaystyle\\frac{1}{cos\\ n\u03b8+isin\\ n\u03b8}\\)\\(=cos(-n\u03b8)+isin(-n\u03b8)\\)<\/li>\n<\/ul>\n
\u3053\u306e\u6027\u8cea\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u8a08\u7b97\u554f\u984c\u3092\u89e3\u304d\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u554f\u984c\u306e\u7b54\u3048\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n
- \n
- \u8907\u7d20\u6570\\(z\\)\u304c\\(z+\\displaystyle\\frac{1}{z}=\\sqrt{2}\\)\u3068\u306a\u308b\u3068\u304d\u3001\\(z^{20}+\\displaystyle\\frac{1}{z^{20}}\\)\u306e\u5024\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n
\u307e\u305a\u3001\u8907\u7d20\u6570\\(z\\)\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002\u4e21\u8fba\u306b\\(z\\)\u3092\u304b\u3051\u3066\u8a08\u7b97\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
\\(z+\\displaystyle\\frac{1}{z}=\\sqrt{2}\\)<\/p>\n
\\(z^2+1=\\sqrt{2}z\\)<\/p>\n
\\(z^2-\\sqrt{2}z+1=0\\)<\/p>\n
\u89e3\u306e\u516c\u5f0f\u3088\u308a\u3001\u8907\u7d20\u6570\\(z\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n
- \n
- \\(z=\\displaystyle\\frac{\\sqrt{2}\u00b1\\sqrt{2}i}{2}\\)<\/li>\n<\/ul>\n
\u305d\u3053\u3067\u3001\u8907\u7d20\u6570\\(z\\)\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\\(z=\\displaystyle\\frac{\\sqrt{2}\u00b1\\sqrt{2}i}{2}\\)<\/p>\n
\\(z=\\displaystyle\\frac{1}{\\sqrt{2}}\u00b1\\displaystyle\\frac{1}{\\sqrt{2}}i\\)<\/p>\n
\\(z=cos\\displaystyle\\frac{\u03c0}{4}\u00b1isin\\displaystyle\\frac{\u03c0}{4}\\)<\/p>\n
\u305d\u3053\u3067\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\\(z^{20}+\\displaystyle\\frac{1}{z^{20}}\\)<\/p>\n
\\(=z^{20}+z^{-20}\\)<\/p>\n
\\(=cos\\displaystyle\\frac{20\u03c0}{4}\u00b1i\\displaystyle\\frac{20\u03c0}{4}\\)\\(+cos\\displaystyle\\frac{-20\u03c0}{4}\u00b1i\\displaystyle\\frac{-20\u03c0}{4}\\)<\/p>\n
\\(=cos5\u03c0\u00b1isin5\u03c0\\)\\(+cos(-5\u03c0)\u00b1isin(-5\u03c0)\\)<\/p>\n
\\(=cos\u03c0\u00b1isin\u03c0\\)\\(+cos(-\u03c0)\u00b1isin(-\u03c0)\\)<\/p>\n
\\(=-2\\)<\/p>\n
\u3053\u3046\u3057\u3066\u3001\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u6975\u5f62\u5f0f\u3067\u8868\u3057\u3066\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u305f\u3068\u3048\\(n\\)\u306e\u5024\u304c\u5927\u304d\u304f\u3066\u3082\u5bb9\u6613\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n
3\u500d\u89d2\u306e\u516c\u5f0f\u306e\u5c0e\u51fa\u304c\u53ef\u80fd<\/h3>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u306f3\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5c0e\u51fa\u3059\u308b\u3068\u304d\u3082\u6709\u52b9\u3067\u3059\u3002\u4ee5\u4e0b\u304c3\u500d\u89d2\u306e\u516c\u5f0f\u3067\u3059\u3002<\/p>\n
- \n
- \\(sin3\u03b1=3sin\u03b1-4sin^3\u03b1\\)<\/li>\n
- \\(cos3\u03b1=-3cos\u03b1+4cos^3\u03b1\\)<\/li>\n<\/ul>\n
\u52a0\u6cd5\u5b9a\u7406\u30682\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u30663\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5f97\u308b\u3053\u3068\u306f\u3067\u304d\u308b\u3082\u306e\u306e\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u3088\u308a\u7c21\u5358\u306b3\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u306e\u306f\u5bb9\u6613\u306b\u7406\u89e3\u3067\u304d\u307e\u3059\u3002<\/p>\n
- \n
- \\((cos\u03b8+isin\u03b8)^3=cos3\u03b8+isin3\u03b8\\) – \u2460<\/li>\n<\/ul>\n
\u6b21\u306b\u30013\u4e57\u306e\u5c55\u958b\u516c\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\\((cos\u03b8+isin\u03b8)^3\\)\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\\((cos\u03b8+isin\u03b8)^3\\)<\/p>\n
\\(=cos^3\u03b8+3cos^2\u03b8\u00b7isin\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\)\\(-isin^3\u03b8\\)<\/p>\n
\\(=cos^3\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\)\\(+i(3cos^2\u03b8\u00b7sin\u03b8-sin^3\u03b8)\\) – \u2461<\/p>\n
\u2460\u3068\u2461\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n
- \n
- \\(cos3\u03b8+isin3\u03b8\\)\\(=cos^3\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\)\\(+i(3cos^2\u03b8\u00b7sin\u03b8-sin^3\u03b8)\\)<\/li>\n<\/ul>\n
\u5b9f\u6570\u90e8\u5206\u3068\u865a\u6570\u90e8\u5206\u306f\u540c\u3058\u5024\u306b\u306a\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u306b\u3042\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/span><\/p>\n
- \n
- \\(cos3\u03b8\\)\\(=cos^3\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\) – \u2462<\/li>\n
- \\(sin3\u03b8\\)\\(=3cos^2\u03b8\u00b7sin\u03b8-sin^3\u03b8\\) – \u2463<\/li>\n<\/ul>\n
\u6b21\u306b\u3001\\(sin^2\u03b8+cos^2\u03b8=1\\)\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u30fb\u2462\u3078\\(sin^2\u03b8=1-cos^2\u03b8\\)\u3092\u4ee3\u5165\u3059\u308b<\/strong><\/p>\n
\\(cos3\u03b8\\)\\(=cos^3\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\)<\/p>\n
\\(cos3\u03b8\\)\\(=cos^3\u03b8-3cos\u03b8(1-cos^2\u03b8)\\)<\/p>\n
\\(cos3\u03b8\\)\\(=4cos^3\u03b8-3cos\u03b8\\)<\/p>\n
\u30fb\u2463\u3078\\(cos^2\u03b8=1-sin^2\u03b8\\)\u3092\u4ee3\u5165\u3059\u308b<\/strong><\/p>\n
\\(sin3\u03b8\\)\\(=3cos^2\u03b8\u00b7sin\u03b8-sin^3\u03b8\\)<\/p>\n
\\(sin3\u03b8\\)\\(=3(1-sin^2\u03b8)\u00b7sin\u03b8-sin^3\u03b8\\)<\/p>\n
\\(sin3\u03b8\\)\\(=3sin\u03b8-4sin^3\u03b8\\)<\/p>\n
\u3053\u3046\u3057\u3066\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u30663\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u30023\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5fd8\u308c\u3066\u3082\u3001\u3053\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u516c\u5f0f\u306e\u5c0e\u51fa\u304c\u53ef\u80fd\u3067\u3059\u3002<\/p>\n
1\u306e\\(n\\)\u4e57\u6839\u3092\u8a08\u7b97\u3059\u308b<\/h2>\n
\u5148\u307b\u3069\u30013\u500d\u89d2\u306e\u516c\u5f0f\u3092\u5f97\u308b\u3068\u304d\u3001\u5b9f\u6570\u90e8\u5206\u3068\u865a\u6570\u90e8\u5206\u3092\u6bd4\u8f03\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u8003\u3048\u65b9\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u30011\u306e\\(n\\)\u4e57\u6839\u3092\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\u3001\\(z^n=1\\)\u306e\\(z\\)\u3092\u8a08\u7b97\u3059\u308b\u306e\u3067\u3059\u3002<\/span><\/p>\n
\u4f8b\u3068\u3057\u3066\u3001\\(z^6=1\\)\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002\\(z=\u00b11\\)\u306f\u7b54\u3048\u3068\u3057\u3066\u3059\u3050\u306b\u308f\u304b\u308a\u307e\u3059\u3002\u305d\u308c\u3067\u306f\u3001\u3042\u30684\u3064\u306e\u89e3\u306f\u3069\u306e\u3088\u3046\u306b\u5f97\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\\(z=r(cos\u03b8+isin\u03b8)\\)\u3068\u8003\u3048\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n
\\(z^6=r^6(cos\u03b8+isin\u03b8)^6\\)<\/p>\n
\\(z^6=r^6(cos6\u03b8+isin6\u03b8)\\)<\/p>\n
\u6b21\u306b\\(z^6=1\\)\u306b\u3064\u3044\u3066\u3001\u53f3\u8fba\u306b\u7740\u76ee\u3057\u307e\u3057\u3087\u3046\u30021\u3092\u8907\u7d20\u6570\u306e\u6975\u5f62\u5f0f\u3067\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
- \n
- \\(1=cos0+isin0\\)<\/li>\n<\/ul>\n
\u306a\u304a\u3001\u3053\u306e\u5f0f\u30920\u00b0\u4ee5\u5916\u306e\u89d2\u5ea6\u3067\u3082\u6210\u308a\u7acb\u3064\u3088\u3046\u306b\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
- \n
- \\(1=cos2\u03c0k+isin2\u03c0k\\)\uff08\\(k\\)\u306f\u6574\u6570\uff09<\/li>\n<\/ul>\n
\u3053\u3046\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
- \n
- \\(r^6(cos6\u03b8+isin6\u03b8)\\)\\(=cos2\u03c0k+isin2\u03c0k\\)<\/li>\n<\/ul>\n
\u3053\u306e\u5f0f\u306b\u3064\u3044\u3066\u3001\u7b49\u5f0f\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308b\u306e\u3067\\(r=1\\)\u3068\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u5148\u307b\u3069\u3068\u540c\u3058\u3088\u3046\u306b\u3001\u5b9f\u6570\u90e8\u5206\u3068\u865a\u6570\u90e8\u5206\u3092\u6bd4\u3079\u307e\u3057\u3087\u3046\u3002\u5b9f\u6570\u90e8\u5206\u3068\u865a\u6570\u90e8\u5206\u306f\u540c\u3058\u5024\u306b\u306a\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/span>\u3064\u307e\u308a\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n
- \n
- \\(cos6\u03b8=cos2\u03c0k\\)<\/li>\n
- \\(sin6\u03b8=sin2\u03c0k\\)<\/li>\n<\/ul>\n
\u3053\u3046\u3057\u3066\\(6\u03b8=2\u03c0k\\)\u3067\u3042\u308a\u3001\\(\u03b8=\\displaystyle\\frac{\u03c0}{3}k\\)\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u305d\u3053\u3067\\(0\u2266\u03b8<2\u03c0\\)\u306e\u7bc4\u56f2\u3067\u8a08\u7b97\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u304c\u8a72\u5f53\u3059\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n
- \n
- \\(\u03b8=0,\\displaystyle\\frac{\u03c0}{3},\\displaystyle\\frac{2\u03c0}{3},\u03c0,\\)\\(\\displaystyle\\frac{4\u03c0}{3},\\)\\(\\displaystyle\\frac{5\u03c0}{3}\\)<\/li>\n<\/ul>\n
\u305d\u3053\u3067\\(z=(cos\u03b8+isin\u03b8)\\)\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u304c\u7b54\u3048\u3067\u3042\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n
- \n
- \\(z=\u00b11,\\)\\(\\displaystyle\\frac{1}{2}\u00b1\\displaystyle\\frac{\\sqrt{3}}{2}i,\\)\\(-\\displaystyle\\frac{1}{2}\u00b1\\displaystyle\\frac{\\sqrt{3}}{2}i\\)<\/li>\n<\/ul>\n
\u3053\u3046\u3057\u3066\u3001\u3059\u3079\u3066\u306e\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n
\u56f3\u3092\u5229\u7528\u3057\u3066\u30011\u306e\\(n\\)\u4e57\u6839\u306e\u610f\u5473\u3092\u5b66\u3076<\/h3>\n
\u305d\u308c\u3067\u306f\u3001\u8907\u7d20\u6570\u5e73\u9762\u3067\u8003\u3048\u308b\u30681\u306e\\(n\\)\u4e57\u6839\u306f\u4f55\u3092\u610f\u5473\u3057\u3066\u3044\u308b\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u56f3\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u5b9f\u306f\u8a08\u7b97\u3057\u306a\u304f\u3066\u3082\u4e00\u77ac\u3067\\(z^6=1\\)\u306e\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u5148\u307b\u3069\u8a08\u7b97\u3057\u305f\u901a\u308a\u3001\\(z^6=1\\)\u3067\u89d2\u5ea6\u3092\u8a08\u7b97\u3059\u308b\u3068\\(6\u03b8=2\u03c0k\\)\u306b\u306a\u308a\u307e\u3059\u3002\u3064\u307e\u308a\\(\u03b8\\)\u306f\\(2\u03c0\\)\uff08360\u00b0\uff09\u30926\u7b49\u5206\u3057\u3066\u3044\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/span>\\(k=0\\)\u306e\u3068\u304d\u3001\\(\u03b8=0\\)\u3067\u3059\u3002\u305d\u3053\u3067\\(\u03b8=0\\)\u3092\u8d77\u70b9\u3068\u3057\u3066\\(2\u03c0\\)\u30926\u7b49\u5206\u3059\u308b\u5834\u5408\u3001\u4ee5\u4e0b\u306e\u56f3\u3092\u63cf\u3051\u307e\u3059\u3002<\/p>\n
![\"\"](\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/ddh1.jpg\")
<\/p>\n1\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3059\u3068\\(1=cos2\u03c0k+isin2\u03c0k\\)\u3068\u306a\u308b\u305f\u3081\u3001\\(z^n=1\\)\u3068\u3044\u3046\u306e\u306f\u3001\u4e00\u3064\u306e\u5186\u3092\\(n\\)\u7b49\u5206\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u307e\u3059\u3002<\/span>\u305d\u306e\u305f\u3081\u56f3\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u8a08\u7b97\u3057\u306a\u304f\u3066\u3082\\(z^6=1\\)\u306e\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3059\u3002<\/p>\n
\u8907\u7d20\u6570\u5e73\u9762\u3067\u306f\u3001\u8a08\u7b97\u306b\u3088\u3063\u3066\u7b54\u3048\u3092\u5f97\u3089\u308c\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u56f3\u5f62\u304c\u4f55\u3092\u610f\u5473\u3057\u3066\u3044\u308b\u306e\u304b\u5b66\u3076\u3053\u3068\u3082\u91cd\u8981\u3067\u3059\u3002<\/p>\n
\\(n\\)\u4e57\u6839\u3068\u6975\u5f62\u5f0f\u306e\u95a2\u4fc2<\/h3>\n
\u6b21\u306b\u30011\u4ee5\u5916\u306e\\(n\\)\u4e57\u6839\u3092\u78ba\u8a8d\u3057\u307e\u3057\u3087\u3046\u3002\u57fa\u672c\u7684\u306a\u8003\u3048\u65b9\u306f1\u306e\\(n\\)\u4e57\u6839\u3068\u540c\u3058\u3067\u3042\u308b\u305f\u3081\u3001\u7406\u89e3\u3059\u308b\u306e\u306f\u96e3\u3057\u304f\u3042\u308a\u307e\u305b\u3093\u3002\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n
- \n
- \\(z^4=8-8\\sqrt{3}i\\)\u306e\u3068\u304d\u3001\\(z\\)\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n
\\(z=r(cos\u03b8+isin\u03b8)\\)\u3068\u8003\u3048\u308b\u3068\u3001\\(z^4\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
- \n
- \\(z^4=r^4(cos4\u03b8+isin4\u03b8)\\)<\/li>\n<\/ul>\n
\u6b21\u306b\u3001\\(8-8\\sqrt{3}i\\)\u3092\u6975\u5f62\u5f0f\u3067\u8868\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\\(8-8\\sqrt{3}i\\)<\/p>\n
\\(=16\\left(\\displaystyle\\frac{1}{2}-i\\displaystyle\\frac{\\sqrt{3}}{2}\\right)\\)<\/p>\n
\\(=16\\left(cos\\displaystyle\\frac{5\u03c0}{3}+i\\displaystyle\\frac{5\u03c0}{3}\\right)\\)<\/p>\n
\u9577\u3055\uff08\u7d76\u5bfe\u5024\uff09\u3092\u78ba\u8a8d\u3059\u308b\u3068\u3001\\(r^4=16\\)\u3067\u3059\u3002\\(r>0\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(r=2\\)\u3067\u3059\u3002\u307e\u305f\u5b9f\u6570\u90e8\u5206\u3068\u865a\u6570\u90e8\u5206\u306f\u540c\u3058\u5024\u306b\u306a\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n
- \n
- \\(cos4\u03b8=cos\\displaystyle\\frac{5\u03c0}{3}\\)<\/li>\n
- \\(sin4\u03b8=sin\\displaystyle\\frac{5\u03c0}{3}\\)<\/li>\n<\/ul>\n
\u3064\u307e\u308a\u3001\u504f\u89d2\u306f\u6574\u6570\\(k\\)\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<\/p>\n
\\(4\u03b8=\\displaystyle\\frac{5\u03c0}{3}+2\u03c0k\\)<\/p>\n
\\(\u03b8=\\displaystyle\\frac{5\u03c0}{12}+\\displaystyle\\frac{\u03c0}{2}k\\)<\/p>\n
\\(0\u2266\u03b8<2\u03c0\\)\u306e\u7bc4\u56f2\u3067\u8003\u3048\u308b\u3068\u3001\\(k=0,1,2,3\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u305d\u306e\u305f\u3081\\(z=2(cos\u03b8+isin\u03b8)\\)\u3078\u4ee3\u5165\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e4\u3064\u306e\u5f0f\u304c\u7b54\u3048\u3067\u3059\u3002<\/p>\n
- \n
- \\(z=2\\left(cos\\displaystyle\\frac{5\u03c0}{12}+isin\\displaystyle\\frac{5\u03c0}{12}\\right)\\)<\/li>\n
- \\(z=2\\left(cos\\displaystyle\\frac{11\u03c0}{12}+isin\\displaystyle\\frac{11\u03c0}{12}\\right)\\)<\/li>\n
- \\(z=2\\left(cos\\displaystyle\\frac{17\u03c0}{12}+isin\\displaystyle\\frac{17\u03c0}{12}\\right)\\)<\/li>\n
- \\(z=2\\left(cos\\displaystyle\\frac{23\u03c0}{12}+isin\\displaystyle\\frac{23\u03c0}{12}\\right)\\)<\/li>\n<\/ul>\n
\u307e\u305f\u3001\u56f3\u3067\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
![\"\"](\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/ddh2.jpg\")
<\/p>\n\\(z^4\\)\u3067\u3042\u308b\u305f\u3081\u3001\u5186\u30924\u7b49\u5206\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u8907\u7d20\u6570\u5e73\u9762\u3067\\(n\\)\u4e57\u6839\u3068\u504f\u89d2\u306e\u95a2\u4fc2\u3092\u7406\u89e3\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3059\u308b<\/h2>\n
\u8907\u7d20\u6570\u306e\u8a08\u7b97\u3067\u91cd\u8981\u306a\u516c\u5f0f\u304c\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3067\u3059\u3002\u516c\u5f0f\u3092\u899a\u3048\u308b\u5fc5\u8981\u306f\u306a\u304f\u3001\u8907\u7d20\u6570\u5e73\u9762\u306e\u6027\u8cea\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5c0e\u51fa\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308c\u3070\u3001\\(n\\)\u4e57\u6839\u306e\u8a08\u7b97\u3092\u7c21\u5358\u306b\u884c\u3048\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u30013\u500d\u89d2\u306e\u516c\u5f0f\u306e\u5c0e\u51fa\u3082\u53ef\u80fd\u3067\u3059\u3002<\/p>\n
\u307e\u305f\u30011\u306e\\(n\\)\u4e57\u6839\u3092\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002\u3053\u306e\u3068\u304d\u306f\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u8907\u7d20\u6570\u5e73\u9762\u3067\u3069\u306e\u3088\u3046\u306a\u56f3\u3092\u63cf\u3051\u308b\u306e\u304b\u5b66\u3076\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\\(n\\)\u4e57\u3059\u308b\u5834\u5408\u3001\u5186\u3092\\(n\\)\u7b49\u5206\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u8907\u7d20\u6570\u5e73\u9762\u3068\u6975\u5f62\u5f0f\u306e\u95a2\u4fc2\u3092\u7406\u89e3\u3067\u304d\u307e\u3059\u3002\u6975\u5f62\u5f0f\u3092\u5229\u7528\u3057\u3066\u3001\u8907\u7d20\u6570\u306e\\(n\\)\u4e57\u6839\u306e\u8a08\u7b97\u3092\u884c\u3044\u307e\u3057\u3087\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
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- \\(z^4=r^4(cos4\u03b8+isin4\u03b8)\\)<\/li>\n<\/ul>\n
- \\(z^4=8-8\\sqrt{3}i\\)\u306e\u3068\u304d\u3001\\(z\\)\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n
- \\(z=\u00b11,\\)\\(\\displaystyle\\frac{1}{2}\u00b1\\displaystyle\\frac{\\sqrt{3}}{2}i,\\)\\(-\\displaystyle\\frac{1}{2}\u00b1\\displaystyle\\frac{\\sqrt{3}}{2}i\\)<\/li>\n<\/ul>\n
- \\(\u03b8=0,\\displaystyle\\frac{\u03c0}{3},\\displaystyle\\frac{2\u03c0}{3},\u03c0,\\)\\(\\displaystyle\\frac{4\u03c0}{3},\\)\\(\\displaystyle\\frac{5\u03c0}{3}\\)<\/li>\n<\/ul>\n
- \\(r^6(cos6\u03b8+isin6\u03b8)\\)\\(=cos2\u03c0k+isin2\u03c0k\\)<\/li>\n<\/ul>\n
- \\(1=cos2\u03c0k+isin2\u03c0k\\)\uff08\\(k\\)\u306f\u6574\u6570\uff09<\/li>\n<\/ul>\n
- \\(1=cos0+isin0\\)<\/li>\n<\/ul>\n
- \\(cos3\u03b8+isin3\u03b8\\)\\(=cos^3\u03b8-3cos\u03b8\u00b7sin^2\u03b8\\)\\(+i(3cos^2\u03b8\u00b7sin\u03b8-sin^3\u03b8)\\)<\/li>\n<\/ul>\n
- \\((cos\u03b8+isin\u03b8)^3=cos3\u03b8+isin3\u03b8\\) – \u2460<\/li>\n<\/ul>\n
- \\(z=\\displaystyle\\frac{\\sqrt{2}\u00b1\\sqrt{2}i}{2}\\)<\/li>\n<\/ul>\n
- \u8907\u7d20\u6570\\(z\\)\u304c\\(z+\\displaystyle\\frac{1}{z}=\\sqrt{2}\\)\u3068\u306a\u308b\u3068\u304d\u3001\\(z^{20}+\\displaystyle\\frac{1}{z^{20}}\\)\u306e\u5024\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n
- \\((cos\u03b8+isin\u03b8)^{-n}\\)\\(=\\displaystyle\\frac{1}{(cos\u03b8+isin\u03b8)^{n}}\\)\\(=\\displaystyle\\frac{1}{cos\\ n\u03b8+isin\\ n\u03b8}\\)\\(=cos(-n\u03b8)+isin(-n\u03b8)\\)<\/li>\n<\/ul>\n
- \\((1+\\sqrt{3}i)^6\\)<\/li>\n<\/ul>\n