{"id":349,"date":"2012-04-07T11:37:57","date_gmt":"2012-04-07T02:37:57","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=349"},"modified":"2021-03-10T20:17:11","modified_gmt":"2021-03-10T11:17:11","slug":"tkr201201","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201201\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2012\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u6b21\u306e\u9023\u7acb\u4e0d\u7b49\u5f0f\u3067\u5b9a\u307e\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u9818\u57df \\(D\\) \u3092\u8003\u3048\u308b.\r\n\\[\r\nx^2 +(y-1)^2 \\leqq 1 , \\quad x \\geqq \\dfrac{\\sqrt{2}}{3}\r\n\\]\r\n\u76f4\u7dda \\(\\ell\\) \u306f\u539f\u70b9\u3092\u901a\u308a, \\(D\\) \u3068\u306e\u5171\u901a\u90e8\u5206\u304c\u7dda\u5206\u3068\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u305d\u306e\u7dda\u5206\u306e\u9577\u3055 \\(L\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.\r\n\u307e\u305f, \\(L\\) \u304c\u6700\u5927\u5024\u3092\u3068\u308b\u3068\u304d, \\(x\\) \u8ef8\u3068 \\(\\ell\\) \u306e\u306a\u3059\u89d2 \\(\\theta \\quad \\left( 0 \\lt \\theta \\lt \\dfrac{\\pi}{2} \\right)\\) \u306e\u4f59\u5f26 \\(\\cos \\theta\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n\"\"\r\n

\\(C : \\ x^2 +(y-1)^2 = 1\\) , \\(\\ell : \\ y =( \\tan \\theta ) x\\) \u3068\u304a\u304f.
\r\n\\(C\\) \u3068 \\(\\ell\\) \u3088\u308a, \u3053\u308c\u3089\u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u306f\r\n\\[\\begin{align}\r\nx^2 +\\left\\{ ( \\tan \\theta ) x -1 \\right\\}^2 & = 1 \\\\\r\n\\left( 1 +\\tan^2 \\theta \\right) x^2 -2x \\tan \\theta & = 0\r\n\\end{align}\\]\r\n\\(x \\neq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{\u2234} \\quad x & =\\dfrac{2 \\tan \\theta}{1 +\\tan^2 \\theta} \\\\\r\n& =2\\sin \\theta \\cos \\theta \\\\\r\n& = \\sin 2\\theta\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nL & = \\sqrt{1 +\\tan^2 \\theta} \\left( \\sin 2 \\theta -\\dfrac{\\sqrt{2}}{3} \\right) \\\\\r\n& = 2 \\sin \\theta -\\dfrac{\\sqrt{2}}{3 \\cos \\theta}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 \\(f( \\theta )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'( \\theta ) & = 2\\cos \\theta -\\dfrac{\\sqrt{2} \\sin \\theta}{3 \\cos^2 \\theta} \\\\\r\n& = \\dfrac{\\sqrt{2} \\cos \\theta}{3} \\left( 3 \\sqrt{2} -\\dfrac{\\sin \\theta}{\\cos^3 \\theta} \\right) \\\\\r\n& = \\dfrac{\\sqrt{2} \\cos \\theta}{3} \\left\\{ 3 \\sqrt{2} -\\left( 1 +\\tan^2 \\theta \\right) \\tan \\theta \\right\\} \\\\\r\n& = -\\dfrac{\\sqrt{2} \\cos \\theta}{3} \\left( \\tan \\theta -\\sqrt{2} \\right) \\left( \\tan^2 \\theta +\\sqrt{2} \\tan \\theta +3 \\right)\r\n\\end{align}\\]\r\n\\(\\tan^2 \\theta +\\sqrt{2} \\tan \\theta +3 >0\\) \u306a\u306e\u3067, \\(f'( \\theta ) =0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\n\\tan \\theta =\\sqrt{2}\r\n\\]\r\n\u3053\u3053\u3067 \\(\\tan \\alpha =\\sqrt{2} \\quad \\left( 0 \\lt \\alpha \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u304f\u3068,\r\n\\(f( \\theta )\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & (0) & \\cdots & \\alpha & \\cdots & \\left( \\dfrac{\\pi}{2} \\right) \\\\ \\hline f'( \\theta ) & & + & 0 & - & \\\\ \\hline f( \\theta ) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(L\\) \u3092\u6700\u5927\u306b\u3059\u308b\u3068\u304d\r\n\\[\\begin{align}\r\n\\cos \\alpha & = \\dfrac{1}{\\sqrt{1 +\\tan^2 \\alpha}} =\\underline{\\dfrac{\\sqrt{3}}{3}} , \\\\\r\n\\sin \\alpha & = \\sqrt{1 -\\cos^2 \\theta} =\\dfrac{\\sqrt{6}}{3}\r\n\\end{align}\\]\r\n\u307e\u305f, \u6700\u5927\u5024\u306f\r\n\\[\r\nf( \\alpha ) =2 \\cdot \\dfrac{\\sqrt{6}}{3} -\\dfrac{\\sqrt{2}}{3 \\cdot \\frac{\\sqrt{3}}{3}} =\\underline{\\dfrac{\\sqrt{6}}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u9023\u7acb\u4e0d\u7b49\u5f0f\u3067\u5b9a\u307e\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u9818\u57df \\(D\\) \u3092\u8003\u3048\u308b. \\[ x^2 +(y-1)^2 \\leqq 1 , \\quad x \\geqq \\dfrac{\\sqrt{2}}{3} \\] \u76f4\u7dda \\(\\ell\\) \u306f\u539f\u70b9 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[58],"tags":[139,68],"class_list":["post-349","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2012","tag-tokyo_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/349","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=349"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/349\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=349"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=349"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}