{"id":2125,"date":"2024-04-16T16:40:58","date_gmt":"2024-04-16T07:40:58","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2125"},"modified":"2024-08-07T11:37:54","modified_gmt":"2024-08-07T02:37:54","slug":"%e9%80%86%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86","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%a6b\/%e9%80%86%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206"},"content":{"rendered":"

\u9006\u4e09\u89d2\u95a2\u6570 \\(\\sin^{-1} x , \\ \\cos^{-1} x,\\\u00a0 \\tan^{-1} x \\) \u306e\u5fae\u5206\u3002
\n$$\\frac{d}{dx} \\sin^{-1} x = \\frac{1}{\\sqrt{1-x^2}}, \\quad \\frac{d}{dx} \\cos^{ -1} x = -\\frac{1}{\\sqrt{1-x^2}}, \\quad \\frac{d}{dx} \\tan^{-1} x = \\frac{1}{1 + x^2}$$<\/p>\n

\u3092\u793a\u3059\u3002<\/p>\n

\n

\u4e09\u89d2\u95a2\u6570 \\(\\sin x, \\cos x, \\tan x\\) \u306e\u9006\u95a2\u6570\u3092\u4e00\u822c\u306b\u9006\u4e09\u89d2\u95a2\u6570\u3068\u3044\u3046\u3002\u9006\u4e09\u89d2\u95a2\u6570\u306f\u30b9\u30de\u30db\u30a2\u30d7\u30ea\u306e\u300c\u8a08\u7b97\u6a5f\u300d\u306b\u3082\u642d\u8f09\u3055\u308c\u3066\u3044\u307e\u3059\u3002\uff08\u4ee5\u4e0b\u306f iPhone \u306e\u4f8b\u3002\u6a2a\u5411\u304d\u306b\u3057\u3066 2nd \u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u3002\uff09<\/p>\n

\"\"<\/p>\n

\u4f8b\u3048\u3070\uff0c \\( y = \\sin x \\) \u306e\u9006\u95a2\u6570\u3068\u306f\uff0c\u65b9\u7a0b\u5f0f \\( y = \\sin x \\) \u3092 \\(x\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u3066 \\(x = \\sin^{-1} y \\) \u3068\u66f8\u304d\uff0c \\( \\sin^{-1} \\) \u306e\u90e8\u5206\u3092\u300c\u30a2\u30fc\u30af\u30b5\u30a4\u30f3\u300d\u3068\u8aad\u3080\u3002<\/p>\n

\u3042\u3089\u305f\u3081\u3066 \\(x\\) \u3068 \\( y\\) \u3092\u53d6\u308a\u66ff\u3048\u3066\u66f8\u304d\uff0c\\( y = \\sin^{-1} x \\) \uff08\u30ef\u30a4\u30fb\u30a4\u30b3\u30fc\u30eb\u30fb\u30a2\u30fc\u30af\u30b5\u30a4\u30f3\u30fb\u30a8\u30c3\u30af\u30b9\uff09\u306f \\(y = \\sin x\\) \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u3068\u3044\u3046\u3002\u3053\u306e\u3068\u304d\uff0c\\(x \\) \u3068 \\( y\\) \u304c1\u5bfe1\u5bfe\u5fdc\u3059\u308b\u305f\u3081\u306b\uff0c\\( -1 \\leq x \\leq 1\\) \u306b\u5bfe\u3057\u3066 \\(\\displaystyle -\\frac{\\pi}{2} \\leq y \\leq \\frac{\\pi}{2}\\) \u306e\u7bc4\u56f2\uff08\u300c\u4e3b\u5024\u300d\uff09\u306b\u5236\u9650\u3057\u3066\u3044\u308b\u3002\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n

$\\sin^{-1} x$ \u307e\u305f\u306f \u30a2\u30fc\u30af\u30b5\u30a4\u30f3 $\\arcsin x$<\/h4>\n

$$y = \\sin^{-1} x = \\arcsin x$$\uff08\u30ef\u30a4\u30fb\u30a4\u30b3\u30fc\u30eb\u30fb\u30a2\u30fc\u30af\u30b5\u30a4\u30f3\u30fb\u30a8\u30c3\u30af\u30b9\uff09\u306f \\(y = \\sin x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f\u00a0 \\(x = \\sin y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3067\u3042\u308a\uff0c\u305d\u306e\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f
$$-1 \\leq x \\leq 1, \\qquad -\\frac{\\pi}{2} \\leq y \\leq \\frac{\\pi}{2}$$<\/p>\n

\"\"<\/p>\n

$\\cos^{-1} x$ \u307e\u305f\u306f \u30a2\u30fc\u30af\u30b3\u30b5\u30a4\u30f3 $\\arccos x$<\/h4>\n

$$ y = \\cos^{-1} x = \\arccos x$$ \uff08\u30ef\u30a4\u30fb\u30a4\u30b3\u30fc\u30eb\u30fb\u30a2\u30fc\u30af\u30b3\u30b5\u30a4\u30f3\u30fb\u30a8\u30c3\u30af\u30b9\uff09\u306f \\(y = \\cos x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f\u00a0 \\(x = \\cos y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3067\u3042\u308a\uff0c\u305d\u306e\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f
$$-1 \\leq x \\leq 1, \\qquad 0 \\leq y \\leq \\pi$$<\/p>\n

\"\"<\/p>\n

$\\tan^{-1} x$ \u307e\u305f\u306f \u30a2\u30fc\u30af\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8 $\\arctan x$<\/h4>\n

$$ y = \\tan^{-1} x = \\arctan x$$ \uff08\u30ef\u30a4\u30fb\u30a4\u30b3\u30fc\u30eb\u30fb\u30a2\u30fc\u30af\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8\u30fb\u30a8\u30c3\u30af\u30b9\uff09\u306f \\(y = \\tan x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f\u00a0 \\(x = \\tan y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3067\u3042\u308a\uff0c\u305d\u306e\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f
$$-\\infty < x < \\infty, \\qquad -\\frac{\\pi}{2} < y < \\frac{\\pi}{2}$$<\/p>\n

\"\"<\/p>\n

 <\/p>\n

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

$\\sin^{-1} x$ \u306e\u5fae\u5206<\/h4>\n

$y = \\sin^{-1} x$ \u306e\u5fae\u5206\u306f\uff0c\u307e\u305a\u3053\u308c\u304c $x = \\sin y$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u3092\u601d\u3044\u51fa\u3059\u3002\u3059\u308b\u3068\uff0c$x$ \u306f $y$ \u306e\u95a2\u6570\u3068\u3057\u3066\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u3051\u3066\u3044\u308b\u306e\u3067\uff0c<\/p>\n

$$\\frac{dx}{dy} = \\frac{d}{dy} \\sin y = \\cos y$$<\/p>\n

\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/a><\/p>\n

$$ \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}}$$<\/p>\n

\u3092\u4f7f\u3046\u3068\uff0c$$\\frac{d}{dx} \\sin^{-1} x = \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}} = \\frac{1}{\\cos y} = \\frac{1}{\\sqrt{1-\\sin^2 y}} = \\frac{1}{\\sqrt{1-x^2}}$$<\/p>\n

$\\cos^{-1} x$ \u306e\u5fae\u5206<\/h4>\n

$y = \\cos^{-1} x$ \u306e\u5fae\u5206\u306f\uff0c\u307e\u305a\u3053\u308c\u304c $x = \\cos y$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u3092\u601d\u3044\u51fa\u3059\u3002\u3059\u308b\u3068\uff0c$x$ \u306f $y$ \u306e\u95a2\u6570\u3068\u3057\u3066\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u3051\u3066\u3044\u308b\u306e\u3067\uff0c<\/p>\n

$$\\frac{dx}{dy} = \\frac{d}{dy} \\cos y = -\\sin y$$<\/p>\n

\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/a><\/p>\n

$$ \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}}$$<\/p>\n

\u3092\u4f7f\u3046\u3068\uff0c$$\\frac{d}{dx} \\cos^{-1} x = \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}} = \\frac{1}{-\\sin y} = \\frac{1}{-\\sqrt{1-\\cos^2 y}} = -\\frac{1}{\\sqrt{1-x^2}}$$<\/p>\n

\u307e\u305f\uff0c\\( y = \\cos^{-1} x \\) \u306f\\(x = \\cos y\\) \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\uff0c
$$ x = \\cos y = \\sin\\left(\\frac{\\pi}{2} -y\\right), \\quad\\therefore\\ \\ \\sin^{-1} x = \\frac{\\pi}{2} -y$$
$$\\therefore\\ \\ \\cos^{-1} x + \\sin^{-1} x = y + \\frac{\\pi}{2} -y = \\frac{\\pi}{2}$$\u6539\u3081\u3066\u66f8\u304f\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u4fc2\u304c\u3042\u308b\u3002
$$\\cos^{-1} x + \\sin^{-1} x = \\frac{\\pi}{2}, \\quad\\therefore\\ \\\u00a0 \\cos^{-1} x =\u00a0 -\\sin^{-1} x + \\frac{\\pi}{2}$$<\/p>\n

\\((\\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

$\\tan^{-1} x$ \u306e\u5fae\u5206<\/h4>\n

$y = \\tan^{-1} x$ \u306e\u5fae\u5206\u306f\uff0c\u307e\u305a\u3053\u308c\u304c $x = \\tan y$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u3092\u601d\u3044\u51fa\u3059\u3002\u3059\u308b\u3068\uff0c$x$ \u306f $y$ \u306e\u95a2\u6570\u3068\u3057\u3066\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u3051\u3066\u3044\u308b\u306e\u3067\uff0c<\/p>\n

$$\\frac{dx}{dy} = \\frac{d}{dy} \\tan y = \\frac{1}{\\cos^2 y}$$<\/p>\n

\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/a><\/p>\n

$$ \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}}$$<\/p>\n

\u3092\u4f7f\u3046\u3068\uff0c<\/p>\n

$$\\frac{d}{dx} \\tan^{-1} x = \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}} = \\cos^2 y = \\frac{1}{1 + \\tan^2 y} = \\frac{1}{1 + x^2}$$<\/p>\n

\u307e\u305f\uff0c$y = \\tan^{-1} x$ \u306f $ x = \\tan y$ \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n

\\begin{eqnarray}
\nx &=& \\tan y = \\frac{\\sin y}{\\cos y} \\\\
\n&=& \\frac{\\cos\\left(\\frac{\\pi}{2} -y\\right)}{\\sin\\left(\\frac{\\pi}{2} -y\\right)} \\\\
\n&=& \\frac{1}{\\tan\\left(\\frac{\\pi}{2} -y\\right)}
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c$\\displaystyle x = \\frac{1}{\\tan\\left(\\frac{\\pi}{2} -y\\right)}$ \u3059\u306a\u308f\u3061 $\\displaystyle \\tan\\left(\\frac{\\pi}{2} -y\\right) = \\frac{1}{x}$ \u3088\u308a<\/p>\n

$$ \\frac{\\pi}{2} -y = \\tan^{-1} \\frac{1}{x}$$<\/p>\n

$$\\therefore\\ \\ \\tan^{-1} x + \\tan^{-1} \\frac{1}{x} = y + \\frac{\\pi}{2} -y = \\frac{\\pi}{2}$$<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u306e\u8868\u8a18\u6cd5<\/h3>\n

\\( \\displaystyle (\\sin x)^{-1} = \\frac{1}{\\sin x}\\) \u306a\u306e\u3067\uff0c\u3064\u3044\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u9593\u9055\u3044\u3092\u3057\u3066\u3057\u307e\u3046\u3053\u3068\u3082\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002<\/p>\n

\\(\\displaystyle \\sin^{-1} x = \\frac{x}{\\sin} \\)\uff08\u5927\u9593\u9055\u3044!!<\/strong><\/span>\uff09\u3068\u304b\uff0c\\(\\displaystyle \\sin^{-1} x = \\frac{1}{\\sin x} \\) \uff08\u9593\u9055\u3044!!<\/strong><\/span>\uff09\u306a\u3069\u3068\u8aa4\u89e3\u3057\u3066\u3057\u307e\u3046\u6050\u308c\u304c\u3042\u308b\u306e\u3067\uff0c\u3053\u306e\u3088\u3046\u306a\u304a\u8336\u76ee\u306a\u9593\u9055\u3044\u3092\u8a98\u767a\u3057\u306a\u3044\u3088\u3046\u306b\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8868\u8a18\u6cd5\u3092\u7528\u3044\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/p>\n

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

\u307e\u305f\uff0c\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u8a00\u8a9e\u3067\u306f\uff0c\u4e09\u89d2\u95a2\u6570\u3092<\/p>\n

sin(x), cos(x), tan(x)<\/pre>\n
\u9006\u4e09\u89d2\u95a2\u6570\u3092\u305d\u308c\u305e\u308c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u8a18\u3059\u308b\u5834\u5408\u304c\u3042\u308b\u3002<\/p>\n
asin(x), acos(x), atan(x)<\/pre>\n
\u53c2\u8003\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u306e\u9593\u306e\u95a2\u4fc2<\/h3>\n
\u4e0a\u3067\u5f97\u3089\u308c\u305f\u9006\u4e09\u89d2\u95a2\u6570\u306e\u9593\u306e\u95a2\u4fc2\u3092\u307e\u3068\u3081\u3066\u304a\u304f\u3068\uff0c\uff08\u7279\u306b\u4ee5\u4e0b\u306e2\u3064\u306f\u5b87\u5b99\u8ad6\u306e\u3068\u3053\u308d\u3067\u5b9f\u969b\u306b\u4f7f\u3046\uff09<\/p>\n
$$\\cos^{-1} x + \\sin^{-1} x = \\frac{\\pi}{2}$$<\/p>\n
$$\\tan^{-1} x + \\tan^{-1} \\frac{1}{x} =\u00a0 \\frac{\\pi}{2}$$<\/p>\n
\u534a\u89d2\u306e\u516c\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u95a2\u4fc2<\/h4>\n
Wikipedia \u306e\u9006\u95a2\u6570<\/a>\u306e\u9805\u306b\u306f\uff0c\u305d\u306e\u4ed6\u306b\u3082\u3044\u308d\u3044\u308d\u306a\u9006\u4e09\u89d2\u95a2\u6570\u306e\u9593\u306e\u95a2\u4fc2\u304c\u63b2\u8f09\u3055\u308c\u3066\u3044\u308b<\/a>\u3002<\/p>\n
\u5f8c\u3005\u4f7f\u3046\u53ef\u80fd\u6027\u304c\u3042\u308b\u306e\u306f\uff08\u5b9f\u969b\u306b\u5b87\u5b99\u8ad6\u306e\u9805\u3067\u4f7f\u3046\uff09\uff0c\u4ee5\u4e0b\u306e\u534a\u89d2\u306e\u516c\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\uff0c\u3044\u304f\u3064\u304b\u306e\u95a2\u4fc2\u3002\u307e\u305a\uff0c<\/p>\n
\\begin{eqnarray}
\n\\tan^2 \\frac{\\theta}{2} &=& \\frac{\\sin^2\\frac{\\theta}{2}}{\\cos^2\\frac{\\theta}{2}} \\\\
\n&=& \\frac{1 -\\cos\\theta}{1 + \\cos\\theta} \\\\
\n&=& \\frac{\\sin^2\\theta}{(1 + \\cos\\theta)^2} \\\\ \\ \\\\
\n\\therefore\\ \\ \\tan \\frac{\\theta}{2} &=& \\frac{\\sin\\theta}{1 + \\cos\\theta} \\\\
\n&=& \\frac{\\sqrt{1 -\\cos^2\\theta}}{1 + \\cos\\theta} \\tag{1}\\\\
\n&=& \\frac{\\sin\\theta}{1 + \\sqrt{1 -\\sin^2\\theta}} \\tag{2}\\\\
\n&=& \\frac{\\tan\\theta}{1 + \\sqrt{1 + \\tan^2\\theta}} \\tag{3}
\n\\end{eqnarray}<\/p>\n
$(1)$ \u306e $\\displaystyle \\tan \\frac{\\theta}{2} = \\frac{\\sqrt{1 -\\cos^2\\theta}}{1 + \\cos\\theta} $ \u3088\u308a<\/p>\n
\\begin{eqnarray}
\n\\frac{\\theta}{2} &=& \\tan^{-1}\\left(\\frac{\\sqrt{1 -\\cos^2\\theta}}{1 + \\cos\\theta} \\right)
\n\\end{eqnarray}<\/p>\n
$\\theta \\equiv \\cos^{-1} x$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\cos^{-1} x&=& 2 \\tan^{-1}\\left(\\frac{\\sqrt{1 -x^2}}{1 +x} \\right)
\n\\end{eqnarray}<\/p>\n
\u6b21\u306b\uff0c$(2)$ \u306e $\\displaystyle \\tan \\frac{\\theta}{2} = \\frac{\\sin\\theta}{1 +\\sqrt{1 -\\sin^2\\theta}} $ \u3088\u308a<\/p>\n
\\begin{eqnarray}
\n\\frac{\\theta}{2} &=& \\tan^{-1}\\left(\\frac{\\sin\\theta}{1 +\\sqrt{1 -\\sin^2\\theta}} \\right)
\n\\end{eqnarray}<\/p>\n
$\\theta \\equiv \\sin^{-1} x$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\sin^{-1} x&=& 2 \\tan^{-1}\\left(\\frac{x}{1 +\\sqrt{1 -x^2} } \\right)
\n\\end{eqnarray}<\/p>\n
\u307e\u305f\uff0c$(3)$ \u304b\u3089\u4ee5\u4e0b\u306e\u95a2\u4fc2\u3082\u308f\u304b\u308b\u3060\u308d\u3046\u3002<\/p>\n
\\begin{eqnarray}
\n\\tan^{-1} x&=& 2 \\tan^{-1}\\left(\\frac{x}{1 +\\sqrt{1 + x^2} } \\right)
\n\\end{eqnarray}<\/p>\n
\u500d\u89d2\u306e\u516c\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308b\u95a2\u4fc2<\/h4>\n
\\begin{eqnarray}
\n\\cos(2\\theta) &=& \\cos^2\\theta -\\sin^2\\theta \\\\
\n&=& \\cos^2\\theta (1 -\\tan^2\\theta)\\\\
\n&=& \\frac{1 -\\tan^2\\theta}{1 + \\tan^2\\theta} \\\\
\n\\therefore \\ \\ 2 \\theta &=& \\cos^{-1} \\left(\\frac{1 -\\tan^2\\theta}{1 + \\tan^2\\theta} \\right)
\n\\end{eqnarray}<\/p>\n
\u4e0a\u5f0f\u3067 $\\theta \\equiv \\tan^{-1} x$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n2 \\tan^{-1} x &=& \\cos^{-1} \\left(\\frac{1 -x^2}{1 + x^2} \\right)
\n\\end{eqnarray}<\/p>\n
\u6b21\u306b\uff0c<\/p>\n
\\begin{eqnarray}
\n\\sin(2\\theta) &=& 2 \\sin\\theta \\cos\\theta \\\\
\n&=& 2 \\tan\\theta \\cos^2\\theta\\\\
\n&=& \\frac{ 2\\tan\\theta}{1 + \\tan^2\\theta} \\\\
\n\\therefore \\ \\ 2 \\theta &=& \\sin^{-1} \\left(\\frac{ 2\\tan\\theta}{1 + \\tan^2\\theta} \\right)
\n\\end{eqnarray}<\/p>\n
\u4e0a\u5f0f\u3067 $\\theta \\equiv \\tan^{-1} x$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n2 \\tan^{-1} x &=& \\sin^{-1} \\left(\\frac{ 2x}{1 + x^2} \\right)
\n\\end{eqnarray}<\/p>\n
\u307e\u305f\uff0c<\/p>\n
\\begin{eqnarray}
\n\\tan (2\\theta) &=& \\frac{2 \\tan\\theta}{1 -\\tan^2\\theta} \\\\
\n\\therefore \\ \\ 2 \\theta &=& \\tan^{-1} \\left(\\frac{2 \\tan\\theta}{1 -\\tan^2\\theta} \\right)
\n\\end{eqnarray}<\/p>\n
\u4e0a\u5f0f\u3067 $\\theta \\equiv \\tan^{-1} x$ \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n2 \\tan^{-1} x &=& \\tan^{-1} \\left(\\frac{2 x}{1 -x^2} \\right)
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\u306e\u307e\u3068\u3081\uff1a<\/p>\n
\\begin{eqnarray}
\n2 \\tan^{-1} x &=& \\cos^{-1} \\left(\\frac{1 -x^2}{1 + x^2} \\right)\\\\
\n&=& \\sin^{-1} \\left(\\frac{ 2x}{1 + x^2} \\right) \\\\
\n&=& \\tan^{-1} \\left(\\frac{2 x}{1 -x^2} \\right)
\n\\end{eqnarray}<\/p>\n
\u9006\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n
3\u3064\u307e\u3068\u3081\u3066\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3068…<\/p>\n
\"\"<\/p>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
\u9006\u4e09\u89d2\u95a2\u6570 \\(\\sin^{-1} x , \\ \\cos^{-1} x,\\\u00a0 \\tan^{-1} x \\) \u306e\u5fae\u5206\u3002 $$\\frac{d}{dx} \\sin^{-1} x = \\frac{1}{\\sqrt{1-x^2}}, \\quad \\frac{d}{dx} \\cos^{ -1} x = -\\frac{1}{\\sqrt{1-x^2}}, \\quad \\frac{d}{dx} \\tan^{-1} x = \\frac{1}{1 + x^2}$$<\/p>\n
\u3092\u793a\u3059\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":7,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2125","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\/2125","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=2125"}],"version-history":[{"count":34,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2125\/revisions"}],"predecessor-version":[{"id":2295,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2125\/revisions\/2295"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}