{"id":8721,"date":"2023-05-08T10:11:05","date_gmt":"2023-05-08T01:11:05","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=8721"},"modified":"2023-05-30T10:01:05","modified_gmt":"2023-05-30T01:01:05","slug":"solution-complex-plane-and-polar-form","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/solution-complex-plane-and-polar-form\/","title":{"rendered":"\ubcf5\uc18c\ud3c9\uba74\uacfc \ubcf5\uc18c\uc218\uc758 \uadf9\ud615\uc2dd (\uc720\uc81c \ud574\uc124)"},"content":{"rendered":"<div class=\"box\">\n<p>\uc774 \uae00\uc740 \uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 2023\ud559\ub144\ub3c4 1\ud559\uae30 \ud574\uc11d\uae30\ud558 \uc218\uc5c5 \uad50\uc7ac 4\ub2e8\uc6d0 \u300c\ubcf5\uc18c\ud3c9\uba74\uacfc \ubcf5\uc18c\uc218\uc758 \uadf9\ud615\uc2dd\u300d \uc720\uc81c \ud574\uc124\uc785\ub2c8\ub2e4. \ub208\uc73c\ub85c \uc77d\uae30\ub9cc \ud558\uc9c0 \ub9d0\uace0 \uaf2d \uc190\uc73c\ub85c \uc9c1\uc811 \ud480\uc5b4\ubcf4\uc138\uc694\u2661\n<\/p>\n<\/div>\n<div style=\"visibility: hidden; display: none;\">\n\\[<br \/>\n\\newcommand{\\complexI}{\\boldsymbol{i}}<br \/>\n\\newcommand{\\rpart}{\\operatorname{Re}}<br \/>\n\\newcommand{\\ipart}{\\operatorname{Im}}<br \/>\n\\newcommand{\\Arg}{\\operatorname{Arg}}<br \/>\n\\]\n<\/div>\n<p><!-- ########## 4.2 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.1<\/span><br \/>\n\\(z\\)\ub97c \ubcf5\uc18c\uc218\ub77c \ud560 \ub54c, \ubcf5\uc18c\ud3c9\uba74 \uc704\uc5d0\uc11c \ub2e4\uc74c \ub450 \ubcf5\uc18c\uc218 \uc0ac\uc774\uc758 \uc704\uce58\uad00\uacc4\ub97c \uc870\uc0ac\ud558\uc2dc\uc624.\n<\/p>\n<p>(1) \\(z\\)\uc640 \\(-z.\\)<br \/>\n(2) \\(z\\)\uc640 \\(\\overline{z}.\\)<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(-z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc740 \\(z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \uc6d0\uc810\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc774\ub3d9\ud55c \uc810\uc774\ub2e4.\n<\/p>\n<p>(2) \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(\\overline{z}\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc740 \\(z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc774\ub3d9\ud55c \uc810\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.4 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.4<\/span><br \/>\n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c, \ubcf5\uc18c\uc218 \\((2+\\complexI )^n &#8211; (2- \\complexI )^n\\)\uc758 \uc2e4\uc218\ubd80\ubd84\uc774 \\(0\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. [\uc8fc\uc758: \uad50\uc7ac\uc758 \ubb38\uc81c\uc640 \ub2e4\ub984.]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(z = 2+\\complexI\\)\ub77c\uace0 \ud558\uba74 \\(\\overline{z} = 2-\\complexI\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\n\\rpart \\left(z^n &#8211; \\left( \\overline{z} \\right)^n \\right)<br \/>\n&#038;= \\rpart \\left(z^n &#8211; \\overline{z^n} \\right) \\\\[6pt]<br \/>\n&#038;= \\frac{1}{2} \\left\\{ \\left(z^n &#8211; \\overline{z^n}\\right) + \\overline{\\left(z^n &#8211; \\overline{z^n}\\right)} \\right\\} \\\\[6pt]<br \/>\n&#038;= \\frac{1}{2} \\left\\{ z^n &#8211; \\overline{z^n} + \\overline{z^n} &#8211; z^n \\right\\} \\\\[6pt]<br \/>\n&#038;=0.<br \/>\n\\end{aligned}\\]\n<\/p>\n<p><span class=\"proof\">\ucc38\uace0.<\/span> \\(p\\)\uc640 \\(q\\)\uac00 \uc815\uc218\uc774\uace0 \\(z = p + q \\complexI\\)\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(z^n\\)\uc774 \uc21c\ud5c8\uc218\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \uc790\uc5f0\uc218 \\(n\\)\uc774 \uc874\uc7ac\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc810 \\((p,\\,\\,q)\\)\uac00 \ub124 \uc9c1\uc120 \\(x=0 ,\\) \\(y=0 ,\\) \\(y=x ,\\) \\(y=-x\\) \uc911 \ud558\ub098 \uc774\uc0c1\uc758 \uc704\uc5d0 \ub193\uc5ec \uc788\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ########## 4.6 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.6<\/span><br \/>\n\\(z\\)\uac00 \uc2e4\uc218\uac00 \uc544\ub2cc \ubcf5\uc18c\uc218\uc77c \ub54c, \\(z+\\frac{1}{z}\\)\uc774 \uc2e4\uc218\uac00 \ub418\uae30 \uc704\ud55c \uc870\uac74\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\uc8fc\uc5b4\uc9c4 \ubcf5\uc18c\uc218\uac00 \uc2e4\uc218\uac00 \ub418\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uadf8 \ubcf5\uc18c\uc218\uc758 \ud5c8\uc218\ubd80\ubd84\uc774 \\(0\\)\uc778 \uac83\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\ipart\\left( z+\\frac{1}{z} \\right) =0\\]<br \/>\n\uc774 \ub418\uae30 \uc704\ud55c \uc870\uac74\uc744 \ucc3e\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\n\\ipart\\left( z+\\frac{1}{z} \\right)<br \/>\n&#038;= \\ipart(z) + \\ipart \\left( \\frac{1}{z} \\right) \\\\[6pt]<br \/>\n&#038;= \\ipart(z) &#8211; \\frac{1}{\\lvert z \\rvert^2} \\ipart(z) \\\\[6pt]<br \/>\n&#038;= \\ipart(z) \\left( 1- \\frac{1}{\\lvert z \\rvert^2} \\right)<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\uba70, \\(\\ipart(z) \\ne 0\\)\uc774\ubbc0\ub85c, \uc704 \uc2dd\uc758 \uac12\uc774 \\(0\\)\uc774 \ub418\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[1-\\frac{1}{\\lvert z \\rvert^2 } =0\\]<br \/>\n\uc774\ub2e4. \uc989 \ubb38\uc81c\uc758 \ubcf5\uc18c\uc218\uac00 \uc2e4\uc218\uac00 \ub418\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[\\lvert z \\rvert = 1\\]<br \/>\n\uc774\ub2e4. \uc774 \uc870\uac74\uc744 \uae30\ud558\ud559\uc801\uc73c\ub85c \ud45c\ud604\ud558\uba74, \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc774 \uc911\uc2ec\uc774 \uc6d0\uc810\uc778 \ub2e8\uc704\uc6d0 \uc704\uc5d0 \ub193\uc5ec \uc788\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ########## 4.7 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.7<\/span><br \/>\n\\(\\alpha\\)\uc640 \\(\\beta\\)\uac00 \ubcf5\uc18c\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc784\uc758\uc758 \ubcf5\uc18c\uc218 \\(z\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha z + \\beta \\overline{z}\\)\uac00 \uc2e4\uc218\uc774\uba74, \\(\\alpha = \\overline{\\beta}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(z=1\\)\uc77c \ub54c<br \/>\n\\[\\begin{aligned}<br \/>\n0<br \/>\n&#038;= \\ipart\\left( \\alpha z + \\beta \\overline{z} \\right) \\\\[6pt]<br \/>\n&#038;= \\ipart\\left( \\alpha + \\beta \\right) \\\\[6pt]<br \/>\n&#038;= -\\frac{\\complexI}{2} \\left\\{ (\\alpha + \\beta ) &#8211; \\overline{(\\alpha + \\beta)} \\right\\}<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[ \\alpha + \\beta = \\overline{\\alpha} + \\overline{\\beta} \\tag{a}\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c \\(z=\\complexI\\)\uc77c \ub54c<br \/>\n\\[\\begin{aligned}<br \/>\n0<br \/>\n&#038;= \\ipart\\left( \\alpha z + \\beta \\overline{z} \\right) \\\\[6pt]<br \/>\n&#038;= \\ipart\\left( \\alpha \\complexI &#8211; \\beta \\complexI \\right) \\\\[6pt]<br \/>\n&#038;= -\\frac{\\complexI}{2} \\left\\{ (\\alpha \\complexI &#8211; \\beta \\complexI ) &#8211; \\overline{(\\alpha \\complexI &#8211; \\beta \\complexI)} \\right\\} \\\\[6pt]<br \/>\n&#038;= -\\frac{\\complexI}{2} \\left\\{ (\\alpha \\complexI &#8211; \\beta \\complexI ) + \\complexI \\left( \\overline{\\alpha} &#8211; \\overline{\\beta} \\right) \\right\\}<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[ \\alpha &#8211; \\beta = &#8211; \\overline{\\alpha} + \\overline{\\beta} \\tag{b}\\]<br \/>\n\uc774\ub2e4. (a)\uc640 (b)\ub97c \ubcc0\ub9c8\ub2e4 \ub354\ud558\uc5ec \ud480\uba74<br \/>\n\\[2\\alpha = 2\\overline{\\beta}\\]<br \/>\n\uc989 \\(\\alpha = \\overline{\\beta}\\)\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<p><span class=\"proof\">\ub2e4\ub978 \ubc29\ubc95.<\/span><br \/>\n\\(\\alpha = a+b\\complexI ,\\)<br \/>\n\\(\\beta = c+d\\complexI ,\\)<br \/>\n\\(z = x+y\\complexI \\)\ub77c\uace0 \ud558\uc790. (\ub2e8, \\(a,\\) \\(b,\\) \\(c,\\) \\(d,\\) \\(x,\\) \\(y\\)\ub294 \ubaa8\ub450 \uc2e4\uc218\uc774\ub2e4.) \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc2dd\uc758 \uac12\uc774 \uc2e4\uc218\uc5ec\uc57c \ud55c\ub2e4.<br \/>\n\\[ (a+b\\complexI ) (x+y\\complexI ) + (c+d\\complexI )(x-y\\complexI ) \\]<br \/>\n\uc704 \uc2dd\uc744 \ud480\uba74<br \/>\n\\[(ax-by+cx+dy) + \\complexI (ay+bx+dx-cy )\\]<br \/>\n\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \uc2e4\uc218 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[ay+bx+dx-cy = 0\\]<br \/>\n\uc989<br \/>\n\\[(b+d)x + (a-c)y = 0\\]<br \/>\n\uc774 \uc131\ub9bd\ud574\uc57c \ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\left\\{<br \/>\n\\begin{array}{l}<br \/>\nb+d = 0 \\\\[6pt]<br \/>\na-c = 0<br \/>\n\\end{array}<br \/>\n\\right.\\]<br \/>\n\uc774 \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \ud480\uba74 \\(b=-d,\\) \\(a=c\\)\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[ \\alpha = a+d\\complexI = c-d\\complexI = \\overline{\\beta} \\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ########## 4.9 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.9<\/span><br \/>\n\ub450 \ubcf5\uc18c\uc218 \\(z = 2 + 7 \\complexI ,\\) \\(w = (3+4 \\complexI ) z\\)\uc5d0 \ub300\ud558\uc5ec, \ub2e4\uc74c \ubcf5\uc18c\uc218\uc758 \uc808\ub313\uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<p>(1) \\(w\\)<br \/>\n(2) \\(z^3 w\\)<br \/>\n(3) \\(\\frac{1}{z}\\)<br \/>\n(4) \\(\\frac{1}{zw}\\)<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \\(\\lvert w \\rvert = \\sqrt{3^2 + 4^2} \\times \\sqrt{2^2 + 7^2} = \\sqrt{9+16} \\sqrt{4+49} = \\sqrt{25} \\sqrt{53} = 5 \\sqrt{53}.\\)<\/p>\n<p>(2) \\(\\lvert z \\rvert = \\sqrt{53}\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\lvert z^3 w \\right\\rvert = \\lvert z \\rvert^3 \\times \\lvert w \\rvert = \\left( \\sqrt{53} \\right)^3 \\times 5\\sqrt{53} = 14045.\\]<\/p>\n<p>(3) \\[\\left\\lvert \\frac{1}{z} \\right\\rvert = \\frac{1}{\\lvert z \\rvert} = \\frac{1}{\\sqrt{53}}.\\]<\/p>\n<p>(4) \\[\\left\\lvert \\frac{1}{zw} \\right\\rvert = \\frac{1}{\\lvert z \\rvert \\lvert w \\rvert} = \\frac{1}{<br \/>\n5\\sqrt{53} \\times \\sqrt{53} } = \\frac{1}{265}.\\]<\/p>\n<\/div>\n<p><!-- ########## 4.10 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.10<\/span><br \/>\n\\(z\\)\uac00 \ubcf5\uc18c\uc218\uc77c \ub54c, \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n\\[\\sqrt{2} \\lvert z \\rvert \\ge \\lvert \\rpart (z) \\rvert + \\lvert \\ipart (z) \\rvert .\\]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(z= a+b\\complexI ,\\) \\(a\\in\\mathbb{R} ,\\) \\(b\\in\\mathbb{R}\\)\uc774\ub77c\uace0 \ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\n\\, &#038; \\, &#038; \\sqrt{2} \\lvert z \\rvert &#038;\\ge \\lvert \\rpart (z) \\rvert + \\lvert \\ipart (z) \\rvert  \\\\[6pt]<br \/>\n\\, &#038; \\Longleftrightarrow &#038;\\quad \\sqrt{2} \\sqrt{a^2 + b^2} &#038;\\ge \\lvert a \\rvert + \\lvert b \\rvert \\\\[6pt]<br \/>\n\\, &#038; \\Longleftrightarrow &#038;\\quad \\left( \\sqrt{2} \\sqrt{a^2 + b^2} \\right)^2 &#038;\\ge \\left( \\lvert a \\rvert + \\lvert b \\rvert \\right)^2 \\\\[6pt]<br \/>\n\\, &#038; \\Longleftrightarrow &#038;\\quad 2 \\left( a^2 + b^2 \\right) &#038;\\ge a^2 + 2\\lvert a \\rvert \\lvert b \\rvert + b^2 \\\\[6pt]<br \/>\n\\, &#038; \\Longleftrightarrow &#038;\\quad \\lvert a \\rvert^2 &#8211; 2 \\lvert a \\rvert \\lvert b \\rvert + \\lvert b \\rvert^2 &#038;\\ge 0 \\\\[6pt]<br \/>\n\\, &#038; \\Longleftrightarrow &#038;\\quad \\left( \\lvert a \\rvert + \\lvert b \\rvert \\right)^2 &#038;\\ge 0 .<br \/>\n\\end{aligned}\\]<br \/>\n\uc704 \uad00\uacc4\uc2dd\uc5d0\uc11c \ub9c8\uc9c0\ub9c9 \ubd80\ub4f1\uc2dd\uc774 \ucc38\uc774\ubbc0\ub85c, \ucc98\uc74c \ubd80\ub4f1\uc2dd\ub3c4 \ucc38\uc774\ub2e4.\n<\/p>\n<p><span class=\"proof\">\ub2e4\ub978 \ubc29\ubc95.<\/span> \\(z= a+b\\complexI ,\\) \\(a\\in\\mathbb{R} ,\\) \\(b\\in\\mathbb{R}\\)\uc774\ub77c\uace0 \ud558\uc790. \\(z\\ne 0,\\) \\(a \\ge 0, \\) \\(b\\ge 0\\)\uc778 \uacbd\uc6b0\ub9cc \uc0dd\uac01\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<p>\\(r = \\lvert z \\rvert ,\\) \\(z = r (\\cos\\theta + \\complexI \\sin\\theta )\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\begin{aligned}<br \/>\n\\lvert \\rpart (z) \\rvert + \\lvert \\ipart (z) \\rvert<br \/>\n&#038;= a+b \\\\[6pt]<br \/>\n&#038;= r\\cos\\theta + r\\sin\\theta \\\\[6pt]<br \/>\n&#038;= r ( \\cos\\theta + \\sin \\theta ) \\\\[6pt]<br \/>\n&#038;= \\sqrt{2} r \\left( \\frac{1}{\\sqrt{2}} \\cos\\theta + \\frac{1}{\\sqrt{2}} \\sin\\theta \\right) \\\\[6pt]<br \/>\n&#038;= \\sqrt{2} r \\left( \\sin \\frac{\\pi}{4} \\cos\\theta + \\cos \\frac{\\pi}{4} \\sin\\theta \\right) \\\\[6pt]<br \/>\n&#038;= \\sqrt{2} r \\sin\\left( \\frac{\\pi}{4} +\\theta \\right ) \\\\[6pt]<br \/>\n&#038;\\le \\sqrt{2} r = \\sqrt{2} \\lvert z \\rvert .<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<p><!-- ########## 4.11 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.11<\/span><br \/>\n\\(z\\)\uc640 \\(w\\)\uac00 \ubcf5\uc18c\uc218\uc77c \ub54c, \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<br \/>\n\\[\\left\\lvert 1+\\overline{z} w \\right\\rvert^2 + \\left\\lvert z-w \\right\\rvert^2<br \/>\n= \\left( 1+ \\lvert z \\rvert^2 \\right) \\left( 1+ \\lvert w \\rvert ^2 \\right) .\\]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\ubb38\uc81c\uc758 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc744 \ud480\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert 1+\\overline{z} w \\right\\rvert^2 + \\left\\lvert z-w \\right\\rvert^2<br \/>\n&#038;= \\left( 1+ \\overline{z} w \\right) \\overline{ \\left( 1+\\overline{z} w \\right) }<br \/>\n+ \\left( z-w \\right) \\overline{ \\left( z-w \\right) } \\\\[6pt]<br \/>\n&#038;= \\left( 1+ \\overline{z} w \\right) \\left( 1+ z \\overline{w} \\right)<br \/>\n+ \\left( z-w \\right) \\left( \\overline{z}-\\overline{w} \\right) \\\\[6pt]<br \/>\n&#038;= 1 + \\overline{z} w + z\\overline{w} + \\overline{z} w z \\overline{w}<br \/>\n+ z\\overline{z} &#8211; z\\overline{w} &#8211; w\\overline{z} + w\\overline{w} \\\\[6pt]<br \/>\n&#038;= 1 + \\overline{z} z \\overline{w} w + z\\overline{z} + w\\overline{w} \\\\[6pt]<br \/>\n&#038;= 1 + \\lvert z \\rvert^2 \\lvert w \\rvert^2 + \\lvert z \\rvert^2 + \\lvert w \\rvert^2 \\\\[6pt]<br \/>\n&#038;= \\left( 1+ \\lvert z \\rvert^2 \\right) \\left( 1+ \\lvert w \\rvert ^2 \\right).<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<p><!-- ########## 4.12 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.12<\/span><br \/>\n\ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \ub450 \uc810 \\(z_1 ,\\) \\(z_2\\) \uc0ac\uc774\uc758 \uac70\ub9ac\ub97c \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<p>(1) \\(z_1 = 2+5 \\complexI ,\\,\\, z_2 = -4+ \\complexI .\\)<br \/>\n(2) \\(z_1 = 3-4 \\complexI ,\\,\\, z_2 = -2+\\complexI .\\)<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \\(\\left\\lvert z_1 &#8211; z_2 \\right\\rvert = \\sqrt{ (2+4)^2 + (5-1)^2 } = \\sqrt{52} = 2\\sqrt{13}.\\)<\/p>\n<p>(2) \\(\\left\\lvert z_1 &#8211; z_2 \\right\\rvert = \\sqrt{ (3+2)^2 + (-4-1)^2 } = \\sqrt{50} = 5\\sqrt{2}.\\)\n<\/p>\n<\/div>\n<p><!-- ########## 4.13 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.13<\/span><br \/>\n\\(z\\)\uc640 \\(w\\)\uac00 \ubcf5\uc18c\uc218\uc774\uace0 \ub2e4\uc74c\uacfc \uac19\uc740 \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[w = \\frac{z-\\complexI }{z+ \\complexI } . \\quad (z\\ne &#8211; \\complexI ).\\]<br \/>\n\\(z\\)\uac00 \uc2e4\uc218\ucd95 \uc704\ub97c \uc6c0\uc9c1\uc77c \ub54c, \\(w\\)\uc758 \uc790\ucde8\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(u = z+\\complexI \\)\ub77c\uace0 \ud558\uba74 \\(u\\)\ub294 \uc2e4\uc218\ubd80\uac00 \\(z\\)\uc774\uace0 \ud5c8\uc218\ubd80\uac00 \\(1\\)\uc778 \ubcf5\uc18c\uc218\uc774\uba70<br \/>\n\\[w = \\frac{\\overline{u}}{u}\\]<br \/>\n\uc774\ub2e4. \uc774\ub54c<br \/>\n\\[\\lvert w \\rvert = \\left\\lvert \\frac{\\overline{u}}{u} \\right\\rvert = \\frac{\\left\\lvert \\overline{u} \\right\\rvert}{ \\lvert u \\rvert} = \\frac{\\lvert u \\rvert}{\\lvert u \\rvert} = 1\\]<br \/>\n\uc774\ubbc0\ub85c, \\(w\\)\ub294 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \ub2e8\uc704\uc6d0 \uc704\uc5d0 \ub193\uc778 \uc810\uc774\ub2e4.<\/p>\n<p>\ud55c\ud3b8 \\(u\\)\uc758 \ud3b8\uac01\uc744 \\(\\theta\\)\ub77c\uace0 \ub450\uba74 \\(\\overline{u}\\)\uc758 \ud3b8\uac01\uc740 \\(-\\theta\\)\uc774\ub2e4. \\(z\\)\uac00 \uc2e4\uc218\ucd95 \uc704\ub97c \uc6c0\uc9c1\uc77c \ub54c \\(\\theta\\)\uc758 \ubc94\uc704\ub294 \\(0 < \\theta < \\pi\\)\uc774\uace0,\n\\[\\Arg (w) = \\Arg\\left( \\frac{\\overline{u}}{u}\\right) = \\Arg\\left( \\overline{u} \\right) - \\Arg(u) = -\\theta -\\theta = - 2\\theta\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(w\\)\uc758 \ud3b8\uac01\uc758 \ubc94\uc704\ub294\n\\[-2 \\pi < -2 \\theta < 0\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(w\\)\uc758 \ud3b8\uac01\uc740 \\(0\\)\uc744 \uc81c\uc678\ud55c \ubaa8\ub4e0 \uac12\uc744 \uac00\uc9c8 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(w\\)\ub294 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \ub2e8\uc704\uc6d0 \uc704\uc5d0 \ub193\uc778 \uc810\uc774\uba70, \\(1\\)\uc744 \uc81c\uc678\ud55c \uc810\uc774\ub2e4. \uc989 \\(w\\)\uc758 \uc790\ucde8\uc758 \ubc29\uc815\uc2dd\uc740\n\\[\\lvert w \\rvert = 1 ,\\quad w \\ne 1\\]\n\uc774\ub2e4.<\/p>\n<p><span class=\"proof\">\ub2e4\ub978 \ubc29\ubc95.<\/span><br \/>\n\\(x,\\) \\(y\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(w = x+y\\complexI\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[x+y\\complexI = \\frac{z-\\complexI}{z+\\complexI}.\\]<br \/>\n\\(z\\)\uac00 \uc2e4\uc218\uc77c \ub54c, \uc704 \uc2dd\uc758 \uc6b0\ubcc0\uc744 \ubcc0\ud615\ud558\uc5ec \uc2e4\uc218\ubd80\uc640 \ud5c8\uc218\ubd80\ub97c \ubd84\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[x+y\\complexI = \\frac{z^2 -1}{z^2 +1} + \\frac{-2z}{z^2 +1} \\complexI.\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(x\\)\uc640 \\(y\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ub9e4\uac1c\ubcc0\uc218 \\(z\\)\uc5d0 \ub300\ud55c \uc2dd\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4.<br \/>\n\\[x=\\frac{z^2 -1}{z^2 +1} ,\\quad y= \\frac{-2z}{z^2 +1} .\\tag{a}\\]<br \/>\n\uc5ec\uae30\uc11c \uccab \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \ubcc0\ud615\ud558\uc5ec \\(x\\)\ub97c \\(z\\)\uc5d0 \ub300\ud55c \uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\nx &#038;= \\frac{z^2 +1 -2}{z^2 +1} = 1- \\frac{2}{z^2 +1} ,\\\\[6pt]<br \/>\nx-1 &#038;= &#8211; \\frac{2}{z^2 +1} , \\\\[6pt]<br \/>\nz^2 +1 &#038;= &#8211; \\frac{2}{x-1} , \\\\[6pt]<br \/>\nz^2 &#038;= \\frac{2}{1-x} -1 = \\frac{1+x}{1-x} \\quad ( -1 \\le x < 1 ) ,\\\\[6pt]\nz &#038;= \\pm\\sqrt{ \\frac{1+x}{1-x} } \\quad (-1 \\le x < 1 ).\n\\end{aligned}\\]\n\uc5ec\uae30\uc11c \uad6c\ud55c \\(z^2\\)\uacfc \\(z\\)\ub97c (a)\uc758 \ub450 \ubc88\ub300 \ub4f1\uc2dd\uc5d0 \ub300\uc785\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\ny \n&#038;= \\pm 2 \\sqrt{ \\frac{1+x}{1-x}} \\times \\frac{x-1}{2} \\\\[6pt]\n&#038;= \\mp \\sqrt{ (1+x)(1-x) } \\\\[6pt]\n&#038;= \\mp \\sqrt{1-x^2} \\quad (-1 \\le x < 1 ).\n\\end{aligned}\\]\n\uadf8\ub7ec\ubbc0\ub85c \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(x+y\\complexI\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc790\ucde8\ub294 \uc911\uc2ec\uc774 \uc6d0\uc810\uc778 \ub2e8\uc704\uc6d0\uc774\uba70, \ud55c \uc810 \\(1\\)\ub9cc \uc81c\uc678\ub41c\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ########## 4.14 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.14<\/span><br \/>\n\\(z_1 ,\\) \\(z_2 ,\\) \\(z_3\\)\uc774 \ubcf5\uc18c\uc218\uc77c \ub54c, \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<p>(1) \\(\\left\\lvert z_1 &#8211; z_2 \\right\\rvert \\le \\left\\lvert z_1 \\right\\rvert + \\left\\lvert z_1 \\right\\rvert . \\)<br \/>\n(2) \\(\\left\\lvert z_1 \\right\\rvert &#8211; \\left\\lvert z_2 \\right\\rvert \\le \\left\\lvert z_1 &#8211; z_2 \\right\\rvert . \\)<br \/>\n(3) \\(\\left\\lvert z_1 +z_2 +z_3 \\right\\rvert \\le \\left\\lvert z_1 \\right\\rvert + \\left\\lvert z_2 \\right\\rvert + \\left\\lvert z_3 \\right\\rvert .\\)\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert z_1 &#8211; z_2 \\right\\rvert<br \/>\n&#038;= \\left\\lvert z_1 + \\left( -z_2 \\right) \\right\\rvert \\\\[6pt]<br \/>\n&#038;\\le \\left\\lvert z_1 \\right\\rvert + \\left\\lvert -z_2 \\right\\rvert \\\\[6pt]<br \/>\n&#038;= \\left\\lvert z_1 \\right\\rvert + \\left\\lvert z_2 \\right\\rvert .<br \/>\n\\end{aligned}\\]<\/p>\n<p>(2) \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert z_1 \\right\\rvert<br \/>\n&#038;= \\left\\lvert \\left( z_1 &#8211; z_2 \\right) + z_2 \\right\\rvert \\\\[6pt]<br \/>\n&#038;\\le \\left\\lvert z_1 &#8211; z_2 \\right\\rvert + \\left\\lvert z_2 \\right\\rvert .<br \/>\n\\end{aligned}\\]<br \/>\n\ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0\uc11c \\(\\left\\lvert z_2 \\right\\rvert\\)\ub97c \ube7c\uba74 \ubc14\ub77c\ub294 \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>(3) \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert z_1 + z_2 + z_3 \\right\\rvert<br \/>\n&#038;= \\left\\lvert z_1 + \\left( z_2 + z_3 \\right) \\right\\rvert \\\\[6pt]<br \/>\n&#038;\\le \\left\\lvert z_1 \\right\\rvert + \\left\\lvert z_2 + z_3 \\right\\rvert \\\\[6pt]<br \/>\n&#038;\\le \\left\\lvert z_1 \\right\\rvert + \\left\\lvert z_2 \\right\\rvert + \\left\\lvert z_3 \\right\\rvert .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<p><!-- ########## 4.15 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.15<\/span><br \/>\n\\(z\\)\uac00 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \ub2e8\uc704\uc6d0 \uc704\ub97c \uc6c0\uc9c1\uc77c \ub54c, \ub2e4\uc74c\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<p>(1) \\(\\left\\lvert z^2 +z+1 \\right\\rvert\\)\uc758 \ucd5c\ub313\uac12<br \/>\n(2) \\(\\left\\lvert z^3 &#8211; 2 \\right\\rvert\\)\uc758 \ucd5c\uc19f\uac12<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \\(\\lvert z \\rvert = 1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert z^2 +z+1 \\right\\rvert \\le \\left\\lvert z \\right\\rvert^2 + \\lvert z \\rvert + \\lvert 1 \\rvert = 1+1+1 = 3\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c \\(z=1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert z^2 +z+1 \\right\\rvert = \\lvert 1+1+1 \\rvert = 3\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \ucd5c\ub313\uac12\uc740 \\(3\\)\uc774\ub2e4.<\/p>\n<p>(2) \\(\\lvert z \\rvert = 1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert z^3 &#8211; 2 \\right\\rvert \\ge \\left\\lvert \\left\\lvert z^3 \\right\\rvert &#8211; \\lvert 2 \\rvert \\right\\rvert = \\lvert 1-2 \\rvert = 1\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c \\(z=1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert z^3 &#8211; 2 \\right\\rvert = \\lvert 1-2 \\rvert = 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \ucd5c\uc19f\uac12\uc740 \\(1\\)\uc774\ub2e4.<\/p>\n<p><span class=\"proof\">\ub2e4\ub978 \ubc29\ubc95.<\/span><br \/>\n(1) \ub4f1\uc2dd \\(\\lvert z \\rvert = 1\\)\uc744 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \ud45c\ud604\ud558\uba74<br \/>\n\\[z = \\cos \\theta + \\complexI \\sin \\theta ,\\quad \\theta \\in \\mathbb{R}\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\begin{aligned}<br \/>\nz^2 + z + 1<br \/>\n&#038;= (\\cos \\theta + \\complexI \\sin \\theta )(\\cos \\theta + \\complexI \\sin \\theta ) + \\cos \\theta + \\complexI \\sin \\theta + 1 \\\\[6pt]<br \/>\n&#038;= \\left( \\cos^2 \\theta &#8211; \\sin^2 \\theta + \\cos \\theta + 1 \\right)<br \/>\n+\\complexI \\left( 2\\sin\\theta \\cos\\theta + \\sin\\theta \\right) \\\\[6pt]<br \/>\n&#038;= \\left( 2\\cos^2 \\theta + \\cos\\theta \\right) = \\complexI \\sin\\theta (2\\cos\\theta +1 ) \\\\[6pt]<br \/>\n&#038;= \\cos\\theta ( 2\\cos\\theta +1) + \\complexI \\sin\\theta (2\\cos\\theta +1)<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\uace0,<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert z^2+z+1 \\right\\rvert^2<br \/>\n&#038;= \\cos^2 \\theta (2\\cos\\theta +1)^2 +\\sin^2 \\theta (2\\cos\\theta +1)^2 \\\\[6pt]<br \/>\n&#038;= \\left( \\cos^2 \\theta + \\sin^2 \\theta \\right) (2\\cos\\theta +1)^2  \\\\[6pt]<br \/>\n&#038;= (2\\cos\\theta +1)^2 , \\\\[6pt]<br \/>\n\\left\\lvert z^2+z+1 \\right\\rvert &#038;= \\left\\lvert 2\\cos\\theta +1 \\right\\rvert<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[-2 \\le 2\\cos\\theta \\le 2\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[-1 \\le 2\\cos\\theta +1 \\le 3\\]<br \/>\n\uc989<br \/>\n\\[\\left\\lvert 2\\cos\\theta +1 \\right\\rvert \\le 3\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(\\theta = 2k\\pi ,\\) \\(k\\in\\mathbb{Z}\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert 2\\cos\\theta +1 \\right\\rvert = 3\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \ucd5c\ub313\uac12\uc740 \\(3\\)\uc774\ub2e4.<\/p>\n<p>(2) \\(z = \\cos \\theta + \\complexI \\sin \\theta ,\\) \\(\\theta \\in \\mathbb{R}\\)\ub77c\uace0 \ud558\uba74<br \/>\n\\[z^3 = \\cos 3\\theta + \\complexI \\sin 3\\theta\\]<br \/>\n\uc774\ubbc0\ub85c, \\(z^3\\) \ub610\ud55c \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \ub2e8\uc704\uc6d0 \uc704\ub97c \uc6c0\uc9c1\uc774\uba70, \ub2e8\uc704\uc6d0 \uc704\uc758 \ubaa8\ub4e0 \uc810\uc744 \uc9c0\ub09c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\left\\lvert z^3 &#8211; 2 \\right\\rvert\\]<br \/>\n\uc758 \uac12\uc740 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \ub2e8\uc704\uc6d0 \uc704\uc758 \uc810 \\(z^3\\)\uacfc \\(2\\) \uc0ac\uc774\uc758 \uac70\ub9ac\ub97c \ub098\ud0c0\ub0b8\ub2e4. \ub2e8\uc704\uc6d0 \uc704\uc758 \uc810 \uc911\uc5d0\uc11c \\(2\\)\uc640 \uac00\uc7a5 \uac00\uae4c\uc6b4 \uc810\uc740 \\(1\\)\uc774\ub2e4. \uc989 \\(z^3 = 1\\)\uc77c \ub54c<br \/>\n\\[\\left\\lvert z^3 &#8211; 2 \\right\\rvert = \\lvert 1-2 \\rvert = 1\\]<br \/>\n\uc774\uba70, \uc774 \uac12\uc774 \uad6c\ud558\ub294 \ucd5c\uc19f\uac12\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.16 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.16<\/span><br \/>\n\ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubcf5\uc18c\uc218 \\(z\\)\uc758 \uc601\uc5ed\uc744 \ubcf5\uc18c\ud3c9\uba74\uc5d0 \ub098\ud0c0\ub0b4\uc2dc\uc624.<br \/>\n\\[\\left\\lvert \\frac{2z-3}{z+1} \\right\\rvert \\le 2 .\\]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\uc8fc\uc5b4\uc9c4 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\left\\lvert 2z-3 \\right\\rvert &#038; \\le \\left\\lvert 2z+2 \\right\\rvert , \\\\[6pt]<br \/>\n\\left\\lvert z- \\frac{3}{2} \\right\\rvert &#038; \\le \\left\\lvert z+1 \\right\\rvert .<br \/>\n\\end{aligned}\\]<br \/>\n\uc774 \ubd80\ub4f1\uc2dd\uc758 \uacbd\uacc4\ub294 \ub450 \uc810 \\(\\frac{3}{2}\\)\uacfc \\(-1\\)\ub85c\ubd80\ud130 \uac70\ub9ac\uac00 \uac19\uc740 \uc810\ub4e4\uc758 \ubaa8\uc784\uc774\ubbc0\ub85c, \uc9c1\uc120 \\(x = \\frac{1}{4}\\)\uc774\ub2e4. \uc774 \uacbd\uacc4\ub97c \ud3ec\ud568\ud558\uc5ec, \\(\\frac{3}{2}\\)\uc5d0 \ub354 \uac00\uae4c\uc6b4 \uc810\ub4e4\uc758 \ubaa8\uc784, \uc989 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \uc624\ub978\ucabd \uc601\uc5ed\uc774 \uad6c\ud558\ub294 \uc601\uc5ed\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.17 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.17<\/span><br \/>\n\ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubcf5\uc18c\uc218 \\(z\\)\uc758 \uc601\uc5ed\uc744 \ubcf5\uc18c\ud3c9\uba74\uc5d0 \ub098\ud0c0\ub0b4\uc2dc\uc624.<br \/>\n\\[\\left\\lvert z \\right\\rvert \\le 3 \\le \\left\\lvert \\frac{3z-5}{z-\\complexI} \\right\\rvert .\\tag{a}\\]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\ubb38\uc81c\uc758 \ubd80\ub4f1\uc2dd\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub450 \uac1c\uc758 \ubd80\ub4f1\uc2dd\uc744 \uc5f0\ub9bd\ud55c \uac83\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790.<br \/>\n\\[\\begin{align}<br \/>\n\\lvert z \\rvert &#038; \\le 3 , \\tag{b} \\\\[6pt]<br \/>\n3 &#038; \\le \\left\\lvert \\frac{3z-5}{z-\\complexI} \\right\\rvert . \\tag{c}<br \/>\n\\end{align}\\]<br \/>\n\uc6b0\uc120 (b)\ub294 \uc911\uc2ec\uc774 \uc6d0\uc810\uc774\uace0 \ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 \\(3\\)\uc778 \uc6d0\uc758 \ub0b4\ubd80\uc640 \uacbd\uacc4\ub97c \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c (c)\ub97c \ubcc0\ud615\ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\n\\lvert 3z-3\\complexI \\rvert \\le \\lvert 3z-5 \\rvert ,\\\\[6pt]<br \/>\n\\left\\lvert z- \\complexI \\right\\rvert \\le \\left\\lvert z- \\frac{5}{3} \\right\\rvert<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c, (c)\ub294 \uc810 \\(\\complexI\\)\uc640 \\(\\frac{5}{3}\\)\ub97c \uc787\ub294 \uc120\ubd84\uc758 \uc218\uc9c1\uc774\ub4f1\ubd84\uc120\uc744 \uacbd\uacc4\ub85c \ud558\uace0, \uc67c\ucabd \uc717\ubd80\ubd84\uacfc \uacbd\uacc4\uc120\uc744 \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc774\ub2e4.<\/p>\n<p>\ub05d\uc73c\ub85c (a)\ub294 (b)\uc640 (c)\uc758 \uacf5\ud1b5\uc601\uc5ed(\uad50\uc9d1\ud569)\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.20 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.20<\/span><br \/>\n\ub2e4\uc74c \ubcf5\uc18c\uc218\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.\n<\/p>\n<p>(1) \\(z=-2 \\left( \\cos \\frac{\\pi}{4} + \\complexI \\sin\\frac{\\pi}{4} \\right) .\\)<br \/>\n(2) \\(z = \\sin\\frac{\\pi}{13} + \\complexI \\cos\\frac{\\pi}{13}\\)<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1)<br \/>\n\\[ z<br \/>\n= 2 \\left( &#8211; \\cos \\frac{\\pi}{4} &#8211; \\complexI \\sin \\frac{\\pi}{4} \\right)<br \/>\n= 2 \\left( \\cos \\frac{5 \\pi}{4} + \\complexI \\sin \\frac{5 \\pi}{4} \\right) . \\]\n<\/p>\n<p>\n(2)<br \/>\n\\[ z<br \/>\n= \\cos\\left( \\frac{\\pi}{2} &#8211; \\frac{\\pi}{13} \\right) + \\complexI \\sin\\left( \\frac{\\pi}{2} &#8211; \\frac{\\pi}{13} \\right)<br \/>\n= \\cos \\frac{11 \\pi}{26} + \\complexI \\sin\\frac{11 \\pi}{26} .<br \/>\n\\]\n<\/p>\n<\/div>\n<p><!-- ########## 4.21 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.21<\/span><br \/>\n\\(z_1 = \\cos 2\\theta + \\complexI \\sin 2\\theta ,\\) \\(z_2 = \\cos \\theta + \\complexI \\sin \\theta \\)\uc77c \ub54c, \ub2e4\uc74c \ubcf5\uc18c\uc218\uc758 \uc808\ub313\uac12\uacfc \ud3b8\uac01\uc758 \ud06c\uae30\ub97c \uad6c\ud558\uc2dc\uc624. [\uc8fc\uc758: \uad50\uc7ac\uc758 \ubb38\uc81c\uc640 \ub2e4\ub984.]<\/p>\n<p>(1) \\(z_1 + z_2\\) (\ub2e8, \\(-\\frac{\\pi}{2} < \\theta < \\frac{\\pi}{2}\\))<br \/>\n(2) \\(z_1 &#8211; z_2\\) (\ub2e8, \\(0 < \\theta < \\pi\\))\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(z_1\\)\uc774 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \\(\\mathrm{A}\\)\uc774\ub77c\uace0 \ud558\uace0 \\(z_2\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \\(\\mathrm{B}\\)\ub77c\uace0 \ud558\uba70, \\(z_1 + z_2\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \\(\\mathrm{C}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p> \uadf8\ub7ec\uba74 \\(\\theta \\ne 0\\)\uc77c \ub54c \uc0ac\uac01\ud615 \\(\\mathrm{AOBC}\\)\ub294 \ub9c8\ub984\ubaa8\uc774\ub2e4. \ub9c8\ub984\ubaa8\uc758 \ub450 \ub300\uac01\uc120\uc740 \uc11c\ub85c \uc218\uc9c1\uc774\ub4f1\ubd84\ud558\ubbc0\ub85c<br \/>\n\\[\\angle\\mathrm{BOC} = \\frac{\\theta}{2}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\left\\lvert z_1 + z_2 \\right\\rvert = \\overline{ \\mathrm{OC} } = 2\\cos\\frac{\\theta}{2}\\]<br \/>\n\uc774\uba70,<br \/>\n\\[\\Arg\\left(z_1 + z_2 \\right) = \\frac{3}{2}\\theta\\]<br \/>\n\uc774\ub2e4. \\(\\theta = 0\\)\uc77c \ub54c\ub3c4 \\(z_1 + z_2\\)\uc758 \uc808\ub313\uac12\uacfc \ud3b8\uac01\uc740 \ub3d9\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4.<\/p>\n<p>(2) \uc55e\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc810 \\(\\mathrm{A},\\) \\(\\mathrm{B}\\)\ub97c \uc815\ud558\uc790. \uadf8\ub7ec\uba74 \ubcf5\uc18c\uc218 \\(z_1 &#8211; z_2\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \ubca1\ud130\ub294 \ubca1\ud130 \\(\\overrightarrow{\\mathrm{BA}}\\)\uc640 \uac19\ub2e4. \uc774 \ubca1\ud130\uc758 \ud06c\uae30\ub294 \uc120\ubd84 \\(\\mathrm{AB}\\)\uc758 \uae38\uc774\uc640 \uac19\ub2e4. \uadf8\ub7f0\ub370 \uc0bc\uac01\ud615 \\(\\mathrm{AOB}\\)\uc5d0\uc11c \uc0ac\uc778 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\frac{\\overline{\\mathrm{AB}}}{\\sin\\theta} = \\frac{1}{\\sin\\frac{\\pi-\\theta}{2}}.\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\overline{\\mathrm{AB}} = \\frac{\\sin\\theta}{\\sin\\left( \\frac{\\pi}{2} &#8211; \\frac{\\theta}{2} \\right)} = \\frac{ 2\\sin \\frac{\\theta}{2} \\cos\\frac{\\theta}{2} }{\\cos\\frac{\\theta}{2}}<br \/>\n= 2\\sin \\frac{\\theta}{2}.\\]<br \/>\n\ud55c\ud3b8 \\(z_1 &#8211; z_2\\)\uc758 \ud3b8\uac01\uc740<br \/>\n\\[\\theta + \\angle\\mathrm{BAO} = \\theta + \\frac{\\pi &#8211; \\theta}{2} = \\frac{\\pi + \\theta}{2}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.23 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.23<\/span><br \/>\n\\(\\alpha = 2+5\\complexI ,\\) \\(\\beta = 2+4\\complexI ,\\) \\(\\gamma = 5+8\\complexI \\)\uc77c \ub54c,<br \/>\n\\[z = \\frac{\\beta &#8211; \\alpha}{\\gamma &#8211; \\alpha}\\]<br \/>\n\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624. (\ub2e8, \ud3b8\uac01\uc740 \\(0\\le\\theta < 2\\pi\\)\uc758 \ubc94\uc704\uc5d0\uc11c \ud0dd\ud55c\ub2e4.)\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><\/p>\n<p>\uc6b0\uc120<br \/>\n\\[\\begin{aligned}<br \/>\n\\beta &#8211; \\alpha &#038;= -\\complexI = \\cos \\frac{3}{2}\\pi + \\complexI \\sin \\frac{3}{2}\\pi , \\\\[6pt]<br \/>\n\\gamma &#8211; \\alpha &#038;= 3+3\\complexI = 3\\sqrt{2} \\left( \\cos\\frac{\\pi}{4} + \\complexI \\sin\\frac{\\pi}{4}\\right)<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\nz<br \/>\n&#038;= \\frac{\\beta &#8211; \\alpha}{\\gamma &#8211; \\alpha}  \\\\[6pt]<br \/>\n&#038;= \\frac{1}{3\\sqrt{2}}\\left( \\cos\\left( \\frac{3}{2}\\pi &#8211; \\frac{\\pi}{4} \\right) + \\complexI \\sin \\left( \\frac{3}{2}\\pi &#8211; \\frac{\\pi}{4} \\right) \\right) \\\\[6pt]<br \/>\n&#038;= \\frac{\\sqrt{2}}{6} \\left( \\cos\\frac{5}{4}\\pi + \\complexI \\sin \\frac{5}{4}\\pi \\right)<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## 4.24 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.24<\/span><br \/>\n\\(a\\)\uc640 \\(b\\)\uac00 \uc2e4\uc218\uc774\uace0, \ub450 \ubcf5\uc18c\uc218<br \/>\n\\[z_1 = 2-\\sqrt{3} a+a\\complexI ,\\quad z_2 = \\sqrt{3} b &#8211; 1 + \\left( \\sqrt{3} -b \\right)\\complexI\\]<br \/>\n\uc5d0 \ub300\ud558\uc5ec \\(z_1\\)\uacfc \\(z_2\\)\uc758 \uc808\ub313\uac12\uc774 \uac19\uc73c\uba70, \\(\\frac{z_2}{z_1}\\)\uc758 \ud3b8\uac01 \uc911 \ud558\ub098\uac00 \\(\\frac{\\pi}{2}\\)\uc774\ub2e4. \uc774\ub54c \uc2e4\uc218 \\(a,\\) \\(b\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(\\frac{z_2}{z_1}\\)\uc758 \ud3b8\uac01\uc774 \\(\\frac{\\pi}{2}\\)\uc774\uace0 \\(z_1\\)\uc758 \uc808\ub313\uac12\uacfc \\(z_2\\)\uc758 \uc808\ub313\uac12\uc774 \uac19\uc73c\ubbc0\ub85c<br \/>\n\\[z_2 = \\complexI z_1\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n\\sqrt{3} b-1 + \\left( \\sqrt{3} -b \\right) \\complexI &#038;= \\complexI \\left( 2-\\sqrt{3} a + a\\complexI \\right) , \\\\[6pt]<br \/>\n\\sqrt{3} b-1 + \\left( \\sqrt{3} -b \\right) \\complexI &#038;= -a + \\left( 2-\\sqrt{3} a\\right)\\complexI<br \/>\n\\end{aligned}\\]<br \/>\n\uc5ec\uae30\uc11c \uc2e4\uc218\ubd80\ubd84\uacfc \ud5c8\uc218\ubd80\ubd84\uc744 \ube44\uad50\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\left\\{<br \/>\n\\begin{array}{l}<br \/>\n\\sqrt{3} b-1 =-a \\\\[6pt]<br \/>\n\\sqrt{3} -b = 2-\\sqrt{3} a<br \/>\n\\end{array}<br \/>\n\\right.\\]<br \/>\n\uc870\uc2ec\uc2a4\ub7fd\uac8c \uc774 \uc5f0\ub9bd\ubc29\uc815\uc2dd\uc744 \ud480\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[a=\\frac{\\sqrt{3}-1}{2} ,\\quad b=\\frac{\\sqrt{3}-1}{2}.\\]<br \/>\n(\uaf2d \uc190\uc73c\ub85c \uc9c1\uc811 \ud480\uc5b4\ubcf4\uc138\uc694.)<\/p>\n<\/div>\n<p><!--\n\\(z_1\\)\uacfc \\(z_2\\)\uc758 \uc808\ub313\uac12\uc744 \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\n\\left\\lvert z_1 \\right\\rvert\n&= \\left( 2-\\sqrt{3} a \\right)^2 + a^2 \\\\[6pt]\n&= 4-4\\sqrt{3} a +3a^2 +a^2 \\\\[6pt]\n&= 4a^2 -4\\sqrt{3} a +4 ,\\\\[6pt]\n\\left\\lvert z_2 \\right\\rvert\n&= \\left(\\sqrt{3}b -1\\right)^2 + \\left( \\sqrt{3} -b \\right)^2 \\\\[6pt]\n&= 3b^2 - 2\\sqrt{3} b +1 +3 -2\\sqrt{3} b +b^2 \\\\[6pt]\n&= 4b^2 -4\\sqrt{3}b +4 .\n\\end{aligned}\\]\n--><\/p>\n<p><!-- ########## 4.25 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.25<\/span><br \/>\n\ubcf5\uc18c\uc218<br \/>\n\\[z = \\frac{1+\\sqrt{3} \\complexI}{1+\\complexI}\\]<br \/>\n\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ubb3c\uc74c\uc5d0 \ub2f5\ud558\uc2dc\uc624.<\/p>\n<p>(1) \\(z\\)\ub97c \\(a+b\\complexI\\) \uaf34\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<br \/>\n(2) \\(1+\\sqrt{3} \\complexI\\)\uc640 \\(1+\\complexI\\)\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uace0, \uc774\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(z\\)\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<br \/>\n(3) \\(\\sin\\frac{\\pi}{12} ,\\) \\(\\cos\\frac{\\pi}{12}\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1)<br \/>\n\\[\\begin{aligned}<br \/>\nz<br \/>\n&#038;= \\frac{ ( 1+\\sqrt{3}\\complexI ) ( 1-\\complexI ) }{ ( 1+\\complexI )( 1-\\complexI ) } \\\\[6pt]<br \/>\n&#038;= \\frac{ 1+\\sqrt{3} + (\\sqrt{3} &#8211; 1 )\\complexI }{ 2 } \\\\[6pt]<br \/>\n&#038;= \\frac{ 1+\\sqrt{3}}{2} + \\frac{\\sqrt{3} &#8211; 1 }{2} \\complexI . \\\\[6pt]<br \/>\n\\end{aligned}\\]<\/p>\n<p>(2)<br \/>\n\\[\\begin{aligned}<br \/>\n1+\\sqrt{3}\\complexI &#038;= 2\\left( \\cos\\frac{\\pi}{3} + \\complexI \\sin\\frac{\\pi}{3} \\right) ,\\\\[6pt]<br \/>\n1+\\complexI &#038;= \\sqrt{2} \\left( \\cos\\frac{\\pi}{4} + \\complexI \\sin\\frac{\\pi}{4} \\right) .<br \/>\n\\end{aligned}\\]<\/p>\n<p>(3) \\(z\\)\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[z = \\frac{1+\\sqrt{3} \\complexI}{1+\\complexI} =<br \/>\n\\sqrt{2} \\left( \\cos\\frac{\\pi}{12} + \\complexI \\sin\\frac{\\pi}{12} \\right).\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\n\\cos\\frac{\\pi}{12} &#038;= \\frac{1+\\sqrt{3}}{2\\sqrt{2}} , \\\\[6pt]<br \/>\n\\sin\\frac{\\pi}{12} &#038;= \\frac{\\sqrt{3} -1}{2\\sqrt{2}} .<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<p><!-- ########## 4.26 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.26<\/span><br \/>\n\ubcf5\uc18c\ud3c9\uba74\uc5d0 \uc138 \uc810 \\(\\mathrm{P} (-1),\\) \\(\\mathrm{Q} (1),\\) \\(\\mathrm{R} (z)\\)\uac00 \uc788\ub2e4. \ubcf5\uc18c\uc218<br \/>\n\\[w=\\frac{(-1+\\complexI )(z-1)}{z+1}\\tag{a}\\]<br \/>\n\uc774 \uc74c\uc758 \uc2e4\uc218\uc77c \ub54c, \\(\\angle\\mathrm{PRQ}\\)\uc758 \ud06c\uae30\ub97c \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\ubcf5\uc18c\uc218 \\(-1+\\complexI\\)\uc758 \ud3b8\uac01\uc774 \\(\\frac{3}{4}\\pi\\)\uc774\ubbc0\ub85c,<br \/>\n\\[w = (-1+\\complexI) \\times \\frac{z-1}{z+1}\\]<br \/>\n\uc774 \uc74c\uc758 \uc2e4\uc218\uac00 \ub418\ub824\uba74 \ubcf5\uc18c\uc218<br \/>\n\\[\\frac{z-1}{z+1}\\]<br \/>\n\uc758 \ud3b8\uac01\uc774 \\(\\frac{\\pi}{4}\\)\uac00 \ub418\uc5b4\uc57c \ud55c\ub2e4. \ud55c\ud3b8<br \/>\n\\[z-1 = \\overrightarrow{\\mathrm{QR}} ,\\quad<br \/>\nz+1 = \\overrightarrow{\\mathrm{PR}} \\]<br \/>\n\uc774\ubbc0\ub85c, \uc0bc\uac01\ud615 \\(\\mathrm{RPQ}\\)\uc5d0\uc11c<br \/>\n\\[\\angle\\mathrm{PRQ} = \\left\\lvert \\Arg(z-1) &#8211; \\Arg(z+1) \\right\\rvert = \\frac{\\pi}{4}\\]<br \/>\n\uc774\ub2e4. (\ubcf5\uc18c\ud3c9\uba74\uc5d0 \uc0bc\uac01\ud615 \\(\\mathrm{RPQ}\\)\uc758 \uadf8\ub9bc\uc744 \uadf8\ub824 \ubcf4\uc790.)\n<\/p>\n<\/div>\n<p><!-- ########## 4.28 ########## --><\/p>\n<p class=\"problem marginbottom0\"><span class=\"definition\">\uc720\uc81c 4.28<\/span><br \/>\n\ub2e4\uc74c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<p>(1) \\(z=\\left( \\frac{1}{2} &#8211; \\frac{\\sqrt{3}}{6} \\complexI \\right)^{-12} \\)<br \/>\n(2) \\(w=\\left( \\frac{1}{1-\\complexI} \\right)^5\\)<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n(1) \uadf9\ud615\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc790.<br \/>\n\\[ \\frac{1}{2} &#8211; \\frac{\\sqrt{3}}{6}\\complexI = \\frac{1}{\\sqrt{3}} \\left( \\frac{\\sqrt{3}}{2} &#8211; \\frac{1}{2}\\complexI \\right) = \\frac{1}{\\sqrt{3}} \\left( \\cos\\frac{\\pi}{6} + \\complexI \\sin\\frac{\\pi}{6} \\right)\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\nz^{-12}<br \/>\n&#038;= \\left( \\frac{1}{\\sqrt{3}} \\right)^{-12} \\left( \\cos\\frac{-12\\pi}{6} + \\complexI \\sin\\frac{-12\\pi}{6} \\right) \\\\[6pt]<br \/>\n&#038;= 3^6 \\left( \\cos 0 + \\complexI \\sin 0 \\right)<br \/>\n= 3^6 .<br \/>\n\\end{aligned}\\]<\/p>\n<p> (2)<br \/>\n\\[\\begin{aligned}<br \/>\nw<br \/>\n&#038;= (1-\\complexI )^{-5} \\\\[6pt]<br \/>\n&#038;= \\left[ \\sqrt{2} \\left\\{ \\cos\\left( &#8211; \\frac{\\pi}{4} \\right) +\\complexI \\sin \\left( &#8211; \\frac{\\pi}{4} \\right) \\right\\} \\right]^{-5} \\\\[6pt]<br \/>\n&#038;= \\frac{1}{4\\sqrt{2}} \\left( \\cos \\frac{5\\pi}{4} + \\complexI \\sin \\frac{5\\pi}{4} \\right) \\\\[6pt]<br \/>\n&#038;= \\frac{1}{4\\sqrt{2}} \\left( -\\frac{1}{\\sqrt{2}} &#8211; \\frac{1}{\\sqrt{2}} \\complexI \\right) \\\\[6pt]<br \/>\n&#038;= &#8211; \\frac{1}{8} &#8211; \\frac{1}{8}\\complexI .<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<p><!-- ########## 4.30 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.30<\/span><br \/>\n\\(\\omega = \\cos\\frac{\\pi}{7} + \\complexI \\sin\\frac{\\pi}{7}\\)\uc77c \ub54c, \ub2e4\uc74c \uc2dd\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.<br \/>\n\\[(2+\\omega )(2+ \\omega^2 ) \\cdots (2+ \\omega^{13} )\\tag{a}\\]\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(\\omega\\)\ub294 \ub2e4\uc74c \ubc29\uc815\uc2dd\uc758 \ud5c8\uadfc\uc774\ub2e4.<br \/>\n\\[z^{14} = 1.\\]<br \/>\n\ud2b9\ud788 \uc704 \ubc29\uc815\uc2dd\uc758 \ubaa8\ub4e0 \uadfc\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[1,\\,\\, \\omega ,\\,\\, \\omega^2 ,\\,\\, \\omega^3 ,\\,\\, \\cdots ,\\,\\, \\omega^{13} .\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[(z-1)(z-\\omega )(z-\\omega^2) \\cdots (z-\\omega^{13}) = z^{14}-1 \\]<br \/>\n\uc774\ub2e4. \\(\\omega^7 = -1\\)\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc704 \uc2dd\uc758 \uc88c\ubcc0\uc744 \ubcc0\ud615\ud558\uba74<br \/>\n\\[ ( z+\\omega^7 )( z+\\omega^8 ) \\cdots ( z+\\omega^{13} )( z + \\omega^{14} ) \\cdots ( z+\\omega^{20} ) = z^{14}-1\\]<br \/>\n\uc989<br \/>\n\\[ ( z+\\omega^7 )( z+\\omega^8 ) \\cdots ( z+\\omega^{13} )( z + 1 ) \\cdots ( z+\\omega^{6} ) = z^{14}-1\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc5d0 \\(z=2\\)\ub97c \ub300\uc785\ud558\uba74<br \/>\n\\[ ( 2+\\omega^7 )( 2+\\omega^8 ) \\cdots ( 2+\\omega^{13} ) \\times 2 \\times ( 2+\\omega^1 ) \\cdots ( 2+\\omega^6 ) = 2^{14}-1\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\n( 2+\\omega )( 2+\\omega^2 )\\cdots ( 2+\\omega^{13} )<br \/>\n= \\frac{1}{2} (2^{14} -1 ) = \\frac{16383}{2} .<br \/>\n\\]<br \/>\n\uc774\ub2e4.<br \/>\n<!--\n\uc989\n\\[(z-\\omega)(z-\\omega^2)\\cdots(z-\\omega^{13}) = z^{13} + z^{12} + \\cdots + 1\\tag{b}\\]\n\uc774\ub2e4. \ud55c\ud3b8\n\\[\\omega^7 = -1 ,\\,\\, \\omega^8 = -\\omega ,\\,\\, \\omega^9 = -\\omega^2 ,\\,\\, \\cdots ,\\,\\, \\omega^{13} = -\\omega^6\\]\n\uc774\ubbc0\ub85c (b)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.\n\\[(z+\\omega^8 )(z+\\omega^9) \\cdots (z+ \\omega^{13}) (z+\\omega^7 ) (z+\\omega) (z+\\omega^2) \\cdots (z+\\omega^6 )\\]\n-->\n<\/p>\n<\/div>\n<p><!-- ########## 4.32 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.32<\/span><br \/>\n\\(8\\complexI\\)\uc758 \\(6\\)\uc81c\uacf1\uadfc\uc744 \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(8\\complexI\\)\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[8\\complexI = 8 \\left( \\cos\\frac{\\pi}{2} + \\complexI \\sin\\frac{\\pi}{2} \\right)\\]<br \/>\n\uc774\ubbc0\ub85c \\(8\\complexI\\)\uc758 \\(6\\)\uc81c\uacf1\uadfc\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\sqrt{2} \\left\\{<br \/>\n\\cos \\left( \\frac{\\pi}{12} + \\frac{2k\\pi}{6} \\right) +<br \/>\n\\complexI \\sin \\left( \\frac{\\pi}{12} + \\frac{2k\\pi}{6} \\right)<br \/>\n\\right\\} \\quad (k=0,\\,\\,1,\\,\\,2,\\,\\,3,\\,\\,4,\\,\\,5.)\\]\n<\/p>\n<\/div>\n<p><!-- ########## 4.33 ########## --><\/p>\n<p class=\"problem\"><span class=\"definition\">\uc720\uc81c 4.33<\/span><br \/>\n\\(-128 (1+\\sqrt{3} \\complexI )\\)\uc758 \\(8\\)\uc81c\uacf1\uadfc\uc744 \ubaa8\ub450 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<div class=\"solution\">\n<p><span class=\"proof\">\ud574\uc124.<\/span><br \/>\n\\(-128 (1+\\sqrt{3} \\complexI )\\)\ub97c \uadf9\ud615\uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[-128 ( 1+\\sqrt{3} \\complexI ) = 256 \\left( \\cos\\frac{4\\pi}{3} + \\complexI\\sin\\frac{4\\pi}{3} \\right)\\]<br \/>\n\uc774\ubbc0\ub85c \\(-128 (1+\\sqrt{3} \\complexI )\\)\uc758 \\(8\\)\uc81c\uacf1\uadfc\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[ 2 \\left\\{ \\cos\\left( \\frac{\\pi}{6} + \\frac{2k\\pi}{8} \\right) + \\complexI \\sin\\left( \\frac{\\pi}{6} + \\frac{2k\\pi}{8} \\right) \\right\\} \\quad (k=0,\\,\\,1,\\,\\,2,\\,\\,\\cdots,\\,\\,7.) \\]\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 2023\ud559\ub144\ub3c4 1\ud559\uae30 \ud574\uc11d\uae30\ud558 \uc218\uc5c5 \uad50\uc7ac 4\ub2e8\uc6d0 \u300c\ubcf5\uc18c\ud3c9\uba74\uacfc \ubcf5\uc18c\uc218\uc758 \uadf9\ud615\uc2dd\u300d \uc720\uc81c \ud574\uc124\uc785\ub2c8\ub2e4. \ub208\uc73c\ub85c \uc77d\uae30\ub9cc \ud558\uc9c0 \ub9d0\uace0 \uaf2d \uc190\uc73c\ub85c \uc9c1\uc811 \ud480\uc5b4\ubcf4\uc138\uc694\u2661 \\( \\newcommand{\\complexI}{\\boldsymbol{i}} \\newcommand{\\rpart}{\\operatorname{Re}} \\newcommand{\\ipart}{\\operatorname{Im}} \\newcommand{\\Arg}{\\operatorname{Arg}} \\) \uc720\uc81c 4.1 \\(z\\)\ub97c \ubcf5\uc18c\uc218\ub77c \ud560 \ub54c, \ubcf5\uc18c\ud3c9\uba74 \uc704\uc5d0\uc11c \ub2e4\uc74c \ub450 \ubcf5\uc18c\uc218 \uc0ac\uc774\uc758 \uc704\uce58\uad00\uacc4\ub97c \uc870\uc0ac\ud558\uc2dc\uc624. (1) \\(z\\)\uc640 \\(-z.\\) (2) \\(z\\)\uc640 \\(\\overline{z}.\\) \ud574\uc124. (1) \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(-z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc740 \\(z\\)\uac00 \ub098\ud0c0\ub0b4\ub294 \uc810\uc744 \uc6d0\uc810\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc774\ub3d9\ud55c \uc810\uc774\ub2e4. (2) \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \\(\\overline{z}\\)\uac00&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[56],"tags":[],"class_list":["post-8721","post","type-post","status-publish","format-standard","hentry","category-analytic-geometry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8721","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=8721"}],"version-history":[{"count":150,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8721\/revisions"}],"predecessor-version":[{"id":8873,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8721\/revisions\/8873"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8721"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8721"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8721"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}