{"id":22679,"date":"2021-10-13T17:44:08","date_gmt":"2021-10-13T12:14:08","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=22679"},"modified":"2024-07-05T21:48:44","modified_gmt":"2024-07-05T16:18:44","slug":"unit-ii-complex-numbers-2","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=22679","title":{"rendered":"DE-MOIVRE&#8217;S THEOREM (Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {De-Moivre\u2019s\\ Theorem( Statement\\ only)}:\\ \\hspace{20cm}\\] <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\">\\[( 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<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\">\\[\\color {purple} {Example:\\ 1}\\ \\color {red} {Simplify}\\ (cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ =\\ 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<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{(cos\\ 15^0  +  i sin\u2061\\ 15^0 )^6\\ =\\ i}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2}:\\ \\color {red} {Simplify:}\\  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ =\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ \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 )^{2 +3 }\\ \\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{(cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^2\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ =\\ cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3}:\\ \\color {red} {Simplify:}\\  (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ (cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8 )\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ (cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ (cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8 )\\ =\\ (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^3\\ (cos\\ \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 )^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{(cos\\ 3\u03b8 +  i sin\u2061\\ 3\u03b8 )\\ (cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8 )\\ =\\ cos\\ 5\u03b8 +  i sin\u2061\\ 5\u03b8}\\]<\/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\">\\[\\color {purple} {Example\\ 4:}\\ If\\  a =  cos\u2061\\ x +  i sin\u2061\\ x,\\ b =  cos\u2061\\ y +  i sin\u2061\\ y,\\ \\color {red} {find\\ ab\\ and\\ \\frac{1}{ab}}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ a =  cos\u2061\\ x +  i sin\u2061\\ x,\\ b =  cos\u2061\\ y +  i sin\u2061\\ y\\ \\hspace{16cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ab =  (cos\u2061\\ x +  i sin\u2061\\ x)\\ (cos\u2061\\ y +  i sin\u2061\\ y)\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ab\\ =  cos\u2061\\ (x + y) +  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{ab}\\ =  \\frac{1}{cos\u2061\\ (x + y) +  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{1}{ab}\\ =  cos\u2061\\ (x + y) &#8211;  i sin\u2061\\ (x + y)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ If\\  a =  cos\\ \u03b1 +  i sin\\ \u03b1,\\ b =  cos\\ \u03b2 +  i sin\\ \u03b2 ,\\ \\color {red} {find\\ ab\\ &#8211; \\frac{1}{ab}}\\ \\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\">\\[ab =  (cos\\ \u03b1 +  i sin\\ \u03b1)\\ (cos\\ \u03b2 +  i sin\\ \u03b2)\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ab\\ =  cos\u2061\\ (\u03b1 + \u03b2) +  i sin\u2061\\ (\u03b1 + \u03b2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{ab}\\ =  \\frac{1}{cos\u2061\\ (\u03b1 + \u03b2) +  i sin\u2061\\ (\u03b1 + \u03b2)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{ab}\\ =  cos\u2061\\ (\u03b1 + \u03b2) &#8211;  i sin\u2061\\ (\u03b1 + \u03b2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ab\\ &#8211; \\frac{1}{ab}  =  cos\u2061\\ (\u03b1 + \u03b2) +  i sin\u2061\\ (\u03b1 + \u03b2)\\ &#8211; [ cos\u2061\\ (\u03b1 + \u03b2) &#8211;  i sin\u2061\\ (\u03b1 + \u03b2)] \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  =  cos\u2061\\ (\u03b1 + \u03b2) +  i sin\u2061\\ (\u03b1 + \u03b2)\\ &#8211;  cos\u2061\\ (\u03b1 + \u03b2) + i sin\u2061\\ (\u03b1 + \u03b2) \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   2i sin\u2061\\ (\u03b1 + \u03b2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{ab\\ &#8211; \\frac{1}{ab}\\ =\\  2i sin\u2061\\ (\u03b1 + \u03b2)}\\ \\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; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6565358754\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ If\\  x =  cos\u2061\\ \u03b1 +  i sin\u2061\\ \u03b1,\\ \\color {red} {find\\ the\\ value\\ of\\  x^2\\ +\\  \\frac{1}{x^2}}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ x =  cos\u2061\\ \u03b1 +  i sin\u2061\\ \u03b1 \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2 =  (cos\u2061\\ \u03b1 +  i sin\u2061\\ \u03b1)^2 =\\  cos\u2061\\ 2\u03b1 +  i sin\u2061\\ 2\u03b1 \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{1}{x^2}\\ = \\frac{1}{cos\u2061\\ 2\u03b1 +  i sin\u2061\\ 2\u03b1}\\ = cos\u2061\\ 2\u03b1 &#8211;  i sin\u2061\\ 2\u03b1  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2 +\\ \\frac{1}{x^2}\\ =\\  cos\u2061\\ 2\u03b1 +  i sin\u2061\\ 2\u03b1\\ +cos\u2061\\ 2\u03b1 &#8211;  i sin\u2061\\ 2\u03b1\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=   2cos\u2061\\ 2\u03b1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{x^2\\ +\\  \\frac{1}{x^2}\\ =\\ 2cos\u2061\\ 2\u03b1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 7:}\\ \\color {red} {Simplify\\ :}\\ \\frac{cos\u2061\\ 5 \u03b8 +  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 +  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 )^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{cos\u2061\\ 5 \u03b8 +  i sin\u2061\\ 5 \u03b8} {cos\u2061\\ 3 \u03b8 +  i sin\u2061\\ 3 \u03b8}\\ =\\ cos\\ 2\u03b8 +  i sin\u2061\\ 2\u03b8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\ \\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\">\\[\\hspace{18cm}\\ Feb\\ 2022\\ April\\ 2024\\]<\/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\\ 9:}\\ \\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\\ 10:}\\ \\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\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/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<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11.}\\ \\color {red} {Find\\ the\\ value\\ of}\\ \\frac{(cos\u2061\\  x\\ -\\  i sin\u2061\\  x)^3\\ (cos\u2061\\  3x\\ +\\  i sin\u2061\\  3x)^5} {(cos\u2061\\  2x\\ &#8211; i sin\u2061\\  2x)^5\\ (cos\u2061\\  5x\\ +  i sin\u2061\\  5x)^7}\\ \\color {red} {when\\ x\\ =\\ \\frac{2\\pi}{13}}\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ June\\ 2022\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\frac{(cos\u2061\\  x\\ -\\  i sin\u2061\\  x)^3\\ (cos\u2061\\  3x\\ +\\  i sin\u2061\\  3x)^5} {(cos\u2061\\  2x\\ &#8211; i sin\u2061\\  2x)^5\\ (cos\u2061\\  5x\\ +  i sin\u2061\\  5x)^7}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  x +  i sin\u2061\\  x)^{3 \\times -1}\\ (cos\u2061\\  x +  i sin\u2061\\  x)^{5 \\times 3}} {(cos\u2061\\  x + i sin\u2061\\  x)^{5 \\times -2}\\ (cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^{5 \\times 7}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{(cos\u2061\\  x +  i sin\u2061\\  x)^{-3}\\ (cos\u2061\\  x +  i sin\u2061\\  x)^{15}} {(cos\u2061\\  x + i sin\u2061\\  x)^{-10}\\ (cos\u2061\\  x +  i sin\u2061\\  x)^{35}}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ x +  i sin\u2061\\ x )^{-3 + 15 + 10 &#8211; 35 }\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ x +  i sin\u2061\\ x)^{13}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 13x\\  +\\  i sin\u2061\\ 13x\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 13(\\frac{2\\pi}{13})\\  +\\  i sin\u2061\\ 13(\\frac{2\\pi}{13})\\  \\hspace{2cm}\\ when\\ x\\ =\\ \\frac{2\\pi}{13}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  cos\\ 2\\ \\pi\\  +\\  i sin\u2061\\ 2\\ \\pi\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  1\\  \\hspace{2cm}\\ \\because\\ cos\\ n\\ \\pi\\ = {-\\ 1}^n\\ and\\ sin\\ n\\ \\pi\\ =\\ 0\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\frac{(cos\u2061\\  x\\ -\\  i sin\u2061\\  x)^3\\ (cos\u2061\\  3x\\ +\\  i sin\u2061\\  3x)^5} {(cos\u2061\\  2x\\ &#8211; i sin\u2061\\  2x)^5\\ (cos\u2061\\  5x\\ +  i sin\u2061\\  5x)^7}\\ =\\ 1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ \\color {red} {Prove\\ that:}\\  [\\frac{cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8} {sin\u2061\\  \u03b8 &#8211;  i cos\\  \u03b8}]^4\\ = 1\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ LHS =  [\\frac{cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8} {sin\u2061\\  \u03b8 &#8211;  i cos\\  \u03b8}]^4\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ [\\frac{cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8} {-i(cos\u2061\\  \u03b8  +  i sin\\  \u03b8)}]^4\\ \\hspace{5cm}\\ \\because sin\u2061\\  \u03b8 &#8211;  i cos\\  \u03b8\\ =\\ -i(cos\u2061\\  \u03b8  +  i sin\\  \u03b8)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  [\\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^4} {(-i)^4(cos\u2061\\  \u03b8  +  i sin\\  \u03b8)^4}]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^4} {(1)(cos\u2061\\  \u03b8  +  i sin\\  \u03b8)^4}\\ \\hspace{5cm}\\ \\because (-i)^4\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =  \\frac{(cos\u2061\\  \u03b8 +  i sin\u2061\\  \u03b8)^4} {(cos\u2061\\  \u03b8  +  i sin\\  \u03b8)^4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  (cos\\ \u03b8 +  i sin\u2061\\ \u03b8 )^{4 -4 }\\ \\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 +  i sin\u2061\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  1\\ + i(0)\\ \\hspace{5cm}\\ \\because cos\\ 0\\ =\\ 1,\\ sin\\ 0\\ =\\ 0\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=  1\\ =\\ RHS\\ \\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=\"fluid\" data-ad-layout-key=\"-6x+db-3g-5s+t1\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3292591486\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-vivid-purple-color has-text-color\">Exercise Problems<\/h3>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/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<script async=\"\" src=\"https:\/\/cse.google.com\/cse.js?cx=2f7b32630c6dfd4fb\"><\/script>\n<div class=\"gcse-search\"><\/div>\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 {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 {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<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display s -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4398623749\" 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- 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 {purple} {6.}\\ \\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<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display V -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3566826177\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Exercise Problems<\/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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