{"id":56627,"date":"2025-02-13T19:25:55","date_gmt":"2025-02-13T13:55:55","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=56627"},"modified":"2026-02-24T21:53:21","modified_gmt":"2026-02-24T16:23:21","slug":"complex-numbers-unit-iii","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=56627","title":{"rendered":"COMPLEX NUMBERS (UNIT &#8211; III)"},"content":{"rendered":"\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c22ec3607bb6321651489bcfd9999123\">Syllabus:<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Definition of a complex number- Real and imaginary parts &#8211; Modulus and}\\ \\hspace{10cm}\\\\ \\text{argument &#8211; Polar form of a complex number &#8211; Conjugate of a complex number -}\\ \\hspace{10cm}\\\\ \\text{Representation of complex numbers on Argand plane &#8211; Addition, subtraction, multiplication}\\ \\hspace{10cm}\\\\ \\text{and division of complex numbers &#8211; De-Moivre&#8217;s theorem (without proof) &#8211; Simple problems.}\\ \\hspace{5cm}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Introduction}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[In\\ earlier\\ classes,\\ we\\ have\\ studied\\ linear\\ equations\\ in\\ one\\ and\\ two\\ variables\\ and\\ quadratic\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[equations\\ in\\ one\\ variable.\\ We\\ have\\ seen\\ that\\ the\\ equation\\ x^2 + 1  = 0\\ has\\ no\\ real solution\\ as\\ x^2 + 1 = 0\\]  <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[gives\\ x^2 = &#8211; 1\\ and square\\ of\\ every\\ real\\ number\\ is\\ non-negative.\\ So,\\ we\\ need\\ to\\ extend\\ the\\ real\\ number\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[system\\ to\\ a\\ larger\\ system\\ so\\ that\\ we\\ can\\ find\\ the\\ solution\\ of\\ the\\ equation\\ x^2 = &#8211; 1.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ us\\ denote\\  \\sqrt{-1}\\ by\\ the\\ symbol\\ \u2018i \u2018.\\ Then,\\ we\\ have\\ i^2 = -1.\\   This\\ means\\ that\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u2018i \u2018\\ is\\ a\\ solution\\ of\\ the\\ equation\\ x^2 + 1  = 0.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 1:}\\ \\color {red}{Find\\  the\\ value\\ of}\\ i^3\\ +\\ i^5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ i^3\\ +\\ i^5\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ i^2\\ .\\ i\\ +\\ i^4\\ .\\ i\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ (-1)\\ .\\ i\\ +\\ (1)\\ .\\ i\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -i\\  +\\ i\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 0\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition\\ of\\ complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ number\\ which\\ is\\ of\\ the\\ form\\ a + ib\\  where\\ a, b \u2208 R\\ and\\  i^2 = -1\\ is\\ called\\ a\\ complex\\]  <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[number\\ and\\ it\\ is\\ denoted\\ by\\ Z.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ Z = a + ib ,\\  then\\ a\\ is\\ called\\ the\\ real\\ part\\ of\\ Z\\  and\\ b\\ is\\ called\\ the\\ imaginary\\ part\\ of\\ Z.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re ( Z ) =  a\\   and\\   Im ( Z ) =  b\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Ex:\\ If\\ Z = 3 + 4i\\  then\\ Re ( Z ) =  3\\ and\\ Im ( Z ) =  4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Conjugate\\ of\\ a\\ complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\  Z = a + ib\\ then\\ the\\ conjugate\\ of\\ Z\\ is\\ denoted\\ by\\  \\bar z\\  and\\ is\\ defined\\ as\\  \\bar z\\ =  a &#8211; ib\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ \\color {red} {Find\\ the\\ Real\\ and\\ Imaginary\\ parts\\ of}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1cm}\\ a)\\ \\frac{2-3i}{5}\\  \\hspace{0.5cm}\\ b)\\ i\\ \\hspace{0.5cm}\\ c)\\ 2\\  \\hspace{0.5cm}\\ d)\\ (1-i)(2-i)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a).\\ \\hspace{0.5cm}\\ Let\\ z\\ =\\ \\frac{2-3i}{5}\\ =\\ \\frac{2}{5}\\ -\\ \\frac{3}{5}\\ i\\ \\implies\\ RP\\ =\\ \\frac{2}{5}\\ and\\ IP\\ =\\ -\\ \\frac{3}{5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b).\\ \\hspace{0.5cm}\\ Let\\ z\\ =\\ i\\ =\\ 0\\ + 1\\ i\\ \\implies\\ RP\\ =\\ 0\\ and\\ IP\\ =\\ 1\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[c).\\ \\hspace{0.5cm}\\ Let\\ z\\ =\\ 2\\ =\\ 2\\ + 0\\ i\\ \\implies\\ RP\\ =\\ 2\\ and\\ IP\\ =\\ 0\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[d).\\ \\hspace{0.5cm}\\ Let\\ z\\ =\\ (1-i)(1-2i)\\ =\\ 1\\ -\\ 2i\\ -i\\ +\\ 2\\ i^2\\ =\\ 1\\ -\\ 3i\\ -\\ 2\\ \\hspace{1cm}\\\\ \\hspace{1cm}\\ =\\ -\\ 1\\ -\\ 3i\\ \\implies\\ RP\\ =\\ -\\ 1:\\ and\\ IP\\ =\\ -\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\color {red} {Find\\ the\\ conjugate\\ of}\\ (a)\\ 2\\ -\\ 3\\ i\\  \\hspace{0.5cm}\\ (b)\\ \\frac{3-5i}{2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a)\\ \\hspace{1cm}\\ Let\\ z\\ =\\ 2\\ -\\ 3i\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1.3cm}\\ \\therefore\\ \\bar z\\ =\\ 2\\ +\\ 3i\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[b)\\ \\hspace{1cm}\\ Let\\ z\\ =\\ \\frac{3\\ -\\ 5i}{2}\\ =\\ \\frac{3}{2}\\ -\\ \\frac{5}{2}\\ i\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{1.3cm}\\ \\therefore\\ \\bar z\\ =\\ \\frac{3}{2}\\ +\\ \\frac{5}{2}\\ i \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {Addition\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  +  Z_2  =   a + ib\\  +\\   c + id\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ a + c + i(b + d )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {Difference\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  &#8211;  Z_2  =   a + ib\\  -\\   (c + id)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =   a + ib  &#8211; c &#8211; id\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ a &#8211; c + i(b &#8211; d )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\ If\\  Z_1 =  (-1 , 2),\\  Z_2 =  (-3 , 4),\\ \\color {red} {find\\  2\\ Z_1  -\\  3\\ Z_2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Z_1 =  (-1\\ ,\\ 2)\\ =\\ -1\\ +\\ 2i\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Z_2 =  (-3\\ ,\\ 4)\\ =\\ -3\\ +\\ 4i\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2Z_1  &#8211;  3Z_2 =    2(-1 + 2i ) &#8211; 3 (-3 + 4i)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2Z_1  &#8211; 3 Z_2\\  =\\    -2\\ +\\ 4\\ i  +\\ 9\\ &#8211; 12i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2Z_1\\  -\\  3Z_2\\ =\\    -2\\ +\\  9\\  +\\  4i\\ -\\  12i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2Z_1  &#8211;  3Z_2\\ =\\    7\\  -\\  8i\\ =\\ (7,\\ -\\ 8)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {Multiplication\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z_1  Z_2  =   (a + ib) (c + id)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   ac + iad + ibc + i^2bd\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   ac + i(ad + bc) &#8211; bd\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =   (ac &#8211; bd) + i(ad + bc)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {(ii)\\ Division\\ of\\ two\\ Complex\\ numbers}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Z_1=  a + ib,\\  Z_2 =  c + id\\  be\\ any\\ two\\ complex\\ numbers.\\ Then\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{z_1}{z_2}\\ =\\frac{a+ib}{c+id}\\ \u00d7\\ \\frac{c-id}{c-id}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac- iad + ibc &#8211; i^2bd}{c^2 &#8211; i^2d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac + i (bc &#8211; ad) + bd}{c^2 &#8211; i^2d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac +  bd + i(bc &#8211; ad)}{c^2 + d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{ac +  bd} {c^2 + d^2}\\ +\\  i\\ \\frac{bc &#8211; ad}{c^2 + d^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ \\color {red} {Find\\ the\\ Real\\ and\\ Imaginary\\ parts\\ of}\\ \\frac{1}{2+ 3i} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ z = \\frac{1}{2+ 3i}\\ \u00d7\\ \\frac{2 &#8211; 3i}{2 &#8211; 3i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{2 &#8211; 3i}{(2)^2 + (3)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{2 &#8211; 3i}{4+ 9}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{2 &#8211; 3i}{13}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Z = \\frac{2}{13}\\ -\\ i\\  \\frac{3}{13}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re(Z)\\ =\\ \\frac{2}{13};\\  Im(Z)\\ =\\ -\\frac{3}{13}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ \\color {red} {Find\\ the\\ Real\\ and\\ Imaginary\\ parts\\ of}\\ \\frac{1+ i}{1- i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ z = \\frac{1+ i}{1- i}\\ \u00d7\\ \\frac{1+ i}{1+i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{1 + i + i + (i)^2}{(1)^2 + (1)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{1 + 2i  &#8211; 1}{2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{2i}{2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = i\\ =\\ 0\\ +\\ 1\\ i\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re(Z)\\ =\\ 0;\\  Im(Z)\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7:}\\ \\color {red} {Find\\ the\\ conjugate\\ of}\\ \\frac{7}{5+ 2i} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ z = \\frac{7}{5+ 2i}\\ \u00d7\\ \\frac{5 &#8211; 2i}{5 &#8211; 2i}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{35 &#8211; 14i}{(5)^2 + (2)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{35 &#8211; 14i}{25+ 4}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{35 &#8211; 14i}{29}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Z = \\frac{35}{29}\\ -\\ i\\  \\frac{14}{29}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ conjugate\\ of\\ Z\\ is\\ \\bar z\\ = \\frac{35}{29}\\ +\\ i\\  \\frac{14}{29}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\ \\color {red} {Express}\\ \\frac{(1+ i)(1 + 2i)}{1+ 3i}\\ \\color{red}{in\\ the\\ form\\ of\\ a\\ +\\ ib}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ z = \\frac{(1+ i)(1 + 2i)}{1+ 3i}\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{1+ 2i + i + 2i^2}{1+ 3i}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{1+ 3i &#8211; 2}{1+ 3i}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{-1+ 3i}{1+ 3i}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{-1+3i}{1+ 3i}\\ \u00d7\\ \\frac{1 &#8211; 3i}{1 &#8211; 3i}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{-1 + 3i + 3i &#8211; 9i^2}{(1)^2 + (3)^2}\\ \\hspace{3cm}\\ \\because [(a +ib)(a &#8211; ib)\\ =\\ (a)^2 + (b)^2]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{-1 + 6i + 9 }{1+ 9}\\ \\hspace{4cm}\\ \\because [i^2\\ =\\ -1]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\frac{8 + 6i}{10}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ Z = \\frac{4}{5}\\ +\\ i\\ \\frac{3}{5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Re(Z)\\ =\\ \\frac{4}{5};\\  Im(Z)\\ =\\ \\frac{3}{5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Polar\\ form\\ of\\ a\\ Complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"258\" height=\"259\" data-attachment-id=\"22439\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=22439\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2021\/09\/image-1.png?fit=258%2C259&amp;ssl=1\" data-orig-size=\"258,259\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2021\/09\/image-1.png?fit=258%2C259&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2021\/09\/image-1.png?resize=258%2C259&#038;ssl=1\" alt=\"\" class=\"wp-image-22439\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2021\/09\/image-1.png?w=258&amp;ssl=1 258w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2021\/09\/image-1.png?resize=200%2C200&amp;ssl=1 200w\" sizes=\"(max-width: 258px) 100vw, 258px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Z = r cos\u03b8 + i r sin \u03b8   = r ( cos\u03b8 + i  sin \u03b8 )\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Modulus\\ and\\ Amplitude\\ (or)\\ Argument\\ of\\ a\\ Complex\\ number}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ Z = a + ib\\ then\\ Modulus is |z| = \\sqrt{a^2 + b^2}\\ and\\ Amplitude\\ is\\ \u03b8  = tan^{-1} (\\frac{b}{a})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9:}\\ \\color {red} {Find\\ the\\ modulus\\ and\\ amplitude\\ of}\\ 1 + \\sqrt{3}i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ z = 1 + \\sqrt{3}i\\ = a\\ + ib\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a = 1,\\ b\\ = \\sqrt{3}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {T0\\ find\\ modulus}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = \\sqrt{a^2 + b^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1)^2 + (\\sqrt{3})^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 3}\\ =\\  \\sqrt{4}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {To\\ find\\ amplitude}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = tan^{-1} (\\frac{b}{a})\\ =\\ tan^{-1} \\frac{\\sqrt{3}}{1}\\ =\\  tan^{-1} \\sqrt{3}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = 60^0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\ \\color {red} {Find\\ the\\ modulus\\ and\\ amplitude\\ of}\\ 1 +  i\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ z = 1 + i\\ = a\\ + ib\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a = 1,\\ b\\ = 1\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {T0\\ find\\ modulus}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| = \\sqrt{a^2 + b^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{(1)^2 + (1)^2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = \\sqrt{1 + 1}\\ =\\  \\sqrt{2}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[|z| =\\sqrt{2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {To\\ find\\ amplitude}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = tan^{-1} (\\frac{b}{a})\\ =\\ tan^{-1} \\frac{1}{1}\\ =\\  tan^{-1} {1}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\u03b8  = 45^0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<h4 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c9ed0734ae2d1aa20627a443e074d5fb\">DE-MOIVRE\u2019S THEOREM<\/h4>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {De-Moivre\u2019s\\ Theorem( Statement\\ only)}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( i )\\ If\\ n\\ is\\ an\\ integer\\ positive\\ or\\ negative\\ then\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^n  =   cos\u2061\\ n \u03b8 +  i sin\u2061\\ n \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[( ii )\\ If\\  n\\  is\\  a\\  fraction,\\  then\\  cos\u2061\\ n \u03b8 +  i sin\u2061\\ n \u03b8\\ is\\ one\\ of\\ the\\ values\\ of\\  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^n\\] \n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Results}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1 )\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-n}  =   cos\u2061\\ n \u03b8-  i sin\u2061\\ n \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2)\\ \\frac{1}{cos\u2061\\ \u03b8 + i sin\u2061\\ \u03b8}   =   (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-1 } =   cos\u2061\\ \u03b8-  i sin\u2061\\ \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3)\\ \\frac{1}{cos\u2061\\ \u03b8 &#8211; i sin\u2061\\ \u03b8}   =   (cos\\ \u03b8 &#8211;  i sin\u2061\\ \u03b8 )^{-1 } =   cos\u2061\\ \u03b8 +  i sin\u2061\\ \u03b8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Note}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1)\\ (cos\\ \u03b8_1  + i sin\u2061\\ \u03b8_1) (cos\u2061\\ \u03b8_2  + i sin\\ \u03b8_2)  =  cos(\u2061\u03b8_1+\u03b8_2) + i sin\u2061 (\u03b8_1 +\u03b8_2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2)\\ \\frac{cos\u2061\\ \u03b8_1  + i sin\u2061\\ \u03b8_1}{cos\u2061\\ \u03b8_2  + i sin\u2061\\ \u03b8_2} = cos(\u03b8_1-\u03b8_2) + i sin\u2061 (\u03b8_1 -\u03b8_2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\ If\\  a =  cos\\ \u03b1 +  i sin\\ \u03b1,\\ b =  cos\\ \u03b2 +  i sin\\ \u03b2 ,\\ \\color {red} {find\\ \\frac{a}{b}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ a  =  cos\\ \u03b1 +  i sin\\ \u03b1,\\ b =  cos\\ \u03b2 +  i sin\\ \u03b2\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{a}{b}\\ =  \\frac{cos\\ \u03b1 +  i sin\\ \u03b1}{cos\\ \u03b2 +  i sin\\ \u03b2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{a}{b}\\ =  cos\u2061\\ (\u03b1 &#8211; \u03b2) +  i sin\u2061\\ (\u03b1 &#8211; \u03b2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ If\\  x =  cos\u2061\\ \\theta +  i sin\u2061\\ \\theta,\\ \\color {red} {find\\ the\\ value\\ of\\  x^m\\ +\\  \\frac{1}{x^m}}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ x =  cos\u2061\\ \\theta +  i sin\u2061\\ \\theta, \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^m =  (cos\u2061\\ \\theta +  i sin\u2061\\ \\theta,)^m =\\  cos\u2061\\ m\\ \\theta +  i sin\u2061\\ m\\ \\theta\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{x^m}\\ = \\frac{1}{cos\u2061\\ m\\ \\theta +  i sin\u2061\\ m\\ \\theta}\\ = cos\u2061\\ m\\ \\theta &#8211;  i sin\u2061\\ m\\ \\theta \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^m +\\ \\frac{1}{x^m}\\ =\\  cos\u2061\\ m \\theta +  i sin\u2061\\ m\\ \\theta\\ +\\ cos\u2061\\  m \\theta &#8211;  i sin\u2061\\ m\\ \\theta\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   2 cos\\ m \\theta\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{x^m\\ +\\  \\frac{1}{x^m}\\ =\\ 2\\ cos\\ m \\theta}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 13:}\\ \\color {red} {Simplify\\ :}\\ \\frac{cos\u2061\\ 5 \u03b8 &#8211;  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{cos\u2061\\ 5 \u03b8 &#8211;  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}=  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^-5} {(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^3} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-5 -3 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^-8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 8\u03b8 &#8211;  i sin\u2061\\ 8\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{cos\u2061\\ 5 \u03b8 &#8211;  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}\\ =\\ cos\\ 8\u03b8 &#8211;  i sin\u2061\\ 8\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14:}\\ \\color {red} {Simplify:}\\ \\frac{cos\u2061\\ 5 \u03b8 + i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 &#8211; i sin\u2061\\ 3 \u03b8}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{cos\u2061\\ 5 \u03b8 +  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}=  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^5} {(cos\u2061\\  \u03b8 &#8211;  i sin\u2061\\  \u03b8)^{-3}} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{5 +3 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 8\u03b8 +  i sin\u2061\\ 8\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15:}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{5 \\times -3}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{4 \\times 4}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{2 \\times 7}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{6 \\times -3 }}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-15}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{16}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{14}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-18}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-15 + 16 + 14 &#8211; 18 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^5\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{(cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^5\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^4} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 &#8211;  i sin\u2061\\  3\u03b8)^6}\\ =\\ cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16:}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^{-5}\\ (cos\u2061\\  2\u03b8 +  i sin\u2061\\  2\u03b8)^4} {(cos\u2061\\  4\u03b8 &#8211; i sin\u2061\\  4\u03b8)^{-2}\\ (cos\u2061\\  5\u03b8 &#8211;  i sin\u2061\\  5\u03b8)^3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{(cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^{-5}\\ (cos\u2061\\  2\u03b8 +  i sin\u2061\\  2\u03b8)^4} {(cos\u2061\\  4\u03b8 &#8211; i sin\u2061\\  4\u03b8)^{-2}\\ (cos\u2061\\  5\u03b8 &#8211;  i sin\u2061\\  5\u03b8)^3}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-5 \\times 3}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{4 \\times 2}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{-2 \\times -4}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{3 \\times -5 }}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-15}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^8} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^8\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-15}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-15 + 8 &#8211; 8 + 15 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 0\u03b8 +  i sin\u2061\\ 0\u03b8\\ = 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{(cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^{-5}\\ (cos\u2061\\  2\u03b8 +  i sin\u2061\\  2\u03b8)^4} {(cos\u2061\\  4\u03b8 &#8211; i sin\u2061\\  4\u03b8)^{-2}\\ (cos\u2061\\  5\u03b8 &#8211;  i sin\u2061\\  5\u03b8)^3}\\ =\\ 1}\\]<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Syllabus: DE-MOIVRE\u2019S THEOREM<\/p>\n","protected":false},"author":187055548,"featured_media":56633,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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