{"id":2835,"date":"2025-08-22T03:56:57","date_gmt":"2025-08-21T18:56:57","guid":{"rendered":"https:\/\/math-friend.com\/?p=2835"},"modified":"2025-08-22T04:13:31","modified_gmt":"2025-08-21T19:13:31","slug":"%e3%80%90%e5%8c%97%e6%b5%b7%e9%81%93%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e9%83%a8%e5%88%86%e5%92%8c%e3%81%ae%e4%b8%8e%e3%81%88%e3%82%89%e3%82%8c%e3%81%9f%e6%95%b0%e5%88%97%e3%81%ae%e4%b8%80","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2835","title":{"rendered":"\u3010\u5317\u6d77\u9053\u5927\u5b66\u5165\u8a66\u3011\u90e8\u5206\u548c\u306e\u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306e\u4e00\u822c\u9805\u3068\u9006\u6570\u548c\u306e\u8a08\u7b97\uff082021)"},"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

\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(A=\\sin{x}\\)\u3068\u304a\u304f. \\(\\sin{5x}\\)\u3092\\(A\\)\u306e\u6574\u5f0f\u3067\u8868\u305b.
(2) \\(\\displaystyle \\sin^2{\\frac{\\pi}{5}}\\)\u306e\u5024\u3092\u6c42\u3081\u3088.
(3) \u66f2\u7dda\\(y=\\cos{3x}\\), \\(y=\\cos{7x}\\)\u306e\\(x\\geq 0\\)\u306b\u304a\u3051\u308b\u5171\u6709\u70b9\u306e\\(x\\)\u5ea7\u6a19\u3092\u5c0f\u3055\u3044\u65b9\u304b\u3089\u9806\u306b, \\(x_1, x_2, x_3, \\cdots\\)\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u95a2\u6570\\(y=\\cos{3x}\\)\u306e\\(x_5\\leq x\\leq x_6\\)\u306b\u304a\u3051\u308b\u5024\u57df\u3092\u6c42\u3081\u3088.
(2021 \u5e83\u5cf6\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

(1) \u4ee5\u4e0b, \\(\\sin\\), \\(\\cos\\)\u306e\u52a0\u6cd5\u5b9a\u7406\u3092\u7528\u3044\u3066\u5f0f\u5909\u5f62\u3092\u884c\u3046. \u306a\u304a, \\(\\sin{x}\\)\u306f\u9806\u6b21\\(A\\)\u306b\u7f6e\u304d\u63db\u3048\u308b.$$
\\begin{align}
\\sin{5x}&=\\sin{(2x+3x)}\\\\[1.5ex]
&=\\sin{2x}\\cos{3x}+\\cos{2x}\\sin{3x}\\\\[1.5ex]
&=2\\sin{x}\\cos{x}\\cos{(x+2x)}+(1-2\\sin^2{x})\\sin{(x+2x)}\\\\[1.5ex]
&=2A\\cos{x}(\\cos{x}\\cos{2x}-\\sin{x}\\sin{2x})+(1-2A^2)(\\sin{x}\\cos{2x}+\\cos{x}\\sin{2x})\\\\[1.5ex]
&=2A\\cos^2{x}(1-2\\sin^2{x}-2\\sin^2{x})+(1-2A^2)\\left\\{A(1-2\\sin^2{x})+2\\sin{x}\\cos^2{x}\\right\\}\\\\[1.5ex]
&=2A(1-A^2)(1-4A^2)+(1-2A^2)\\left\\{A(1-2A^2)+2A(1-A^2)\\right\\}\\\\[1.5ex]
&=2A(1-5A^2+4A^4)+(1-2A^2)(-4A^3+3A)\\\\[1.5ex]
&=2A-10A^3+8A^5-4A^3+3A+8A^5-6A^3
&=16A^5-20A^2+5A
\\end{align}
$$<\/p>\n\n\n\n

(2) (1)\u3067\u5f97\u3089\u308c\u305f\\(\\displaystyle \\sin{5x}=16A^5-20A^2+5A\\)\u306b\u3066, \\(\\displaystyle x=\\frac{\\pi}{5}\\)\u3068\u3057\u3066,$$
\\sin{\\pi}=A(16A^4-20A^2+5)
$$\u3068\u306a\u308b. \u3053\u3053\u3067, \u5de6\u8fba\u306f\\(0\\)\u3067\u3042\u308a, \\( \\displaystyle 0<\\frac{\\pi}{5}<\\frac{\\pi}{2}\\)\u3088\u308a, \\(\\displaystyle A=\\sin{\\frac{\\pi}{5}}\\neq 0\\)\u3068\u306a\u308b\u304b\u3089, \u4e21\u8fba\u3092\\(A\\)\u3067\u5272\u308b\u3068,$$
16A^4-20A^2+5=0
$$\u3092\u5f97\u308b. \u6c42\u3081\u305f\u3044\u306e\u306f\\(\\displaystyle \\sin^2{\\frac{\\pi}{5}}=A^2\\)\u3088\u308a, \u4e0a\u306e\u65b9\u7a0b\u5f0f\u3067\\(A^2\\)\u30921\u3064\u306e\u584a\u3068\u3057\u3066\u898b\u308b\u3053\u3068\u3067, \u3053\u308c\u306f\\(A^2\\)\u306b\u95a2\u3059\u308b2\u6b21\u65b9\u7a0b\u5f0f\u3068\u306a\u3063\u3066\u3044\u308b. \u3088\u3063\u3066, $$
A^2=\\frac{10\\pm \\sqrt{(-10)^2-16\\cdot 5}}{16}=\\frac{5\\pm\\sqrt{5}}{8}
$$\u3068\u306a\u308b.

\u3053\u3053\u3067, \\( \\displaystyle 0<\\frac{\\pi}{5}<\\frac{\\pi}{2}\\)\u3088\u308a, \\(\\displaystyle 0<\\sin{\\frac{\\pi}{5}}<\\frac{1}{\\sqrt{2}}\\)\u306a\u306e\u3067, \\(\\displaystyle 0<A^2<\\frac{1}{2}\\)\u3067\u3042\u308b. \\(\\displaystyle \\frac{5+\\sqrt{5}}{8}>\\frac{1}{2}\\)\u3088\u308a, \\(\\displaystyle \\frac{5+\\sqrt{5}}{8}\\)\u306f\u4e0d\u9069\u3067\u3042\u308b. \u307e\u305f, \\(\\displaystyle 0<\\frac{5-\\sqrt{5}}{8}<\\frac{1}{2}\\)\u3082\u78ba\u8a8d\u3067\u304d\u308b\u306e\u3067, $$
\\sin^2{\\frac{\\pi}{5}}=A^2=\\frac{5-\\sqrt{5}}{8}
$$\u3067\u3042\u308b.<\/p>\n\n\n\n

(3) \u4ee5\u4e0b\u306e\\(\\cos\\)\u306e\u5b9a\u7fa9\u306b\u7528\u3044\u308b\u5358\u4f4d\u5186\u306e\u56f3\u304b\u3089, \u4e00\u822c\u306b\\(\\cos{\\theta_1}=\\cos{\\theta_2}\\)\u3068\u306a\u308b\u306e\u306f, \\(n\\)\u3092\u6574\u6570\u3068\u3057\u3066, \\(\\theta_2=\\theta_1+2n\\pi\\), \u3082\u3057\u304f\u306f, \\(\\theta_2=-\\theta_1+2n\\pi\\)\u3068\u8868\u305b\u308b\u3068\u304d\u3067\u3042\u308b. <\/p>\n\n\n\n

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

\u307e\u305f, \u3053\u306e\u9006\u3082\u6210\u308a\u7acb\u3064. \u3088\u3063\u3066, \\(\\cos{3x}=\\cos{7x}\\)\u3092\u6e80\u305f\u3059\\(x\\)\u306b\u3064\u3044\u3066, $$
\\begin{align}
7x&=3x+2n\\pi\\\\[1.5ex]
7x&=-3x+2n\\pi\\\\[1.5ex]
\\end{align}
$$\u3068\u306a\u308a, \u3053\u308c\u3092\u6574\u7406\u3057\u3066, $$
\\begin{align}
x&=\\frac{n\\pi}{2}\\\\[1.5ex]
x&=\\frac{n\\pi}{5}\\\\[1.5ex]
\\end{align}
$$\u306e\u3044\u305a\u308c\u304b\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b.

\\(x\\geq 0\\)\u306b\u6ce8\u610f\u3057\u3066, \\(\\cos{3x}=\\cos{7x}\\)\u3068\u306a\u308b, \\(x\\)\u3092\u5c0f\u3055\u3044\u65b9\u304b\u3089\u9806\u306b\u6c42\u3081\u308b\u3068,$$
x_1=0,\\,\\,x_2=\\frac{\\pi}{5},\\,\\,x_3=\\frac{2\\pi}{5},\\,\\,x_4=\\frac{\\pi}{2},\\,\\,x_5=\\frac{3\\pi}{5},\\,\\,x_6=\\frac{4\\pi}{5}
$$\u3068\u306a\u308b. \u3088\u3063\u3066, \\(x_5\\leq x\\leq x_6\\)\u306e\u3068\u304d, \\(\\displaystyle\\frac{9\\pi}{5}\\leq 3x\\leq \\frac{12\\pi}{5}\\)\u3068\u306a\u308a, \u4ee5\u4e0b\u306e\u56f3\u304b\u3089\\(\\cos{3x}\\)\u304c\u52d5\u304f\u7bc4\u56f2\u306f, \\(\\displaystyle \\cos{\\frac{12\\pi}{5}}\\leq\\cos{3x}\\leq 1\\)\u3068\u306a\u308b.<\/p>\n\n\n\n

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

\u3053\u3053\u3067, $$
\\cos{\\frac{12\\pi}{5}}=\\cos{\\frac{2\\pi}{5}}=1-2\\sin^2{\\frac{\\pi}{5}}=\\frac{-1+\\sqrt{5}}{4}
$$\u3068\u306a\u308b\u304b\u3089, \u6c42\u3081\u308b\u5024\u57df\u306f,$$
\\frac{-1+\\sqrt{5}}{4}\\leq y \\leq 1
$$\u3067\u3042\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

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