{"id":25671,"date":"2022-02-02T20:21:02","date_gmt":"2022-02-02T14:51:02","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=25671"},"modified":"2024-05-25T16:25:28","modified_gmt":"2024-05-25T10:55:28","slug":"de-moivres-theorem-excercise-problems","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=25671","title":{"rendered":"DE-MOIVRE&#8217;S THEOREM (Excercise Problems)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<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 {purple} {1.}\\ \\color {red} {Find\\ the\\ value\\ of}\\ (cos\\ 30^0  +  i sin\u2061\\ 30^0 )^3 \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ 30^0  +  i sin\u2061\\ 30^0 )^3\\ =\\ cos\\ 90^0  +  i sin\u2061\\ 90^0 \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   0 + i (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   i\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display h -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3583972194\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2.}\\ If\\  x =  cos\u2061\\ \\theta +  i sin\u2061\\ \\theta,\\ \\color {red} {find\\ 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\\ -\\  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\\ -\\  i sin\u2061\\ m \\theta\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   2cos\u2061\\ m \\theta\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3:}\\ \\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} {4.}\\ \\color {red} {Simplify}\\ \\frac{cos\u2061\\ 3 \u03b8 &#8211;  i sin\u2061\\ 3 \u03b8} {cos\u2061\\ 2 \u03b8 &#8211;  i sin\u2061\\ 2 \u03b8}\\  \\hspace{18cm}\\]<\/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} {cos\u2061\\ 2 \u03b8 &#8211;  i sin\u2061\\ 2 \u03b8}=  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-3}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{-2}} \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-3\\ +\\ 2 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-1}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ \u03b8 &#8211;  i sin\u2061\\ \u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5.}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  3\u03b8 + i sin\u2061\\  3\u03b8)^2\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^5} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^4\\ (cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^2}\\ \\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)^2\\ (cos\u2061\\  4\u03b8 +  i sin\u2061\\  4\u03b8)^5} {(cos\u2061\\  2\u03b8 + i sin\u2061\\  2\u03b8)^4\\ (cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^2}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{3 \\times 2}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{4 \\times 5}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{2 \\times 4}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{3 \\times 2 }}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^6\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{20}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^8\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^6}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{6\\ +\\  20\\ -\\ 8\\ -\\ 6 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{12}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 12\u03b8\\ +\\  i sin\u2061\\ 12\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6.}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  2\u03b8\\ -\\  i sin\u2061\\  2\u03b8)^4\\ (cos\u2061\\  4\u03b8\\ +  i sin\u2061\\  4\u03b8)^{-5}} {(cos\u2061\\  3\u03b8\\ +\\ i sin\u2061\\  3\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\\  2\u03b8\\ -\\  i sin\u2061\\  2\u03b8)^4\\ (cos\u2061\\  4\u03b8\\ +  i sin\u2061\\  4\u03b8)^{-5}} {(cos\u2061\\  3\u03b8\\ +\\ i sin\u2061\\  3\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)^{-2 \\times 4}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{4 \\times -5}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^{3 \\times 2}\\ (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)^{-8\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{-20}} {(cos\u2061\\  \u03b8 + i sin\u2061\\  \u03b8)^6\\ (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 )^{-8\\ -\\ 20\\ -\\ 6\\ -\\ 15 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{-49}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 49\u03b8\\ -\\  i sin\u2061\\ 49\u03b8\\  \\hspace{910cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/fRWh8VBWCh8\" title=\"DE-MOIVRE'S THEOREM - PART - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7.}\\ \\color {red} {Simplify\\ using\\ DeMoivre\u2019s\\ theorem:}\\ \\frac{(cos\u2061\\  2\u03b8 &#8211;  i sin\u2061\\  2\u03b8)^7\\ (cos\u2061\\  3\u03b8 +  i sin\u2061\\  3\u03b8)^{-5}} {(cos\u2061\\  4\u03b8 + i sin\u2061\\  4\u03b8)^2\\ (cos\u2061\\  5\u03b8 &#8211;  i sin\u2061\\  5\u03b8)^{-6}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Question\\ No.\\  2\\ Solution\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<p><a href=\"https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/\">https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/<\/a><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" 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href=\"https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/\">https:\/\/yanamtakshashila.com\/2021\/12\/16\/de-movires-theorem-revision\/<\/a><\/p>\n","protected":false},"author":187055548,"featured_media":0,"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":"","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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