{"id":2861,"date":"2025-08-24T03:02:51","date_gmt":"2025-08-23T18:02:51","guid":{"rendered":"https:\/\/math-friend.com\/?p=2861"},"modified":"2025-08-24T03:02:54","modified_gmt":"2025-08-23T18:02:54","slug":"%e3%80%90%e4%ba%ac%e9%83%bd%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e4%b8%89%e8%a7%92%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e8%a7%a3%e3%81%ae%e5%80%8b%e6%95%b0%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b2","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2861","title":{"rendered":"\u3010\u4eac\u90fd\u5927\u5b66\u5165\u8a66\u3011\u4e09\u89d2\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u500b\u6570\u3092\u6c42\u3081\u308b(2008)"},"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

\\(0\\leq x < 2\\pi\\)\u306e\u3068\u304d, \u65b9\u7a0b\u5f0f$$
2\\sqrt{2}(\\sin^3{x}+\\cos^3{x})+3\\sin{x}\\cos{x}=0
$$\u3092\u6e80\u305f\u3059\\(x\\)\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.
(2008 \u4eac\u90fd\u5927\u5b66 \u6587\u7cfb [4])<\/span><\/p>\n\n\n\n

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

\n

\\(t=\\sin{x}+\\cos{x}\\)\u3068\u304a\u304f\u3068,$$
\\begin{align}
&t^2=(\\sin{x}+\\cos{x})^2=\\sin^2{x}+2\\sin{x}\\cos{x}+\\cos^2{x}\\\\[1.5ex]
\\iff & \\sin{x}\\cos{x}=\\frac{t^2-1}{2}
\\end{align}
$$\u3068\u306a\u308b. \u307e\u305f, $$
\\begin{align}
&t^3=(\\sin{x}+\\cos{x})^3=\\sin^3{x}+3\\sin^2{x}\\cos{x}+3\\sin{x}\\cos^2{x}+\\cos^3{x}\\\\[1.5ex]
&t^3=\\sin^3{x}+3\\sin{x}\\cos{x}(\\sin{x}+\\cos{x})+\\cos^3{x}\\\\[1.5ex]
t&^3=\\sin^3{x}+\\cos^3{x}+3\\cdot\\frac{t^2-1}{2}\\cdot t\\\\[1.5ex]
\\iff & \\sin^3{x}+\\cos^3{x}=-\\frac{t^3}{2}+\\frac{3t}{2}
\\end{align}
$$\u3068\u306a\u308b\u304b\u3089, \u3053\u308c\u3092\u4e0e\u3048\u3089\u308c\u305f\\(x\\)\u306e\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3057\u3066,$$
\\begin{align}
&2\\sqrt{2}\\left(-\\frac{t^3}{2}+\\frac{3t}{2}\\right)+3\\cdot\\frac{t^2-1}{2}=0\\\\[1.5ex]
\\iff & 2\\sqrt{2}t^3-3t^2-6\\sqrt{2}t+3=0
\\end{align}
$$\u3068\u306a\u308a, \u4e0e\u3048\u3089\u308c\u305f\u65b9\u7a0b\u5f0f\u3092, \\(t\\)\u306e\\(3\\)\u6b21\u65b9\u7a0b\u5f0f\u3067\u8868\u305b\u308b.

\\(t\\)\u304c\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u308b\u3068, $$
t=\\sin{x}+\\cos{x}=\\sqrt{2}\\sin{\\left(x+\\frac{\\pi}{4}\\right)}
$$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d, \\(\\displaystyle \\frac{\\pi}{4}\\leq x+\\frac{\\pi}{4}<\\frac{9\\pi}{4}\\)\u3088\u308a,$$
-\\sqrt{2}\\leq t \\leq \\sqrt{2}
$$\u3068\u308f\u304b\u308b.

\\(f(t)\\)\u3092\\(t\\)\u306e\\(3\\)\u6b21\u65b9\u7a0b\u5f0f\u306e\u5de6\u8fba, $$
f(t)=2\\sqrt{2}t^3-3t^2-6\\sqrt{2}t+3
$$\u3067\u5b9a\u7fa9\u3059\u308b. \\(f(t)\\)\u3092\u5fae\u5206\u3059\u308b\u3068,$$
\\begin{align}
f^\\prime(t)&=6\\sqrt{2}t^2-6t-6\\sqrt{2}\\\\[1.5ex]
&=6(\\sqrt{2}t^2-t-\\sqrt{2})\\\\[1.5ex]
&=6(\\sqrt{2}t+1)(t-\\sqrt{2})
\\end{align}
$$\u3068\u306a\u308a, \\(f^\\prime(t)=0\\)\u3068\u3059\u308b\u3068, \\(\\displaystyle t=-\\frac{1}{\\sqrt{2}}\\), \u307e\u305f\u306f, \\(t=\\sqrt{2}\\)\u3067\u3042\u308b.

\u3053\u308c\u304b\u3089, \\(t\\)\u306e\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\\(-\\sqrt{2}\\leq t\\leq \\sqrt{2}\\)\u3067, \\(f(t)\\)\u306e\u5897\u6e1b\u8868\u3092\u66f8\u304f\u3068,$$
\\begin{array}{|c|c|c|c|c|}
\\hline
t & -\\sqrt{2} & \\cdots & -\\frac{1}{\\sqrt{2}} & \\cdots & \\sqrt{2} \\\\[1.5ex]
\\hline
f'(t) & & + & 0 & – & 0 \\\\[1.5ex]
\\hline
f(t) & 1 & \\nearrow & \\frac{13}{2} &\\searrow& -7\\\\[1.5ex]
\\hline
\\end{array}$$
\u3068\u306a\u308a, \u4ee5\u4e0b\u306e\u30b0\u30e9\u30d5\u304b\u3089\\(t\\)\u306e\\(3\\)\u6b21\u65b9\u7a0b\u5f0f\u306f, \\(-\\sqrt{2}\\leq t\\leq \\sqrt{2}\\)\u306e\u7bc4\u56f2\u3067, \u305f\u30601\u3064\u306e\u5b9f\u6570\u89e3\\(\\alpha\\)\u3092\u3082\u3064\u3053\u3068\u304c\u308f\u304b\u308a, \\(\\alpha\\)\u306f\\(0<\\alpha<\\sqrt{2}\\)\u3092\u6e80\u305f\u3059\u3053\u3068\u3082\u308f\u304b\u308b.<\/p>\n\n\n\n

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

\\(t=\\sqrt{2}\\sin{\\left(x+\\frac{\\pi}{4}\\right)}\\)\u306e\u30b0\u30e9\u30d5\u3092\u66f8\u304f\u3068, \\(0<\\alpha<\\sqrt{2}\\)\u306e\u3068\u304d, \\(t=\\alpha\\)\u3068\u306a\u308b\\(x\\)\u306f, \\(0\\leq x<2\\pi\\)\u306e\u7bc4\u56f2\u306b\\(2\\)\u3064\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

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

\u3088\u3063\u3066, \\(0\\leq x<2\\pi\\)\u306e\u7bc4\u56f2\u3067\u4e0e\u3048\u3089\u308c\u305f\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\\(x\\)\u306e\u500b\u6570\u306f\\(2\\)\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f.
<\/p>\n<\/div><\/div>\n\n\n\n

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