{"id":3145,"date":"2024-04-25T15:00:43","date_gmt":"2024-04-25T06:00:43","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=3145"},"modified":"2024-04-25T15:20:04","modified_gmt":"2024-04-25T06:20:04","slug":"%e9%80%86%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%a8%e9%80%86%e5%8f%8c%e6%9b%b2%e7%b7%9a%e9%96%a2%e6%95%b0%e3%81%ae%e6%9b%b8%e3%81%8d%e6%96%b9%e8%aa%ad%e3%81%bf%e6%96%b9","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/3145\/","title":{"rendered":"\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u66f8\u304d\u65b9\u8aad\u307f\u65b9"},"content":{"rendered":"

\u9006\u4e09\u89d2\u95a2\u6570\u306f<\/p>\n

$$\\sin^{-1} x = \\arcsin x, \\quad \\cos^{-1} x = \\arccos x, \\quad \\tan^{-1}x = \\arctan x$$<\/p>\n

\u3068 arc<\/strong><\/span> \u3068\u66f8\u3044\u3066\u300c\u30a2\u30fc\u30af<\/span><\/strong><\/span>\u300d\u3068\u8aad\u3080\u306e\u306b\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f<\/p>\n

$$\\sinh^{-1} x = \\mbox{arsinh}\\\u00a0 x, \\quad \\cosh^{-1} x = \\mbox{arcosh}\\\u00a0 x, \\quad \\tanh^{-1}x = \\mbox{artanh}\\\u00a0 x$$<\/p>\n

\u306e\u3088\u3046\u306b\uff0carea<\/strong><\/span> \u306e\u7565\u306e ar<\/strong><\/span> \u3067\u3042\u308a\uff0carc<\/strong><\/span> \u3068\u66f8\u304f\u3079\u304d\u3067\u306f\u306a\u3044\u3057\uff0c\u300c\u30a2\u30fc\u30af<\/strong><\/span>\u300d\u3068\u8aad\u3080\u3079\u304d\u3067\u3082\u306a\u3044\u7406\u7531\u306b\u3064\u3044\u3066\u3002\u300c\u9006\u300d\u95a2\u6570\u3060\u304b\u3089\u300carc\u300d\u3092\u3064\u3051\u308b\uff0c\u3068\u3044\u3046\u306e\u3067\u306f\u306a\u3044\u3002\u305d\u3082\u305d\u3082\u300carc\u300d\u306b\u300c\u9006\u300d\u306e\u610f\u5473\u306f\u306a\u3044\u3002<\/p>\n

<\/p>\n

\n

\u300c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u306e\u5fae\u5206<\/a>\u300d\u306e\u30da\u30fc\u30b8\u3067\u65e2\u306b\u66f8\u3044\u3066\u3044\u308b\u306e\u3060\u304c\uff0c\u5b87\u5b99\u5e74\u9f62\u306e\u30da\u30fc\u30b8<\/a>\u306a\u3069\u3067\u3082\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u591a\u7528\u3057\u3066\u3044\u308b\u306e\u3067\uff0c\u5ff5\u306e\u305f\u3081\u3002<\/p>\n

\u5f27\u5ea6\u6cd5\uff08\u30e9\u30b8\u30a2\u30f3\uff09<\/h3>\n

\"\"<\/p>\n

\u305d\u3082\u305d\u3082\u89d2\u5ea6\u3092\u5f27\u5ea6\u6cd5\u306e\u5358\u4f4d\u3067\u3042\u308b\u30e9\u30b8\u30a2\u30f3\u3067\u3042\u3089\u308f\u3059\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u4e0a\u306e\u56f3\u306e\u3088\u3046\u306b\uff0c\u89d2\u5ea6 \\(\\theta\\) \u304c\u898b\u8fbc\u3080\u534a\u5f84 \\(1\\) \u306e\u5186\u4e0a\u306e\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u3067\u8868\u3059\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002<\/p>\n

\u4f8b\u3048\u3070 \\(\\theta = 360^{\\circ}\\) \u3067\u3042\u308c\u3070\uff0c\u5358\u4f4d\u5186\u306e\u5168\u5468\u3092\u8868\u3059\u304b\u3089\uff0c\u534a\u5f84 \\(r = 1\\) \u306e\u5186\u306e\u5186\u5468\u306f \\(2\\pi r = 2\\pi \\) \u3060\u304b\u3089 \\(\\theta = 2 \\pi \\ \\mbox{(rad)}\\) \u3067\u3042\u308b\u3057\uff0c \\(\\displaystyle\\theta = \\frac{360}{6} = 60^{\\circ}\\) \u3067\u3042\u308c\u3070\uff0c\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u306f\u5186\u5468\u306e \\(1\/6\\) \u306b\u306a\u308b\u304b\u3089\uff0c\\(\\displaystyle\\theta = \\frac{2\\pi}{6} = \\frac{\\pi}{3} \\mbox{(rad)}\\) \u3068\u306a\u308b\u3002<\/p>\n

\u5f27\u5ea6\u6cd5\u3068\u4e09\u89d2\u95a2\u6570<\/h3>\n

\u89d2\u5ea6 \\(\\theta\\) \u3092\u5f27\u5ea6\u6cd5\uff08\u30e9\u30b8\u30a2\u30f3\uff09\u3067\u3042\u3089\u308f\u3059\u3068\uff0c$$\\cos \\theta = x$$\u3068\u306f\u3064\u307e\u308a\uff0c\u5358\u4f4d\u5186\u4e0a\u306e\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u304c \\(\\theta\\) \u3068\u306a\u308b\u3088\u3046\u306a\u70b9 \\(P(x, y(x))\\) \uff08\u3053\u3053\u3067 \\(y\\) \u306f \\(y^2 = 1 – x^2\\) \u3092\u6e80\u305f\u3059\uff09\u306e \\(x\\) \u5ea7\u6a19\u3092\u4e0e\u3048\u308b\u95a2\u6570\u304c \\(\\cos \\theta\\) \u3067\u3042\u308b\uff0c\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n

\u540c\u69d8\u306b\u3057\u3066\uff0c$$\\sin \\theta = y$$\u3068\u306f\u3064\u307e\u308a\uff0c\u5358\u4f4d\u5186\u4e0a\u306e\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u304c \\(\\theta\\) \u3068\u306a\u308b\u3088\u3046\u306a\u70b9 \\(P(x(y), y)\\) \uff08\u3053\u3053\u3067 \\(x\\) \u306f \\(x^2 = 1 – y^2\\) \u3092\u6e80\u305f\u3059\uff09\u306e \\(y\\) \u5ea7\u6a19\u3092\u4e0e\u3048\u308b\u95a2\u6570\u304c \\(\\sin \\theta\\) \u3067\u3042\u308b\uff0c\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n

\u5f27\u5ea6\u6cd5\u3068\u9006\u4e09\u89d2\u95a2\u6570<\/h3>\n

$\\cos \\theta = x$ \u306e\u9006\u95a2\u6570\u304c $$\\cos^{-1} x = \\theta$$\u3067\u3042\u308b\u3002\u3053\u308c\u306f\u3064\u307e\u308a\uff0c\u5358\u4f4d\u5186\u4e0a\u306e\u70b9 \\(A (1, 0)\\) \u304b\u3089 \\(P (x, y(x))\\) \uff08\u3053\u3053\u3067 \\(y\\) \u306f \\(y^2 = 1 – x^2\\) \u3092\u6e80\u305f\u3059\uff09\u307e\u3067\u306e\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055 \\(\\theta\\) \u3092\u4e0e\u3048\u308b\u95a2\u6570\u304c \\(\\cos^{-1} x\\) \u3067\u3042\u308b\uff0c\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n

\u540c\u69d8\u306b\u3057\u3066\uff0c$\\sin \\theta = y$ \u306e\u9006\u95a2\u6570\u304c $$\\sin^{-1} y = \\theta$$\u306f\u3064\u307e\u308a\uff0c\u5358\u4f4d\u5186\u4e0a\u306e\u70b9 \\(A (1, 0)\\) \u304b\u3089 \\(P (x(y), y)\\) \uff08\u3053\u3053\u3067 \\(x\\) \u306f \\(x^2 = 1 – y^2\\) \u3092\u6e80\u305f\u3059\uff09\u307e\u3067\u306e\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055 \\(\\theta\\) \u3092\u4e0e\u3048\u308b\u95a2\u6570\u304c \\(\\sin^{-1} y\\) \u3067\u3042\u308b\uff0c\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308b\u3002<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570 \\(\\cos^{-1} x, \\ \\sin^{-1} y\\) \u304c\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u3092\u4e0e\u3048\u308b\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3053\u308c\u3089\u3092\u300c\u30a2\u30fc\u30af\u9577\uff08\u5f27\u306e\u9577\u3055\uff09$\\theta$ \u3092\u4e0e\u3048\u308b\u95a2\u6570<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3057\u3066 arc<\/strong><\/span> \u3092\u3064\u3051\u3066<\/p>\n

$$\\cos^{-1} x = {\\color{red}{\\mbox{arc}}}\\!\\cos\u00a0 x, \\quad \\sin^{-1} y = {\\color{red}{\\mbox{arc}}}\\!\\sin y$$<\/p>\n

\u306a\u3069\u3068\u66f8\u304f\u306e\u306f\u81ea\u7136\u306a\u3053\u3068\u3067\u3042\u308d\u3046\u3002<\/p>\n

 <\/p>\n

\u53cc\u66f2\u7dda\u304c\u5f35\u308b\u9762\u7a4d<\/h3>\n

\"\"<\/p>\n

\\(x^2 – y^2 = 1\\) \u3067\u3042\u308b\uff08\u53f3\u8fba\u304c \\(1\\)\u00a0 \u3068\u3044\u3046\u610f\u5473\u3067\uff09\u300c\u5358\u4f4d<\/strong><\/span>\u300d\u53cc\u66f2\u7dda\u4e0a\u306e\u70b9 \\(A (1, 0)\\) \u3068 \\(P(x, y)\\) \u3092\u8003\u3048\u3066\uff0c\u76f4\u7dda \\(OA\\)\uff0c\\(OP\\) \u304a\u3088\u3073\u5358\u4f4d\u53cc\u66f2\u7dda\u306e\u4e00\u90e8 \\(AP\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62 \\(OAP\\) \u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u3059\u308b\u3068\uff0c\\(S\\) \u306f\u5e95\u8fba\u304c \\(x\\)\uff0c\u9ad8\u3055\u304c \\(y = \\sqrt{x^2 – 1}\\) \u306e\u76f4\u89d2\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u304b\u3089\uff0c\\(y = \\sqrt{x^2-1}, \\ y = 0, \\ x = 1, \\ x = x\\) \u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092\u5f15\u3044\u305f\u3082\u306e\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\nS &=& \\frac{1}{2} x y \\ – \\int_1^x \\sqrt{t^2 – 1} \\,dt \\\\
\n&=& \\frac{1}{2} x \\sqrt{x^2 – 1} \\ – \\int_1^x \\sqrt{t^2 – 1} \\,dt
\n\\end{eqnarray}<\/p>\n

\u90e8\u5206\u7a4d\u5206\u3057\u3066\u7d9a\u3051\u3066\u3082\u3044\u3044\u304c\uff0c\u3044\u3063\u305f\u3093 $x$ \u3067\u5fae\u5206\u3057\uff0c\u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\frac{dS}{dx} &=& \\frac{1}{2} \\left( \\sqrt{x^2 – 1} + \\frac{x^2}{\\sqrt{x^2 – 1}}\\right) – \\sqrt{x^2 – 1} \\\\
\n&=& \\frac{1}{2} \\left(\\frac{x^2}{\\sqrt{x^2 – 1}} –\u00a0 \\sqrt{x^2 – 1}\\right) \\\\
\n&=& \\frac{1}{2} \\frac{1}{\\sqrt{x^2 – 1}}\\\\ \\ \\\\
\n\\therefore\\ \\ S &=& \\int_1^x \\frac{dS}{dt}\\, dt \\\\
\n&=& \\frac{1}{2} \\int_1^x \\frac{1}{\\sqrt{t^2 – 1}}\\, dt \\\\
\n&=& \\frac{1}{2} \\cosh^{-1} x
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5bfe\u6570\u95a2\u6570\u8868\u793a\u3092\u4f7f\u3046\u3068\uff0c\u53cc\u66f2\u7dda \\(x^2 – y^2 = 1\\) \u4e0a\u306e\u70b9 \\(P(x, y)\\) \u306b\u5bfe\u3057\u3066 $y(x) = \\sqrt{x^2 – 1}$ \u3092\u4ee3\u5165\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\sinh^{-1} y &=& \\log \\left(y + \\sqrt{y^2 + 1} \\right) \\\\
\n\\therefore\\ \\ \\sinh^{-1} \\sqrt{x^2 – 1} &=& \\log \\left(\\sqrt{x^2 – 1} + \\sqrt{(x^2 – 1) + 1} \\right) \\\\
\n&=& \\log\\left( x + \\sqrt{x^2 – 1} \\right) \\\\
\n&=& \\cosh^{-1} x \\\\ \\ \\\\
\n\\therefore\\ \\ S &=& \\frac{1}{2}\\sinh^{-1} y
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c<\/p>\n

$$\\cosh^{-1} x = \\sinh^{-1} y = 2 S$$<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570 $\\cosh^{-1} x, \\ \\sinh^{-1} y$ \u304c\u56f3\u5f62 $OAP$ \u306e\u9762\u7a4d $S$ \u306e2\u500d\u3092\u4e0e\u3048\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3053\u308c\u3089\u3092\u300c\u30a8\u30ea\u30a2\uff08\u9762\u7a4d\uff09\uff08\u306e2\u500d\uff09\u3092\u4e0e\u3048\u308b\u95a2\u6570<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3057\u3066\uff0carea<\/strong><\/span> \u306e\u7701\u7565\u5f62\u3067\u3042\u308b ar<\/strong><\/span> \u3092\u3064\u3051\u3066<\/p>\n

$$\\cosh^{-1} x = {\\color{blue}{\\mbox{ar}}}\\!\\cosh x, \\quad \\sinh^{-1} y = {\\color{blue}{\\mbox{ar}}}\\!\\sinh y$$<\/p>\n

\u306a\u3069\u3068\u66f8\u304f\u306e\u306f\u81ea\u7136\u306a\u3053\u3068\u3067\u3042\u308d\u3046\u3002<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570 $\\cosh^{-1} x, \\ \\sinh^{-1} y$ \u306f\u66f2\u7dda $AP$\uff08\u53cc\u66f2\u7dda\u306e\u4e00\u90e8\uff09\u306e\u9577\u3055\u3092\u4e0e\u3048\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u306e\u3067\uff0c\u300c\u30a2\u30fc\u30af\u9577\uff08\u5f27\u9577\uff09\u3092\u4e0e\u3048\u308b\u95a2\u6570\u300d\u3068\u306f\u8a00\u3048\u306a\u3044\u3057 arc \u306a\u3069\u3068\u66f8\u304f\u3079\u304d\u3067\u306f\u306a\u3044\u3002<\/p>\n

\u53cc\u66f2\u7dda $x^2 – y^2 = 1$ \u306f\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u4f7f\u3063\u3066 $x = \\cosh t, \\ y = \\sinh t$ \u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3067\u304d\u308b\u3002\u6df7\u540c\u3057\u3066\u3057\u307e\u3044\u305d\u3046\u3060\u304c\uff0c\u53cc\u66f2\u7dda\u95a2\u6570 $y = \\cosh x$ \u306b\u305d\u3063\u305f\u66f2\u7dda\u306e\u9577\u3055\u306f\u7a4d\u5206\u3067\u304d\u3066 $\\sinh x$ \u3092\u4f7f\u3063\u3066\u66f8\u3051\u308b\u304c\uff0c\u53cc\u66f2\u7dda\u305d\u306e\u3082\u306e\u306e\u9577\u3055 $L$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u304c<\/p>\n

$$ L = \\int_A^P \\sqrt{\\left( \\frac{dx}{dt}\\right)^2 + \\left( \\frac{dy}{dt}\\right)^2} \\, dt$$<\/p>\n

\u3053\u306e\u7a4d\u5206\u7d50\u679c\u3092\u521d\u7b49\u95a2\u6570\u3067\u66f8\u304f\u3053\u3068\u306f\u3067\u304d\u306a\u3044\uff08\u3088\u3046\u3060\uff09\u3002<\/p>\n

\u307e\u3068\u3081\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u66f8\u304d\u65b9\u8aad\u307f\u65b9<\/h3>\n

\u8aad\u307f\u65b9\u306b\u3064\u3044\u3066\u306f\u3053\u3061\u3089<\/a>\u306b\u3082\u66f8\u3044\u305f\u3002<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u306f\u30a2\u30fc\u30af\u9577\u3092\u4e0e\u3048\u308b\u306e\u3067 arc<\/h4>\n

\u9006\u4e09\u89d2\u95a2\u6570 \\(\\cos^{-1} x, \\ \\sin^{-1} y\\) \u304c\u30a2\u30fc\u30af\uff08\u5b64\uff09\\(AP\\) \u306e\u9577\u3055\u3092\u4e0e\u3048\u308b\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3053\u308c\u3089\u3092\u300c\u30a2\u30fc\u30af\u9577\uff08\u5f27\u306e\u9577\u3055\uff09\u3092\u4e0e\u3048\u308b\u95a2\u6570<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3057\u3066 arc<\/strong><\/span> \u3092\u3064\u3051\u3066<\/p>\n

$$\\cos^{-1} x ={\\color{red}{\\mbox{arc}}}\\!\\cos\u00a0 x, \\quad \\sin^{-1} y = {\\color{red}{\\mbox{arc}}}\\!\\sin y$$<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u30a8\u30ea\u30a2\uff08\u9762\u7a4d\uff09\uff08\u306e2\u500d\uff09\u3092\u4e0e\u3048\u308b\u306e\u3067 area \u306e\u7701\u7565\u5f62\u306e ar<\/h4>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570 $\\cosh^{-1} x, \\ \\sinh^{-1} y$ \u304c\u56f3\u5f62 $OAP$ \u306e\u9762\u7a4d $S$ \u306e2\u500d\u3092\u4e0e\u3048\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c\u3053\u308c\u3089\u3092\u300c\u30a8\u30ea\u30a2\uff08\u9762\u7a4d\uff09\uff08\u306e2\u500d\uff09\u3092\u4e0e\u3048\u308b\u95a2\u6570<\/strong><\/span>\u300d\u3067\u3042\u308b\u3068\u3057\u3066\uff0carea<\/strong><\/span> \u306e\u7701\u7565\u5f62\u3067\u3042\u308b ar<\/strong><\/span> \u3092\u3064\u3051\u3066<\/p>\n

$$\\cosh^{-1} x = {\\color{blue}{\\mbox{ar}}}\\!\\cosh x, \\quad \\sinh^{-1} y ={\\color{blue}{\\mbox{ar}}}\\!\\sinh y$$<\/p>\n

\u53c2\u8003\u6587\u732e<\/h4>\n

\u9ad8\u6728\u8c9e\u6cbb\u300c\u89e3\u6790\u6982\u8ad6\u300d\u306e 54. \u7bc0\u306b\uff0c\u9762\u7a4d\u306e\u3053\u3068\u304c\u7c21\u6f54\u306b\u66f8\u304b\u308c\u3066\u3044\u308b\u3002<\/p>\n

\u88dc\u8db3\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u3082\u30a8\u30ea\u30a2\uff08\u9762\u7a4d\uff09\uff08\u306e2\u500d\uff09\u3092\u4e0e\u3048\u308b\u3093\u3060\u3051\u3069…<\/h4>\n

\u3061\u306a\u307f\u306b\uff0c\u3053\u3093\u306a\u3053\u3068\u3092\u66f8\u304f\u3068\u3053\u3093\u304c\u3089\u304b\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u6247\u5f62 \\(OAP\\) \u306e\u9762\u7a4d \\(S\\) \u306f<\/p>\n

$$S = \\frac{1}{2} \\theta$$<\/p>\n

$$\\therefore\\ \\ \\theta = \\arccos x = \\arcsin y = 2 S$$<\/p>\n

\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u306f\u6247\u5f62\u306e\u9762\u7a4d\u306e2\u500d\u3092\u4e0e\u3048\u308b\u95a2\u6570\u3067\u3042\u308b<\/strong><\/span>\u3068\u3044\u3046\u898b\u65b9\u3082\u3067\u304d\u308b\u3002\uff08\u3060\u304b\u3089\u3068\u3044\u3063\u3066\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u3082 arc\u300c\u30a2\u30fc\u30af\uff08\u5f27\u9577\uff09\u300d\u3067\u306f\u306a\u304f area\u300c\u30a8\u30ea\u30a2\uff08\u9762\u7a4d\uff09\u300d\u306b\u5909\u3048\u308d\u3068\u3044\u3046\u3064\u3082\u308a\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3067\u3082\uff0c\u3069\u3063\u3061\u3082 arc \u3063\u3066\u9593\u9055\u3063\u3066\u8aad\u307f\u66f8\u304d\u3057\u3066\u3057\u307e\u3046\u3088\u308a\u306f\uff0c\u3069\u3063\u3061\u3082 ar \u3068\u66f8\u3044\u3066\u3082\u9593\u9055\u3044\u3067\u306f\u306a\u3044\u3088\u3046\u306a\u6c17\u304c\u3057\u307e\u3059\u3002\uff09<\/p>\n

\"\"<\/p>\n

\u88dc\u8db3\uff1a\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u8a00\u8a9e\u306b\u304a\u3051\u308b\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u8868\u8a18<\/h3>\n

\u672c\u30b5\u30a4\u30c8\u3067\u591a\u7528\u3057\u3066\u3044\u308b Maxima \u3092\u306f\u3058\u3081\u3068\u3057\u3066\u591a\u304f\u306e\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u8a00\u8a9e\u3067\u306f\uff0c
\n\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f<\/p>\n

$\\sinh^{-1} x = $ asinh(x)<\/code>\uff0c$\\cosh^{-1} x = $ acosh(x)<\/code>\uff0c$\\tanh^{-1} x = $ atanh(x)<\/code><\/p>\n

\u3068\u8868\u8a18\u3057\u3066\u3044\u308b\u3002\uff08arc \u304b ar \u304b\u306e\u8ad6\u4e89\u3092\u3055\u3051\u3066 a \u3060\u3051\u3092\u3064\u3051\u308b\u3068\u3044\u3046\u306e\u306f\u30ca\u30a4\u30b9\u306a\u843d\u3068\u3057\u3069\u3053\u308d\u3002\uff09<\/p>\n

\u78ba\u4fe1\u72af\u306a\u306e\u306f Mathematica \u306e Wolfram \u8a00\u8a9e\u3067\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3092<\/p>\n

$\\sinh^{-1} x = $ ArcSinh[x]<\/code>\uff0c$\\cosh^{-1} x = $ ArcCosh[x]<\/code>\uff0c$\\tanh^{-1} x = $ ArcTanh[x]<\/code><\/p>\n

\u3068 Arc<\/code> \u3092\u3064\u3051\u3066 \u8868\u8a18\u3057\u3066\u3044\u308b\u3002<\/p>\n

\u307e\u305f\uff0cPython \u306e NumPy \u3067\u3082\uff08SymPy \u306f asinh(x)<\/code>, acosh(x)<\/code>, atanh(x)<\/code> \u306a\u306e\u306b\uff09<\/p>\n

$\\sinh^{-1} x = $ numpy.arcsinh(x)<\/code>\uff0c$\\cosh^{-1} x = $ numpy.arccosh(x)<\/code>\uff0c$\\tanh^{-1} x = $ numpy.arctanh(x)<\/code><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u9006\u4e09\u89d2\u95a2\u6570\u306f<\/p>\n

$$\\sin^{-1} x = \\arcsin x, \\quad \\cos^{-1} x = \\arccos x, \\quad \\tan^{-1}x = \\arctan x$$<\/p>\n

\u3068 arc \u3068\u66f8\u3044\u3066\u300c\u30a2\u30fc\u30af\u300d\u3068\u8aad\u3080\u306e\u306b\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f<\/p>\n

$$\\sinh^{-1} x = \\mbox{arsinh}\\\u00a0 x, \\quad \\cosh^{-1} x = \\mbox{arcosh}\\\u00a0 x, \\quad \\tanh^{-1}x = \\mbox{artanh}\\\u00a0 x$$<\/p>\n

\u306e\u3088\u3046\u306b\uff0carea \u306e\u7565\u306e ar \u3067\u3042\u308a\uff0carc \u3068\u66f8\u304f\u3079\u304d\u3067\u306f\u306a\u3044\u3057\uff0c\u300c\u30a2\u30fc\u30af\u300d\u3068\u8aad\u3080\u3079\u304d\u3067\u3082\u306a\u3044\u7406\u7531\u306b\u3064\u3044\u3066\u3002\u300c\u9006\u300d\u95a2\u6570\u3060\u304b\u3089\u300carc\u300d\u3092\u3064\u3051\u308b\uff0c\u3068\u3044\u3046\u306e\u3067\u306f\u306a\u3044\u3002\u305d\u3082\u305d\u3082\u300carc\u300d\u306b\u300c\u9006\u300d\u306e\u610f\u5473\u306f\u306a\u3044\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[21],"tags":[],"class_list":["post-3145","post","type-post","status-publish","format-standard","hentry","category-21","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"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=3145"}],"version-history":[{"count":54,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3145\/revisions"}],"predecessor-version":[{"id":8476,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/3145\/revisions\/8476"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=3145"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=3145"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=3145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}