{"id":2016,"date":"2021-12-05T19:28:21","date_gmt":"2021-12-05T10:28:21","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=2016"},"modified":"2021-12-05T19:28:21","modified_gmt":"2021-12-05T10:28:21","slug":"thr202105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr202105\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2021\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(z\\) \u3092\u8907\u7d20\u6570\u3068\u3059\u308b.\r\n\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306e \\(3\\) \u70b9 O \\(( 0 )\\) , A \\(( z )\\) , B \\(( z^2 )\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(3\\) \u70b9 O, A, B \u304c\u540c\u4e00\u76f4\u7dda\u4e0a\u306b\u3042\u308b\u305f\u3081\u306e \\(z\\) \u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(3\\) \u70b9 O, A, B \u304c\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u306e\u9802\u70b9\u306b\u306a\u308b\u3088\u3046\u306a \\(z\\) \u5168\u4f53\u3092\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(3\\) \u70b9 O, A, B \u304c\u4e8c\u7b49\u8fba\u4e09\u89d2\u5f62\u306e\u9802\u70b9\u3067\u3042\u308a, \u304b\u3064 \\(z\\) \u306e\u504f\u89d2 \\(\\theta\\) \u304c \\(0 \\leqq \\theta \\leqq \\dfrac{\\pi}{3}\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \u4e09\u89d2\u5f62 OAB \u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u3068\u305d\u306e\u3068\u304d\u306e \\(z\\) \u306e\u5024\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

    \\(z^2 = kz\\) ... [1] \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(k\\) \u304c\u5b58\u5728\u3059\u308b\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
    \r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\nz ( z-k ) & = 0 \\\\\r\n\\text{\u2234} \\quad z = 0 , k\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{z \\ \\text{\u306f\u5b9f\u6570}}\r\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u300c \\(z\\) \u306f\u5b9f\u6570\u3067\u306f\u306a\u3044\u300d... [2] .<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(\\text{OA} = \\text{OB}\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\\begin{align}\r\n| z^2 | & = | z | \\\\\r\n| z |^2 & = | z | \\\\\r\n\\text{\u2234} \\quad | z | & = 1 \\quad ... [3]\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(\\text{OA} = \\text{AB}\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\\begin{align}\r\n| z^2 -z | & = | z | \\\\\r\n| z | | z-1 | & = | z | \\\\\r\n\\text{\u2234} \\quad | z-1 | & = 1 \\quad ... [4]\r\n\\end{align}\\]<\/li>\r\n

    3. 3*<\/strong>\u3000\\(\\text{OB} = \\text{AB}\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\\begin{align}\r\n| z^2 -z | & = | z^2 | \\\\\r\n| z | | z-1 | & = | z |^2 \\\\\r\n\\text{\u2234} \\quad | z | & = | z-1 | \\quad ... [5]\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, \u70b9 \\(0\\) \u3068 \\(1\\) \u306e\u5782\u76f4\u4e8c\u7b49\u5206\u7dda\u3092\u793a\u3059.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> \uff5e 3*<\/strong> \u3068 [2] \u3088\u308a, \u6c42\u3081\u308b\u7bc4\u56f2\u306f\u4e0b\u56f3\u5b9f\u7dda\u90e8\uff08\u3007\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n\"thr20210501\"\r\n

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

      \\(z^2\\) \u306e\u504f\u89d2\u306f \\(2 \\theta\\) \u306a\u306e\u3067, \\(\\angle \\text{AOB} = \\theta\\) \u3067\u3042\u308a\r\n\\[\\begin{align}\r\n\\triangle \\text{OAB} & = \\dfrac{1}{2} | z | | z^2 | \\sin \\theta \\\\\r\n& = \\dfrac{| z |^3}{2} \\sin \\theta\r\n\\end{align}\\]\r\n\\(\\theta\\) \u3092\u56fa\u5b9a\u3059\u308b\u3068, \\(\\triangle \\text{OAB}\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(| z |\\) \u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067, \\(| z-1 | = 1\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
      \r\n\\(z-1\\) \u306e\u504f\u89d2\u306f, \u5186\u5468\u89d2\u306e\u5b9a\u7406\u3088\u308a \\(2 \\theta\\) \u306a\u306e\u3067\r\n\\[\r\nz = 1 +\\cos 2 \\theta +i \\sin \\theta\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\n| z |^2 & = ( 1 +\\cos 2 \\theta )^2 +\\sin^2 \\theta \\\\\r\n& = 2 +2 \\cos 2 theta \\\\\r\n& = 2 +2 ( \\cos^2 \\theta -1 ) \\\\\r\n& = 4 \\cos^2 \\theta\r\n\\end{align}\\]\r\n\\(\\cos \\theta \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\n| z | = 2 \\cos \\theta\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\triangle \\text{OAB} = 4 \\cos^3 \\theta \\sin \\theta\r\n\\]\r\n\\(f ( \\theta ) = \\cos^3 \\theta \\sin \\theta\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf' ( \\theta ) & = 3 \\cos^2 \\theta ( -\\sin \\theta ) \\sin \\theta +\\cos^3 \\theta \\cdot \\cos \\theta \\\\\r\n& = \\cos^2 \\theta ( -3\\sin^2 \\theta +\\cos^2 \\theta ) \\\\\r\n& = \\cos^2 ( 4 \\cos^2 -3 ) \\\\\r\n& = \\cos^2 ( 2 \\cos -\\sqrt{3} ) ( 2 \\cos +\\sqrt{3} )\r\n\\end{align}\\]\r\n\\(f' ( \\theta ) = 0\\) \u3092\u3068\u304f\u3068, \\(\\theta = \\dfrac{\\pi}{6}\\) .
      \r\n\u3057\u305f\u304c\u3063\u3066, \\(f ( \\theta )\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & 0 & \\cdots & \\dfrac{\\pi}{6} & \\cdots & \\dfrac{\\pi}{3} \\\\ \\hline f' ( \\theta ) & & + & 0 & - & \\\\ \\hline f( \\theta ) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\\[\r\nf \\left( \\dfrac{\\pi}{6} \\right) = \\left( \\dfrac{\\sqrt{3}}{2} \\right)^3 \\dfrac{1}{2} = \\dfrac{3 \\sqrt{3}}{16}\r\n\\]\r\n\u3088\u3063\u3066, \\(\\triangle \\text{OAB}\\) \u306f \\(\\theta = \\dfrac{\\pi}{6}\\) \u3059\u306a\u308f\u3061\r\n\\[\r\nz = \\underline{\\dfrac{3}{2} +\\dfrac{\\sqrt{3}}{2} i}\r\n\\]\r\n\u306e\u3068\u304d, \u6700\u5927\u5024\r\n\\[\r\n4 f \\left( \\dfrac{\\pi}{6} \\right) = \\underline{\\dfrac{3 \\sqrt{3}}{4}}\r\n\\]\r\n\u3092\u3068\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(z\\) \u3092\u8907\u7d20\u6570\u3068\u3059\u308b. \u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306e \\(3\\) \u70b9 O \\(( 0 )\\) , A \\(( z )\\) , B \\(( z^2 )\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(3\\) \u70b9 O, A, B … \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":[175],"tags":[148,165],"class_list":["post-2016","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2021","tag-tohoku_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2016","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=2016"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2016\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=2016"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=2016"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=2016"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}