{"id":2508,"date":"2025-08-05T21:58:55","date_gmt":"2025-08-05T12:58:55","guid":{"rendered":"https:\/\/math-friend.com\/?p=2508"},"modified":"2025-08-05T22:15:46","modified_gmt":"2025-08-05T13:15:46","slug":"%e3%80%90%e5%a4%a7%e9%98%aa%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%ae%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%8c%e8%a7%a3%e3%82%92%e6%8c%81%e3%81%a4%e5%bf%85","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2508","title":{"rendered":"\u3010\u5927\u962a\u5927\u5b66\u5165\u8a66\u3011\u4e09\u89d2\u65b9\u7a0b\u5f0f\u304c\u5b9f\u6570\u89e3\u3092\u3082\u3064\u305f\u3081\u306e\u6761\u4ef6\u3092\u6c42\u3081\u308b(2023)"},"content":{"rendered":"\n<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n<p class=\"has-border -border02\">\u5b9f\u6570\\(a, b\\)\u306b\u5bfe\u3057\u3066, \u4ee5\u4e0b\u306e\\(\\theta\\)\u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f\u304c, \u5b9f\u6570\u89e3\u3092\u3082\u3064\u3088\u3046\u306a\\((a,b)\\)\u306e\u5b58\u5728\u7bc4\u56f2\u3092\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.$$<br>\\cos{2\\theta}=a\\sin{\\theta}+b<br>$$<br><span style=\"text-align:right;display:block;\">(2023 \u5927\u962a\u5927\u5b66 \u6587\u7cfb [1])<\/span><\/p>\n\n\n\n<p>\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n<div class=\"wp-block-group is-style-crease\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>\u4e0e\u3048\u3089\u308c\u305f\\(\\theta\\)\u306b\u95a2\u3059\u308b\u65b9\u7a0b\u5f0f\\(\\displaystyle \\cos{2\\theta}=a\\sin{\\theta}+b\\)\u3092\u2460\u3068\u304a\u304f.<br>\\(\\cos{2\\theta}=1-2\\sin^2{\\theta}\\)\u3088\u308a, \u4e0e\u3048\u3089\u308c\u305f\u65b9\u7a0b\u5f0f\u306f,<br>$$<br>2\\sin^2{\\theta}+a\\sin{\\theta}+b-1=0<br>$$\u3068\u5909\u5f62\u3067\u304d\u308b. \u3053\u3053\u3067, \\(t=\\sin{\\theta}\\)\u3068\u304a\u304f\u3068, \u3053\u306e\u65b9\u7a0b\u5f0f\u306f,<br>$$<br>2t^2+at+b-1=0\\,\\,\\,\u30fb\u30fb\u30fb\u2461<br>$$\u3068\u5909\u5f62\u3067\u304d, \u3053\u308c\u306f\\(t\\)\u306b\u95a2\u3059\u308b2\u6b21\u65b9\u7a0b\u5f0f\u3068\u306a\u308b. \\(-1\\leq t\\leq 1\\)\u306a\u308b\\(t\\)\u306b\u5bfe\u3057, \\(\\sin{\\theta}=t\\)\u3068\u306a\u308b\\(\\theta\\)\u304c\u5b58\u5728\u3057, \\(|t|>1\\)\u306a\u308b\\(t\\)\u306b\u95a2\u3057\u3066\u306f\\(\\sin{\\theta}=t\\)\u3068\u306a\u308b\\(\\theta\\)\u304c\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \u3053\u306e\\(t\\)\u306b\u95a2\u3059\u308b\u65b9\u7a0b\u5f0f\u304c\\(-1\\leq t\\leq 1\\)\u306e\u7bc4\u56f2\u306e\u89e3\u3092\u6301\u3064\u3053\u3068\u304c, \u2460\u304c\u5b9f\u6570\u89e3\u3092\u6301\u3064\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n<p>\\(f(t)=2t^2+at+b-1\\)\u3068\u304a\u304f\u3068, $$<br>f(t)=2\\left(t+\\frac{a}{4}\\right)^2-\\frac{a^2}{8}+b-1<br>$$\u3068\u5909\u5f62\u3067\u304d\u308b\u306e\u3067, \u6a2a\u8ef8\u306b\\(t\\), \u7e26\u8ef8\u306b\\(y\\)\u3092\u3068\u308b\u3068, \\(y=f(t)\\)\u306e\u30b0\u30e9\u30d5\u306f, \\(\\displaystyle \\left(-\\frac{a}{4}, -\\frac{a^2}{8}+b-1\\right)\\)\u3092\u9802\u70b9\u3068\u3059\u308b, \u4e0b\u306b\u51f8\u306e\u653e\u7269\u7dda\u3068\u306a\u308b. \u3053\u306e\u653e\u7269\u7dda\u306e\u8ef8\u306f\\(\\displaystyle t=-\\frac{a}{4}\\)\u3067\u3042\u308a, \u8ef8\u306b\u95a2\u3057\u3066\u5834\u5408\u5206\u3051\u3092\u3059\u308b\u3053\u3068\u3067, \u2461\u304c\\(-1\\leq t\\leq 1\\)\u306e\u7bc4\u56f2\u306b\u89e3\u3092\u6301\u3064, \u8a00\u3044\u63db\u3048\u308b\u3068, \\(y=f(t)\\)\u304c\\(t\\)\u8ef8\u3068\\(-1\\leq t\\leq 1\\)\u306e\u7bc4\u56f2\u3067\u5171\u6709\u70b9\u3092\u6301\u3064\\((a,b\\))\u306e\u6761\u4ef6\u3092\u8abf\u3079\u3066\u3044\u304f.<\/p>\n\n\n\n<p>1) \u8ef8\u304c\u76f4\u7dda\\(t=-1\\)\u3088\u308a\u5de6\u306b\u3042\u308b\u3068\u304d, \u3064\u307e\u308a, \\( \\displaystyle -\\frac{a}{4}&lt;-1 \\iff a&gt;4\\)\u306e\u3068\u304d<br>\u3053\u306e\u3068\u304d\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"622\" height=\"595\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/e6311c8dcd200cfac384548d63a3d3d0.png\" alt=\"\" class=\"wp-image-2565\" style=\"width:341px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/e6311c8dcd200cfac384548d63a3d3d0.png 622w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/e6311c8dcd200cfac384548d63a3d3d0-300x287.png 300w\" sizes=\"(max-width: 622px) 100vw, 622px\" \/><\/figure>\n\n\n\n<p>\u3088\u3063\u3066, \\(y=f(t)\\)\u304c\\(t\\)\u8ef8\u3068\\(-1\\leq t\\leq 1\\)\u3067\u5171\u6709\u70b9\u3092\u6301\u3064\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, <br>$$<br>f(-1)\\leq 0 \\,\\,\\,\u304b\u3064\\,\\,\\,f(1)\\geq 0<br>$$\u3068\u306a\u308b.$$<br>\\begin{align}<br>&amp;f(-1)=2-a+b-1=-a+b+1\\leq 0\\\\[1.5ex]<br>\\iff &amp; b\\leq a-1<br>\\end{align}<br>$$\u3067\u3042\u308a, $$<br>\\begin{align}<br>&amp;f(1)=2+a+b-1=a+b+1\\geq 0\\\\[1.5ex]<br>\\iff &amp; b\\geq -a-1<br>\\end{align}<br>$$\u3068\u306a\u308b.<\/p>\n\n\n\n<p>2) \u8ef8\u304c\\(-1\\leq t\\leq 1\\)\u306e\u9818\u57df\u306b\u3042\u308b\u3068\u304d, \u3064\u307e\u308a, \\(\\displaystyle -1\\leq -\\frac{a}{4}\\leq 1 \\iff -4\\leq t\\leq 4\\)\u306e\u3068\u304d<br>\u3053\u306e\u3068\u304d\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"627\" height=\"514\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/201acadea46fe2dd45036b00062803a5.png\" alt=\"\" class=\"wp-image-2566\" style=\"width:427px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/201acadea46fe2dd45036b00062803a5.png 627w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/201acadea46fe2dd45036b00062803a5-300x246.png 300w\" sizes=\"(max-width: 627px) 100vw, 627px\" \/><\/figure>\n\n\n\n<p>\u3088\u3063\u3066, \\(y=f(t)\\)\u304c\\(t\\)\u8ef8\u3068\\(-1\\leq t\\leq 1\\)\u3067\u5171\u6709\u70b9\u3092\u6301\u3064\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, $$<br>f\\left(-\\frac{a}{4}\\right)\\leq 0\\,\\,\\,\u304b\u3064\\,\\,\\,\u300cf(-1)\\geq 0\\,\\,\\,\u307e\u305f\u306f\\,\\,\\,f(1)\\geq 0\u300d<br>$$\u3068\u306a\u308b. \\(\\displaystyle f\\left(-\\frac{a}{4}\\right)\\leq 0\\), \\(f(-1)\\geq 0\\), \\(f(1)\\geq 0\\)\u3092\u305d\u308c\u305e\u308c\u5909\u5f62\u3059\u308b\u3068.<br>$$<br>\\begin{align}<br>&amp;f\\left(-\\frac{a}{4}\\right) = -\\frac{a^2}{8}+b-1 \\leq 0\\\\[1.5ex]<br>\\iff &amp; b\\leq \\frac{a^2}{8}+1<br>\\end{align}<br>$$<br>$$<br>\\begin{align}<br>&amp;f(-1)=2-a+b-1=-a+b+1\\geq 0\\\\[1.5ex]<br>\\iff &amp; b\\geq a-1<br>\\end{align}<br>$$<br>$$<br>\\begin{align}<br>&amp;f(1)=2+a+b-1=a+b+1\\geq 0\\\\[1.5ex]<br>\\iff &amp; b\\geq -a-1<br>\\end{align}<br>$$\u3067\u3042\u308b.<\/p>\n\n\n\n<p>\u2462 \u8ef8\u304c\u76f4\u7dda\\(t=1\\)\u3088\u308a\u53f3\u306b\u3042\u308b\u3068\u304d, \u3064\u307e\u308a, \\( \\displaystyle -\\frac{a}{4}&gt;1 \\iff a&lt;-4\\)\u306e\u3068\u304d<br>\u3053\u306e\u3068\u304d\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"600\" height=\"601\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/b2bb3def13a63dff35df95606093b70f.png\" alt=\"\" class=\"wp-image-2567\" style=\"width:393px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/b2bb3def13a63dff35df95606093b70f.png 600w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/b2bb3def13a63dff35df95606093b70f-300x300.png 300w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/b2bb3def13a63dff35df95606093b70f-150x150.png 150w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n\n<p>\u3088\u3063\u3066, \\(y=f(t)\\)\u304c\\(t\\)\u8ef8\u3068\\(-1\\leq t\\leq 1\\)\u3067\u5171\u6709\u70b9\u3092\u6301\u3064\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, <br>$$<br>f(-1)\\geq 0 \\,\\,\\,\u304b\u3064\\,\\,\\,f(1)\\leq 0<br>$$\u3068\u306a\u308b.$$<br>\\begin{align}<br>&amp;f(-1)=2-a+b-1=-a+b+1\\geq 0\\\\[1.5ex]<br>\\iff b &amp; \\geq a-1<br>\\end{align}<br>$$\u3067\u3042\u308a, $$<br>\\begin{align}<br>&amp;f(1)=2+a+b-1=a+b+1\\leq 0\\\\[1.5ex]<br>\\iff &amp; b\\leq -a-1<br>\\end{align}<br>$$\u3068\u306a\u308b.<\/p>\n\n\n\n<p>1), 2), 3)\u3088\u308a, \\(y=f(t)\\)\u304c\\(t\\)\u8ef8\u3068\\(-1\\leq t\\leq 1\\)\u3067\u5171\u6709\u70b9\u3092\u6301\u3064\\((a,b)\\)\u306e\u9818\u57df\u3092\u56f3\u793a\u3059\u308b\u3068, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"651\" height=\"664\" src=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/1e2c4acd3ecce9183257d5ad63b399e4.png\" alt=\"\" class=\"wp-image-2569\" style=\"width:368px;height:auto\" srcset=\"https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/1e2c4acd3ecce9183257d5ad63b399e4.png 651w, https:\/\/math-friend.com\/wp-content\/uploads\/2025\/08\/1e2c4acd3ecce9183257d5ad63b399e4-294x300.png 294w\" sizes=\"(max-width: 651px) 100vw, 651px\" \/><\/figure>\n\n\n\n<p>\u3053\u308c\u304c, \u2460\u304c\u5b9f\u6570\u89e3\u3092\u6301\u3064\u9818\u57df\u305d\u306e\u3082\u306e\u3067\u3042\u308b.<\/p>\n<\/div><\/div>\n\n\n\n<p>youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n<iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/wTNawSqsxJ4?si=6tkiILtocFYB2FG9\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059. \u5b9f\u6570\\(a, b\\)\u306b\u5bfe\u3057\u3066, \u4ee5\u4e0b\u306e\\(\\theta\\)\u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f\u304c, \u5b9f\u6570\u89e3\u3092\u3082\u3064\u3088\u3046\u306a\\((a,b)\\)\u306e\u5b58\u5728\u7bc4\u56f2\u3092\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.$$\\cos{2\\theta}= [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2571,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[6],"tags":[],"class_list":["post-2508","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tokimakuru"],"_links":{"self":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/2508","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2508"}],"version-history":[{"count":25,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/2508\/revisions"}],"predecessor-version":[{"id":2577,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/2508\/revisions\/2577"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/media\/2571"}],"wp:attachment":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2508"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2508"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2508"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}