{"id":3976,"date":"2019-06-29T02:59:16","date_gmt":"2019-06-28T17:59:16","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=3976"},"modified":"2025-11-19T14:48:49","modified_gmt":"2025-11-19T05:48:49","slug":"calculus-analytic-definition-of-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-analytic-definition-of-trigonometric-functions\/","title":{"rendered":"\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud55c \uc0bc\uac01\ud568\uc218\uc758 \uc815\uc758"},"content":{"rendered":"<p>\uc911\ud559\uad50\uc640 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \uc0bc\uac01\ud568\uc218\ub294 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc0bc\uac01\ud568\uc218\ub97c \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ud558\uba74 \uc5ec\ub7ec \ubaa8\ub85c \ubd88\ud3b8\ud55c \uc810\uc774 \ub9ce\ub2e4. \uba3c\uc800 \uc0bc\uac01\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc740 \uac01(angle)\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub978 \ud568\uc218\uc640 \ud569\uc131\ud560 \ub54c \uac01\uc774 \uc218\uc640 \ud63c\uc6a9\ub418\uc5b4\uc57c \ud55c\ub2e4. \ub610\ud55c \ucef4\ud4e8\ud130 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc0bc\uac01\ud568\uc218\uc758 \uac12\uc744 \uacc4\uc0b0\ud560 \ub54c \uae30\ud558\ud559\uc801\uc778 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\uae30\uac00 \uc5b4\ub835\ub2e4. \uac8c\ub2e4\uac00 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ub41c \uc0bc\uac01\ud568\uc218\ub294 \uadf8 \uc815\uc758\uc5ed\uc744 \ubcf5\uc18c\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud558\uae30\ub3c4 \uc5b4\ub835\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \ubd88\ud3b8\ud568 \ub54c\ubb38\uc5d0 \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud574\uc57c \ud55c\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\ub97c \uc815\uc758\ud558\uace0, \uadf8\ub807\uac8c \uc815\uc758\ub41c \uc0bc\uac01\ud568\uc218\uac00 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub428\uc744 \ubcf4\uc778\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#analyticdefinition\">\uc0bc\uac01\ud568\uc218\uc758 \ud574\uc11d\uc801 \uc815\uc758<\/a><\/li>\n<li><a href=\"#uniqueness\">\uc0bc\uac01\ud568\uc218\uc758 \uc720\uc77c\uc131<\/a><\/li>\n<li><a href=\"#property\">\uba87 \uac00\uc9c0 \uc131\uc9c8<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218 (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_series\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ud14c\uc77c\ub7ec \uae09\uc218 (<a href=\"\/blog\/articles\/calculus-taylor-series-and-maclaurin-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84\ubc95 (<a href=\"\/blog\/articles\/calculus-derivatives-of-trigonometric-functions\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"analyticdefinition\"><\/a><\/p>\n<h3>\uc0bc\uac01\ud568\uc218\uc758 \ud574\uc11d\uc801 \uc815\uc758<\/h3>\n<p>\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \uac83\uc744 <span class=\"defined\">\ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4<\/span>\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc0bc\uac01\ud568\uc218 \uc911 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc0ac\uc778, \ucf54\uc0ac\uc778\uc758 \ud574\uc11d\uc801 \uc815\uc758)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\sin x &#038;:= \\sum _ {n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} ,\\tag{1}\\\\[4pt]<br \/>\n\\cos x &#038;:= \\sum _ {n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} .\\tag{2}<br \/>\n\\end{align}\\]<br \/>\n\ub450 \ud568\uc218\ub97c \uc21c\uc11c\ub300\ub85c \uac01\uac01 <span class=\"defined\">\uc0ac\uc778<\/span>, <span class=\"defined\">\ucf54\uc0ac\uc778<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<p>\ube44 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec (1)\uacfc (2)\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{n\\to\\infty} \\frac{(2n+1)!}{(2n+3)!} =0, \\\\[4pt]<br \/>\n\\lim_{n\\to\\infty} \\frac{(2n)!}{(2n+2)!} =0\\]<br \/>\n\uc774\ubbc0\ub85c (1)\uacfc (2)\uc758 \uc218\ub834\ubc18\uc9c0\ub984\uc740 \ubaa8\ub450 \ubb34\ud55c\ub300\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec (1)\uacfc (2)\ub294 \uc798 \uc815\uc758\ub41c \ud568\uc218\uc774\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"uniqueness\"><\/a><\/p>\n<h3>\uc0bc\uac01\ud568\uc218\uc758 \uc720\uc77c\uc131<\/h3>\n<p>\uc774\uc81c \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc774 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ud55c \uc0bc\uac01\ud568\uc218\uc758 \uc131\uc9c8\uc744 \ubaa8\ub450 \uac00\uc9c0\uace0 \uc788\uc73c\uba70, \ub610\ud55c \uadf8\ub7ec\ud55c \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\ub294 \ud568\uc218\uac00 \uac01\uac01 \uc720\uc77c\ud568\uc744 \ubcf4\uc784\uc73c\ub85c\uc368 \uc774 \uc815\uc758\ub97c \uc815\ub2f9\ud654\ud558\uc790.<\/p>\n<p>\uba3c\uc800 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uba85\ubc31\ud788<br \/>\n\\[\\sin 0=0 ,\\,\\,\\cos 0=1 \\tag{3}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\sin (-x) &#038;= &#8211; \\sin x , \\tag{4}\\\\[6pt]<br \/>\n\\cos (-x) &#038;= \\cos x \\tag{5}<br \/>\n\\end{align}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc0ac\uc778\uc758 \ub3c4\ud568\uc218\uc640 \ucf54\uc0ac\uc778\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\sin x<br \/>\n&#038;= \\frac{d}{dx} \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{d}{dx} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{(-1)^n (2n+1)x^{2n}}{(2n+1)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} = \\cos x,\\\\[6pt]<br \/>\n\\end{align}\\]<\/p>\n<p>\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\cos x<br \/>\n&#038;= \\frac{d}{dx} \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=0}^{\\infty} \\frac{d}{dx} \\frac{(-1)^n x^{2n}}{(2n)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=1}^{\\infty} \\frac{(-1)^n 2n x^{2(n-1)}}{(2n)!} \\\\[4pt]<br \/>\n&#038;= \\sum_{n=1}^{\\infty} \\frac{(-1)^n x^{2n-1}}{(2n-1)!} \\\\[4pt]<br \/>\n&#038;= -\\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{(2n+1)!} = &#8211; \\sin x<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\sin x &#038;= \\cos x ,\\tag{6}\\\\[4pt]<br \/>\n\\frac{d}{dx} \\cos x &#038;= &#8211; \\sin x \\tag{7}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \\(\\phi (x) = \\sin^2 x + \\cos^2 x\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\frac{d}{dx} \\phi (x) = 2 \\sin x \\,\\cos x + 2 \\cos x (-\\sin x )=0\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\phi\\)\ub294 \uc0c1\uc218\ud568\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\phi (0) = 1\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sin^2 x + \\cos^2 x = 1\\tag{8}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ub610\ud55c \uc0ac\uc778\uc758 \uc774\uacc4\ub3c4\ud568\uc218\uc640 \ucf54\uc0ac\uc778\uc758 \uc774\uacc4\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\frac{d^2}{(dx)^2} \\sin x = &#8211; \\sin x ,\\tag{9}\\\\[4pt]<br \/>\n\\frac{d^2}{(dx)^2} \\cos x = &#8211; \\cos x \\tag{10}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc774\ub85c\uc368 (3)~(10)\uc744 \ud1b5\ud574 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc774 \uac01\uac01 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ub41c \uc0bc\uac01\ud568\uc218\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\uc74c\uc744 \uc54c \uc218 \uc788\ub2e4. \uc774\uc81c \uc774\uc640 \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4 \ud568\uc218\uac00 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc73c\ub85c\uc11c \uac01\uac01 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub428\uc744 \ubcf4\uc774\uc790.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc0ac\uc778 \ud568\uc218\uc640 \ucf54\uc0ac\uc778 \ud568\uc218\uc758 \uc720\uc77c\uc131)<\/span><\/p>\n<p>\ud568\uc218 \\(f : \\mathbb{R} \\to \\mathbb{R}\\)\uac00 \ub450 \ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\[f &#8216; &#8216; = &#8211; f ,\\,\\,f(0) =0 ,\\,\\,f &#8216; (0) = b\\tag{11}\\]\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) = b\\sin x\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \ud568\uc218 \\(g : \\mathbb{R} \\to \\mathbb{R}\\)\uac00 \ub450 \ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\[g &#8216; &#8216; = -g ,\\,\\, g(0) = a ,\\,\\, g &#8216; (0) = b\\tag{12}\\]\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) = a\\cos x + b\\sin x\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\n\\eta (x) &#038;= f(x) \\sin x + f &#8216; (x) \\cos x ,\\\\[7pt]<br \/>\n\\xi (x) &#038;= f(x) \\cos x &#8211; f &#8216; (x) \\sin x<br \/>\n\\end{align}\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\eta &#8216; (x) =0 ,\\) \\(\\xi &#8216; (x) =0\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nb&#038;= f(x) \\sin x + f &#8216; (x) \\cos x ,\\\\[7pt]<br \/>\n0&#038;= f(x) \\cos x &#8211; f &#8216; (x) \\sin x .<br \/>\n\\end{align}\\]<br \/>\n\ub530\ub77c\uc11c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nb\\sin x &#038;= \\sin x ( f(x) \\sin x + f &#8216; (x) \\cos x ) \\\\[7pt]<br \/>\n&#038;= f(x) (1- \\cos^2 x ) + f &#8216; (x) \\sin x \\, \\cos x \\\\[7pt]<br \/>\n&#038;= f(x) &#8211; \\cos x ( f(x) \\cos x &#8211; f &#8216; (x) \\sin x ) = f(x).<br \/>\n\\end{align}\\]<br \/>\n\ub2e4\uc74c\uc73c\ub85c \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\phi (x) = g(x) &#8211; a \\cos x\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(\\phi &#8216; &#8216; = &#8211; \\phi ,\\) \\(\\phi (0) = 0,\\) \\(\\phi &#8216; (0) = b\\)\uc774\ubbc0\ub85c \\(\\phi\\)\ub294 (11)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc99d\uba85\uc758 \uc55e\ubd80\ubd84\uc5d0\uc11c \uc5bb\uc740 \uacb0\uacfc\uc5d0 \uc758\ud558\uc5ec \\[\\phi (x) = b\\sin x\\]\ub97c \uc5bb\ub294\ub2e4. \uc989 \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[g(x) = a \\cos x + b \\sin x .\\tag*{\\(\\blacksquare\\)}\\]\n<\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\uc5d0\uc11c \\(a=1,\\) \\(b=1\\)\uc744 \ub300\uc785\ud558\uba74 \\(f &#8216; &#8216; = -f ,\\) \\(f(0) = 0 ,\\) \\(f &#8216; (0) = 1\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(f\\)\ub294<br \/>\n\\[f(x) = \\sin x\\]<br \/>\n\ub85c\uc11c \uc720\uc77c\ud558\uba70, \\(g &#8216; &#8216; = &#8211; g ,\\) \\(g(0) =1,\\) \\(g &#8216; (0) = 0\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(g\\)\ub294<br \/>\n\\[g(x) = \\cos x\\]<br \/>\n\ub85c\uc11c \uc720\uc77c\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"theorem\">\ub530\ub984\uc815\ub9ac 1.<\/span><br \/>\n\ud574\uc11d\uc801\uc73c\ub85c \uc815\uc758\ud55c \uc0ac\uc778, \ucf54\uc0ac\uc778\uc740 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ud55c \uc0ac\uc778, \ucf54\uc0ac\uc778\uacfc \uc644\uc804\ud788 \uc77c\uce58\ud55c\ub2e4.<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"property\"><\/a><\/p>\n<h3>\uba87 \uac00\uc9c0 \uc131\uc9c8<\/h3>\n<p>\uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc758 \uc131\uc9c8\uc744 \uba87 \uac00\uc9c0 \ub354 \uc0b4\ud3b4\ubcf4\uc790. \ub2e4\uc74c \uc815\ub9ac\ub294 (1)\uacfc (2)\ub97c \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\ub294 \uc608\ub97c \ubcf4\uc5ec\uc900\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uc815\ub9ac)<\/span><\/p>\n<p>\uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\sin (x+y) = \\sin x\\, \\cos y + \\cos x \\, \\sin y\\)<\/li>\n<li>\\(\\cos (x+y) = \\cos x\\, \\cos y &#8211; \\sin x \\, \\sin y\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\n\uc2e4\uc218 \\(y\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. (\\(y\\)\ub97c \uace0\uc815\uc2dc\ud0a4\uace0 \\(x\\)\ub9cc \ubcc0\uc218\ub85c \ubcf4\uace0 \uc99d\uba85\ud558\ub294 \uae30\uc220\uc744 \uc0ac\uc6a9\ud558\uaca0\ub2e4.)<br \/>\n\\[f(x) = \\sin (x+y)\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uba74<br \/>\n\\[f &#8216; &#8216; = &#8211; f ,\\,\\, f(0) = \\sin y ,\\,\\, f &#8216; (0) = \\cos y\\]<br \/>\n\uc774\ubbc0\ub85c \\(f\\)\ub294 \uc815\ub9ac 1\uc758 \uc870\uac74 (12)\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\sin (x+y) = f(x) = \\sin y \\,\\cos x + \\sin x \\,\\cos y\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub610\ud55c \uc774 \uc2dd\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74<br \/>\n\\[\\cos (x+y) = f &#8216; (x) = \\cos x \\,\\cos y &#8211; \\sin x\\, \\sin y\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ucf54\uc0ac\uc778\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\cos 2 = 1 &#8211; \\frac{2^2}{2!} + \\frac{2^4}{4!} &#8211; \\frac{2^6}{6!} + &#8211; \\cdots .\\]<br \/>\n\ub610\ud55c \uad50\ub300\uae09\uc218\uc758 \uc624\ucc28\uc758 \ud55c\uacc4 \uacf5\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\left\\lvert \\cos 2 &#8211; \\left( 1 &#8211; \\frac{2^2}{2!} \\right) \\right\\rvert \\le \\frac{2^4}{4!} = \\frac{2}{3}.\\]<br \/>\n\ub530\ub77c\uc11c \\(\\cos 2 < 0\\)\uc774\ub2e4. \ub610\ud55c \\(\\cos 0 = 1\\)\uc774\uace0 \ucf54\uc0ac\uc778\uc740 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(\\cos x=0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(x\\)\uac00 \\(0\\)\uacfc \\(2\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ud55c \\(x\\) \uc911 \uac00\uc7a5 \uc791\uc740 \uac12\uc758 \ub450 \ubc30\ub97c \\(\\pi\\)\ub85c \ub098\ud0c0\ub0b4\uba70 <span class=\"defined\">\uc6d0\uc8fc\uc728<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\uc6d0\uc8fc\uc728)<\/span><\/p>\n<p>\\[\\pi = 2 \\min \\left\\{ x \\,\\vert\\, \\cos x =0 ,\\,\\, 0 \\le x \\le 2 \\right\\}.\\tag{13}\\]\n<\/p><\/div>\n<p>\ucf54\uc0ac\uc778\uc774 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(\\cos 0 \\ne 0\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\{x\\,\\vert\\, \\cos x = 0 ,\\,\\, 0 \\le x \\le 2 \\right\\}\\]<br \/>\n\uc740 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4. \ub530\ub77c\uc11c \uc774 \uc9d1\ud569\uc758 \ucd5c\uc19f\uac12\uc740 \uc798 \uc815\uc758\ub418\uba70, \uadf8 \uac12\uc740 \uc591\uc218\uc774\ub2e4. \uc989 \\(\\pi > 0\\)\uc774\ub2e4.<\/p>\n<p>\uc6d0\uc8fc\uc728\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\cos \\frac{\\pi}{2} = 0\\)\uc774\uace0 \\[\\cos^2 \\frac{\\pi}{2} + \\sin^2 \\frac{\\pi}{2} = 1\\]\uc774\ubbc0\ub85c \\(\\sin \\frac{\\pi}{2}\\)\uc758 \uac12\uc740 \\(1\\)\uc774\uac70\ub098 \\(-1\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc5f4\ub9b0 \uad6c\uac04 \\(\\left(0,\\, \\frac{\\pi}{2} \\right)\\)\uc5d0\uc11c<br \/>\n\\[\\frac{d}{dx}\\sin x = \\cos x > 0\\]<br \/>\n\uc774\ubbc0\ub85c \uc0ac\uc778\uc740 \uc774 \uad6c\uac04\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\sin \\frac{\\pi}{2} = 1 \\tag{14}\\]<br \/>\n\uc774\ub2e4. \uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\cos \\pi = \\cos \\left( \\frac{\\pi}{2} + \\frac{\\pi}{2} \\right) = -\\sin \\frac{\\pi}{2} \\cdot \\sin \\frac{\\pi}{2} = -1\\]<br \/>\n\uc774\uba70, \uac19\uc740 \ubc29\ubc95\uc73c\ub85c<br \/>\n\\[\\sin \\pi =0 ,\\,\\, \\cos 2 \\pi = 1 ,\\,\\, \\sin 2 \\pi = 0\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ub2e4\uc2dc \uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\begin{gather}<br \/>\n\\cos(x+ 2\\pi ) = \\cos x ,\\\\[7pt]<br \/>\n\\sin(x+ 2\\pi) = \\sin x ,\\\\[5pt]<br \/>\n\\cos \\left( x + \\frac{\\pi}{2} \\right) = -\\sin x ,\\\\[5pt]<br \/>\n\\sin \\left( x + \\frac{\\pi}{2} \\right) = \\cos x<br \/>\n\\end{gather}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc0bc\uac01\ud568\uc218\uc758 \uc8fc\uae30\uc131)<\/span><\/p>\n<p>\uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc740 \uac01\uac01 \uc8fc\uae30\uac00 \\(2\\pi\\)\uc778 \uc8fc\uae30\ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc744 \uacb0\ud569\ud558\uc5ec \ub2e4\ub978 \ub124 \uac1c\uc758 \uc0bc\uac01\ud568\uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\uc73c\ubbc0\ub85c \ub2e4\ub978 \uc0bc\uac01\ud568\uc218\uc758 \uc131\uc9c8\uc740 \ub2e4\ub8e8\uc9c0 \uc54a\uaca0\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### --><\/p>\n<h3><\/h3>\n<p><!--\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>\n...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 n.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"remark\">\n\n\n<p><span class=\"remark\">\ucc38\uace0.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n<p><!-- . --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc911\ud559\uad50\uc640 \uace0\ub4f1\ud559\uad50 \uacfc\uc815\uc5d0\uc11c \uc0bc\uac01\ud568\uc218\ub294 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc0bc\uac01\ud568\uc218\ub97c \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ud558\uba74 \uc5ec\ub7ec \ubaa8\ub85c \ubd88\ud3b8\ud55c \uc810\uc774 \ub9ce\ub2e4. \uba3c\uc800 \uc0bc\uac01\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc740 \uac01(angle)\uc758 \uc9d1\ud569\uc774\ubbc0\ub85c \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub978 \ud568\uc218\uc640 \ud569\uc131\ud560 \ub54c \uac01\uc774 \uc218\uc640 \ud63c\uc6a9\ub418\uc5b4\uc57c \ud55c\ub2e4. \ub610\ud55c \ucef4\ud4e8\ud130 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc0bc\uac01\ud568\uc218\uc758 \uac12\uc744 \uacc4\uc0b0\ud560 \ub54c \uae30\ud558\ud559\uc801\uc778 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\uae30\uac00 \uc5b4\ub835\ub2e4. \uac8c\ub2e4\uac00 \uae30\ud558\ud559\uc801\uc73c\ub85c \uc815\uc758\ub41c \uc0bc\uac01\ud568\uc218\ub294 \uadf8 \uc815\uc758\uc5ed\uc744 \ubcf5\uc18c\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud558\uae30\ub3c4 \uc5b4\ub835\ub2e4. \uc774\uc640 \uac19\uc740 \ubd88\ud3b8\ud568 \ub54c\ubb38\uc5d0 \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud574\uc57c \ud55c\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\ub97c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[47,61],"tags":[353,354,357,150,319,358,356,355],"class_list":["post-3976","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","category-differential-equation","tag-353","tag-354","tag-357","tag-150","tag-319","tag-358","tag-356","tag-355"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=3976"}],"version-history":[{"count":36,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3976\/revisions"}],"predecessor-version":[{"id":9620,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/3976\/revisions\/9620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=3976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=3976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=3976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}