\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u56db\u9762\u4f53 ABCD \u306b\u304a\u3044\u3066, P \u3092\u8fba AB \u306e\u4e2d\u70b9\u3068\u3057, \u70b9 Q \u304c\u8fba AC \u4e0a\u3092\u52d5\u304f\u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \\(\\cos \\angle \\text{PDQ}\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(\\text{AQ} = t \\ ( 0 \\leqq t \\leqq 1 )\\) \u3068\u304a\u304f\u3068, \u4f59\u5f26\u5b9a\u7406\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\text{PD} & = \\dfrac{\\sqrt{3}}{2} , \\\\\r\n\\text{PQ}^2 & = \\dfrac{1}{4} +t^2 -2 \\cdot \\dfrac{1}{2} t \\cos 60^{\\circ} \\\\\r\n& = t^2 -\\dfrac{t}{2} +\\dfrac{1}{4} , \\\\\r\n\\text{QD}^2 & = 1 +t^2 -2 \\cdot 1 \\cdot t \\cos 60^{\\circ} \\\\\r\n& = t^2 -t +1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\cos \\angle \\text{PDQ} & = \\dfrac{\\text{PD}^2 +\\text{QD}^2 -\\text{PQ}^2}{2 \\text{PD} \\cdot \\text{QD}} \\\\\r\n& = \\dfrac{3 +4( t^2 -t +1 ) -( 4t^2 -2t +1 )}{4 \\sqrt{3} \\sqrt{t^2 -t +1}} \\\\\r\n& = \\dfrac{1}{2 \\sqrt{3}} \\cdot \\underline{\\dfrac{3-t}{\\sqrt{t^2 -t +1}}} _ {[1]}\r\n\\end{align}\\]\r\n\\([1] \\gt 0\\) \u306a\u306e\u3067, \\(f(t) = \\{ [1] \\}^2\\) \u3068\u304a\u3044\u3066, \\(f(t)\\) \u304c\u6700\u5927\u306b\u306a\u308b\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf'(t) & = \\dfrac{2(t-3) ( t^2 -t +1 ) -(t-3)^2 (2t-1)}{( t^2 -t +1 )^2} \\\\\r\n& = \\dfrac{(t-3)(5t-1)}{( t^2 -t +1 )^2}\r\n\\end{align}\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nt = \\dfrac{1}{5}\r\n\\]\r\n\u306a\u306e\u3067, \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & 0 & \\cdots & \\dfrac{1}{5} & \\cdots & 1 \\\\ \\hline f'(t) & & + & 0 & - & \\\\ \\hline f(t) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{1}{5} \\right) & = \\dfrac{\\left( 3 -\\frac{1}{5} \\right)^2}{\\frac{1}{25} -\\frac{1}{5} +1} \\\\\r\n& = \\dfrac{14^2}{21} = \\dfrac{28}{3}\r\n\\end{align}\\]\r\n\\(\\cos \\angle \\text{PDQ}\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(t = \\dfrac{1}{5}\\) \u306e\u3068\u304d\u3067, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{1}{2 \\sqrt{3}} \\sqrt{f \\left( \\dfrac{1}{5} \\right)} = \\underline{\\dfrac{\\sqrt{7}}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u56db\u9762\u4f53 ABCD \u306b\u304a\u3044\u3066, P \u3092\u8fba AB \u306e\u4e2d\u70b9\u3068\u3057, \u70b9 Q \u304c\u8fba AC \u4e0a\u3092\u52d5\u304f\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\cos \\angle \\text{PDQ}\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.","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":[126],"tags":[140,137],"class_list":["post-1237","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2015","tag-kyoto_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1237","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=1237"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1237\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}