{"id":2026,"date":"2021-12-11T10:03:54","date_gmt":"2021-12-11T01:03:54","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=2026"},"modified":"2021-12-11T10:03:54","modified_gmt":"2021-12-11T01:03:54","slug":"tbr202105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr202105\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2021\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

O \u3092\u539f\u70b9\u3068\u3059\u308b \\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u70b9 A \\(( -1 , 0 )\\) \u3068\u70b9 B \\(( 2 , 0 )\\) \u3092\u3068\u308b.\r\n\u5186 \\(x^2 +y^2 = 1\\) \u306e, \\(x \\geqq 0\\) \u304b\u3064 \\(y \\geqq 0\\) \u3092\u6e80\u305f\u3059\u90e8\u5206\u3092 \\(C\\) \u3068\u3057, \u307e\u305f\u70b9 B \u3092\u901a\u308a \\(y\\) \u8ef8\u306b\u5e73\u884c\u306a\u76f4\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b.\r\n\\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057, \u66f2\u7dda \\(C\\) \u4e0a\u306b\u70b9 P, Q \u3092\r\n\\[\r\n\\angle \\text{POB} = \\dfrac{\\pi}{n} \\ , \\ \\angle \\text{QOB} = \\dfrac{\\pi}{2n}\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3068\u308b.\r\n\u76f4\u7dda AP \u3068\u76f4\u7dda \\(\\ell\\) \u306e\u4ea4\u70b9\u3092 V \u3068\u3057, \u76f4\u7dda AQ \u3068\u76f4\u7dda \\(\\ell\\) \u306e\u4ea4\u70b9\u3092 W \u3068\u3059\u308b.\r\n\u7dda\u5206 AP, \u7dda\u5206 AQ \u304a\u3088\u3073\u66f2\u7dda \\(C\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S(n)\\) \u3068\u3059\u308b.\r\n\u307e\u305f\u7dda\u5206 PV, \u7dda\u5206 QW, \u66f2\u7dda \\(C\\) \u304a\u3088\u3073\u7dda\u5206 VW \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(T(n)\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\displaystyle\\lim_{n \\rightarrow \\infty} n \\{ S(n) +T(n) \\}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\displaystyle\\lim_{n \\rightarrow \\infty} \\dfrac{T(n)}{S(n)}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    (1)<\/strong><\/p>\r\n

    \u70b9 A \u306f\u5186 \\(x^2 +y^2 =1\\) \u4e0a\u306e\u70b9\u306a\u306e\u3067, \u5186\u5468\u89d2\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\r\n\\angle \\text{PAQ} = \\angle \\text{QAB} = \\dfrac{\\pi}{4n}\r\n\\]\r\n\\(\\text{BV} = 3 \\tan \\dfrac{\\pi}{2n}\\) , \\(\\text{BW} = 3 \\tan \\dfrac{\\pi}{4n}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS(n) +T(n) & = \\triangle \\text{VWA} \\\\\r\n& = \\dfrac{1}{2} \\left( 3 \\tan \\dfrac{\\pi}{2n} -3 \\tan \\dfrac{\\pi}{4n} \\right) \\cdot 3 \\\\\r\n& = \\dfrac{9}{2} \\left( \\tan \\dfrac{\\pi}{2n} -\\tan \\dfrac{\\pi}{4n} \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nn \\{ S(n) +T(n) \\} & = \\dfrac{9}{2} \\left( \\dfrac{\\pi}{2} \\cdot \\dfrac{\\sin \\dfrac{\\pi}{2n}}{\\dfrac{\\pi}{2n}} \\cdot \\dfrac{1}{\\cos \\dfrac{\\pi}{2n}} -\\dfrac{\\pi}{4} \\cdot \\dfrac{\\sin \\dfrac{\\pi}{4n}}{\\dfrac{\\pi}{4n}} \\cdot \\dfrac{1}{\\cos \\dfrac{\\pi}{4n}} \\right) \\\\\r\n& \\rightarrow \\dfrac{9}{2} \\left( \\dfrac{\\pi}{2} \\cdot 1 \\cdot 1 -\\dfrac{\\pi}{4} \\cdot 1 \\cdot 1 \\right) \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\\\\r\n& = \\dfrac{9 \\pi}{8}\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\displaystyle\\lim_{n \\rightarrow \\infty} n \\{ S(n) +T(n) \\} = \\underline{\\dfrac{9 \\pi}{8}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nS(n) & = \\triangle \\text{AOP} +( \\text{\u6247\u5f62 OPQ} ) -\\triangle \\text{OPQ} \\\\\r\n& = \\dfrac{1}{2} \\cdot 1 \\cdot \\sin \\dfrac{\\pi}{n} -\\dfrac{1}{2} \\cdot 1^2 \\cdot \\dfrac{\\pi}{2n} +\\dfrac{1}{2} \\cdot 1^2 \\cdot \\sin \\dfrac{\\pi}{2n} \\\\\r\n& = \\dfrac{1}{2} \\left( \\sin \\dfrac{\\pi}{n} +\\sin \\dfrac{\\pi}{2n} -\\dfrac{\\pi}{2n} \\right)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nn S(n) & = \\dfrac{1}{2} \\left( \\pi \\cdot \\dfrac{\\sin \\dfrac{\\pi}{n}}{\\dfrac{\\pi}{n}} +\\dfrac{\\pi}{2} \\cdot \\dfrac{\\sin \\dfrac{\\pi}{2n}}{\\dfrac{\\pi}{2n}} -\\dfrac{\\pi}{2} \\right) \\\\\r\n& \\rightarrow \\dfrac{1}{2} \\left( \\pi +\\dfrac{\\pi}{2} -\\dfrac{\\pi}{2} \\right) \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\\\\r\n& = \\dfrac{\\pi}{2}\r\n\\end{align}\\]\r\n\u3053\u308c\u3068 (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{T(n)}{S(n)} & = \\dfrac{n \\{ S(n) +T(n) \\}}{n S(n)} -1 \\\\\r\n& \\rightarrow \\dfrac{\\dfrac{9 \\pi}{8}}{\\dfrac{\\pi}{2}} -1 \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\\\\r\n& = \\dfrac{5}{4}\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\displaystyle\\lim_{n \\rightarrow \\infty} \\dfrac{T(n)}{S(n)} = \\underline{\\dfrac{5}{4}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"O \u3092\u539f\u70b9\u3068\u3059\u308b \\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u70b9 A \\(( -1 , 0 )\\) \u3068\u70b9 B \\(( 2 , 0 )\\) \u3092\u3068\u308b. \u5186 \\(x^2 +y^2 = 1\\) \u306e, \\(x \\geqq 0\\) \u304b\u3064 \\(y … \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":[178],"tags":[144,165],"class_list":["post-2026","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2021","tag-tsukuba_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2026","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=2026"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2026\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=2026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=2026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=2026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}