{"id":700,"date":"2022-01-11T18:35:02","date_gmt":"2022-01-11T09:35:02","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=700"},"modified":"2024-04-10T17:00:59","modified_gmt":"2024-04-10T08:00:59","slug":"%e8%a3%9c%e8%b6%b3%ef%bc%9a%e9%80%86%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%8c%e3%81%82%e3%82%89%e3%82%8f%e3%82%8c%e3%82%8b%e7%a9%8d%e5%88%86%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/%e7%b0%a1%e5%8d%98%e3%81%aa1%e9%9a%8e%e9%9d%9e%e7%b7%9a%e5%bd%a2%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e4%be%8b\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e9%80%86%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%8c%e3%81%82%e3%82%89%e3%82%8f%e3%82%8c%e3%82%8b%e7%a9%8d%e5%88%86%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6\/","title":{"rendered":"\u88dc\u8db3\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u304c\u3042\u3089\u308f\u308c\u308b\u7a4d\u5206\u306b\u3064\u3044\u3066"},"content":{"rendered":"

<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n

\u9006\u4e09\u89d2\u95a2\u6570 \\(y = \\sin^{-1} x\\) \u306f \u4e09\u89d2\u95a2\u6570 \\(x = \\sin y\\) \u306e\u9006\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u305d\u306e\u5fae\u5206\u306f\uff08\u5b66\u90e81\u5e74\u751f\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6559\u3048\u308b\uff09
\n\\begin{eqnarray}
\n\\frac{d}{dx} \\sin^{-1} x &=& \\frac{dy}{dx} \\\\
\n&=& \\frac{1}{\\frac{dx}{dy}} \\\\
\n&=& \\frac{1}{\\cos y} \\\\
\n&=& \\frac{1}{\\sqrt{1 – \\sin^2 y}}\\\\
\n&=& \\frac{1}{\\sqrt{1-x^2}}\\\\
\n\\therefore\\ \\ \\frac{d}{dx} \\sin^{-1} x &=&\\frac{1}{\\sqrt{1-x^2}}
\n\\end{eqnarray}
\n\u540c\u69d8\u306b\u3057\u3066\uff0c\u4ee5\u4e0b\u3082\u308f\u304b\u308b\u3002
\n\\begin{eqnarray}
\n\\frac{d}{dx} \\cos^{-1} x &=& – \\frac{1}{\\sqrt{1-x^2}}
\n\\end{eqnarray}<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u304c\u3042\u3089\u308f\u308c\u308b\u7a4d\u5206\u516c\u5f0f<\/h3>\n

\u4e0a\u8a18\u306e\u5fae\u5206\u7d50\u679c\u304b\u3089\uff0c\u5927\u5b661\u5e74\u3067\u7fd2\u3046\u7a4d\u5206\u516c\u5f0f\uff08\u7d50\u679c\u304c\u9006\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u3051\u308b\u3084\u3064\uff09\u306f
$$ \\int\\frac{dx}{\\sqrt{1-x^2}} = \\sin^{-1} x, \\quad -\\int\\frac{dx}{\\sqrt{1-x^2}} = \\cos^{-1} x$$<\/p>\n

\u3068\u6559\u308f\u3063\u305f\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u3067\u3082\u3053\u308c\u3060\u3068\uff0c\u540c\u3058\u7a4d\u5206 \\(\\displaystyle \\int\\frac{dx}{\\sqrt{1-x^2}}\\) \u306e\u7b54\u3048\u304c \\(\\sin^{-1} x\\) \u3067\u3082\u3044\u3044\u3057\uff0c\\( -\\cos^{-1} x\\) \u3067\u3082\u3044\u3044\u3053\u3068\u306b\u306a\u308a\uff0c\u4e00\u610f\u306b\u6c7a\u307e\u3089\u306a\u3044\u3088\u3046\u306b\u601d\u3048\u308b\u304c\uff0c\u305d\u306e\u8fba\u306f\u7a4d\u5206\u5b9a\u6570\u3067\u5438\u53ce\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u3053\u3046\u3002<\/p>\n

\u5f31\u91cd\u529b\u5834\u8fd1\u4f3c\uff1a\\(r_g\\) \u306e\u30bc\u30ed\u6b21\u89e3<\/a>\u306e\u3068\u3053\u308d\u3067\u51fa\u3066\u304d\u305f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5909\u6570\u5206\u96e2\u5f62<\/p>\n

$$\\pm\\frac{d(b u_0)}{\\sqrt{1 – (b u_0)^2}} =\u00a0 d\\phi$$<\/p>\n

\u3092\u601d\u3044\u51fa\u3059\u3068\uff0c\u3053\u306e\u89e3\u306f
$$ \\sin^{-1} (bu_0) = \\phi + C_1, \\quad \\therefore\\ \\ u_0 = \\frac{\\sin(\\phi+C_1)}{b}$$ \u307e\u305f\u306f
$$ \\cos^{-1} (bu_0) = \\phi + C_2, \\quad \\therefore\\ \\ u_0 = \\frac{\\cos(\\phi+C_2)}{b}$$ \u306e\u3069\u3061\u3089\u304b\u3068\u306a\u308b\u304c\uff0c\u7a4d\u5206\u5b9a\u6570\u3092\u9069\u5b9c\u3068\u3063\u3066 \\(C_2 = C_1 – \\frac{\\pi}{2} \\) \u3068\u3057\u3066\u3084\u308c\u3070
\n$$ \\frac{\\cos(\\phi+C_2)}{b} = \\frac{\\cos\\left(\\phi+C_1 – \\frac{\\pi}{2}\\right)}{b} = \\frac{\\sin(\\phi + C_1)}{b}$$ \u3068\u306a\u308b\u3053\u3068\u304b\u3089\uff0c\u4e00\u822c\u306b
\n$$ u_0 = \\frac{\\sin(\\phi + C)}{b} $$ \u3068\u3057\u3066\u3088\u3044\u3002<\/p>\n

\u3082\u3063\u3068\u76f4\u63a5\u7684\u306b\u306f\uff0c<\/p>\n

$$\\cos^{-1} x + \\sin^{-1} x = \\frac{\\pi}{2}, \\quad \\cos^{-1} x = \\frac{\\pi}{2} – \\sin^{-1} x$$<\/p>\n

\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308a\uff0c\\(\\cos^{-1} x\\) \u306e\u5fae\u5206\u304c \\(\\sin^{-1} x\\) \u306e\u5fae\u5206\u306b\u30de\u30a4\u30ca\u30b9\u304c\u3064\u3044\u305f\u3082\u306e\u3067\u3042\u308b\u3053\u3068\u306f\uff0c\u3053\u306e\u3053\u3068\u304b\u3089\u660e\u3089\u304b\u3067\u3042\u308d\u3046\u3002<\/p>\n

Maxima-Jupyter \u3067\u306e\u78ba\u8a8d\u4f8b<\/h3>\n
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\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n

Maxima \u3067\u306f\uff0c$\\sin^{-1} x$ \u306f asin(x)<\/code>\uff0c$\\cos^{-1} x$ \u306f acos(x)<\/code>\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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'<\/span>diff<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\frac{d}{d\\,x}\\,\\arcsin x=\\frac{1}{\\sqrt{1-x^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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'<\/span>diff<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\frac{d}{d\\,x}\\,\\arccos x=-\\frac{1}{\\sqrt{1-x^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u9006\u4e09\u89d2\u95a2\u6570\u304c\u3042\u3089\u308f\u308c\u308b\u7a4d\u5206\u516c\u5f0f<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>1\/<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\int {\\frac{1}{\\sqrt{1-x^2}}}{\\;dx}=\\arcsin x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u307e\u305f\uff0c$$\\displaystyle \\cos^{-1}x = \\frac{\\pi}{2}- \\sin^{-1}x $$\u306e\u8a3c\u660e\u306f\uff0c\u4e21\u8fba\u306e $\\cos$ \u3092\u3068\u3063\u3066
\n$$x = \\cos\\left(\\frac{\\pi}{2} – \\sin^{-1} x \\right)$$ \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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cos<\/span>(<\/span>%pi<\/span>\/<\/span>2<\/span> -<\/span> asin<\/span>(<\/span>x<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":7243,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-700","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/700","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=700"}],"version-history":[{"count":15,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/700\/revisions"}],"predecessor-version":[{"id":2973,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/700\/revisions\/2973"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7243"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=700"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}