{"id":25773,"date":"2022-02-05T11:58:02","date_gmt":"2022-02-05T06:28:02","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=25773"},"modified":"2022-02-06T22:15:55","modified_gmt":"2022-02-06T16:45:55","slug":"solutions-to-seminar-2-for-engineering-mathematics-i","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=25773","title":{"rendered":"SOLUTIONS TO SEMINAR &#8211; 2 FOR ENGINEERING MATHEMATICS &#8211; I"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group\\ 1:\\ If\\ Sin\\ A\\ =\\ \\frac{8}{17} \\  and\\ Sin\\ B\\ =\\ \\frac{5}{13},\\ Show\\ that\\ Sin(A\\ +\\ B)\\ =\\ \\frac{171}{221}\\ \\hspace{15cm}\\]<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 {black}{Solution:}\\ Given\\ Sin\\ A\\ =\\ \\frac{8}{17} \\  and\\ Cos\\ B\\ =\\ \\frac{5}{13}\\  \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Sin ( A + B )\\  =\\  Sin A\\  Cos B\\ +\\  Cos A\\  Sin B\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ ?\\ ,\\ Cos\\ B\\ = ?\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\  A\\ =\\ \\sqrt{1\\ -\\ Sin^2\\ A}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ (\\frac{8}{17})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{64}{289}}\\  \\hspace{10cm}\\]<\/div>\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\">\\[ =\\ \\sqrt{\\frac{289\\ -\\ 64}{289}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{225}{289}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ =\\ \\frac{15}{17}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\  B\\ =\\ \\sqrt{1\\ -\\ Sin^2\\ B}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ (\\frac{5}{13})^2}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{1\\ -\\ \\frac{25}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{169\\ -\\ 25}{169}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\sqrt{\\frac{25}{144}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ B\\ =\\ \\frac{5}{12}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin ( A + B )\\  =\\  (\\frac{8}{17})\\  (\\frac{12}{13})\\ +\\  (\\frac{15}{17})\\  (\\frac{5}{13})\\ \\hspace{10cm}\\]<\/div>\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\">\\[=\\ \\frac{96}{221}\\ +\\ \\frac{75}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{96\\ +\\ 75}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{171}{221}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group\\ 2:\\ If\\ Tan\\ A\\ =\\ \\frac{1}{2} \\  and\\ Tan\\ B\\ =\\ \\frac{1}{3},\\  find\\ the\\ value\\ of\\ tan(A\\ +\\ B)\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solution:\\ W.\\ K.\\ T\\ Tan(A + B)\\ =\\ \\frac{Tan A\\ +\\ Tan B}{1\\ -\\ Tan A\\ Tan B}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\frac{1}{2}\\ +\\ \\frac{1}{3}}{1\\ -\\ \\frac{1}{2}\\ \u00d7\\ \\frac{1}{3}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\frac{3\\ +\\ 2}{6}}{1\\ -\\ \\frac{1}{6}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\frac{5}{6}}{\\frac{6\\ -\\ 1}{6}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\frac{5}{6}}{\\frac{5}{6}}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Tan\\ (A\\ +\\ B)\\ =\\ 1\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group\\ 3:\\  If\\ Cos\\ \u03b8\\ =\\  \\frac{1}{3},\\ find\\ Cos\\ 3\u03b8\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ Given\\ Cos\\ \u03b8\\ =\\  \\frac{1}{3}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Cos\\ 3\\ \\theta\\  =\\ 4\\ Cos^3\\ \\theta\\ -\\ 3\\ Cos\\ \\theta\\ &#8211; \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4(\\frac{1}{3})^3\\ -\\ 3(\\frac{1}{3})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{4}{27}\\  -\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{4\\ -\\ 27}{27}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{-23}{27}\\ \\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<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" 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\">\\[Group\\ 4.\\  Prove\\ that\\  \\frac{Sin\\ 3\\ \\theta}{Sin\\ \\theta}\\ -\\ \\frac{Cos\\ 3\\ \\theta}{Cos\\ \\theta}\\ =\\  2\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\  L.\\ H.\\ S\\ =\\  \\frac{Sin\\ 3\\ \\theta}{Sin\\ \\theta}\\ -\\ \\frac{Cos\\ 3\\ \\theta}{Cos\\ \\theta}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\frac{3\\ Sin\\ \\theta\\  -\\ 4\\ Sin^3\\ \\theta}{Sin\\ \\theta}\\ -\\ \\frac{4\\ Cos^3\\ \\theta\\  -\\ 3\\ Cos\\ \\theta}{Cos\\ \\theta}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{Sin\\ \\theta(3\\  -\\ 4\\ Sin^2\\ \\theta)}{Sin\\ \\theta}\\ -\\ \\frac{Cos\\ \\theta(4\\ Cos^2\\ \\theta\\  -\\ 3)}{Cos\\ \\theta}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ -\\ 4\\ Sin^2\\ \\theta\\ -\\ (4\\ Cos^2\\ \\theta\\ -\\ 3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ -\\ 4\\ Sin^2\\ \\theta\\ -\\ 4\\ Cos^2\\ \\theta\\ +\\ 3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ -\\ 4(Sin^2\\ \\theta\\ +\\ Cos^2\\ \\theta)\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ -\\ 4(1)\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ -\\ 4\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{Sin\\ 3\\ \\theta}{Sin\\ \\theta}\\ -\\ \\frac{Cos\\ 3\\ \\theta}{Cos\\ \\theta}\\ =\\  2\\  \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group5.\\ If\\ a\\ =\\ Sin\\ x\\ +\\ Sin\\ y,\\ b\\ =\\ Cos\\ x\\ +\\ Cos\\ y,\\ Prove\\ that\\  a^2\\ +\\ b^2\\ =\\ 4\\ Cos^2\\ (\\frac{x\\ -\\ y}{2})\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ \\hspace{19cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Sin\\ A\\ +\\ Sin\\ B =\\ 2\\ Sin\\ (\\frac{A\\ +\\ B}{2})\\ Cos\\ (\\frac{A\\ -\\ B}{2})\\ and\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cos\\ A\\ +\\ Cos\\ B =\\ 2\\ Cos\\ (\\frac{A\\ +\\ B}{2})\\ Cos\\ (\\frac{A\\ -\\ B}{2})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ a^2\\ +\\ b^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ (Sin\\ x\\ +\\ Sin\\ y)^2\\ +\\ (Cos\\ x\\ +\\ Cos\\ y)^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ [2\\ Sin\\ (\\frac{x\\ +\\ y}{2})\\ Cos\\ (\\frac{x\\ -\\ y}{2})]^2\\ +\\ [2\\ Cos\\ (\\frac{x\\ +\\ y}{2})\\ Cos\\ (\\frac{x\\ -\\ y}{2})]^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ Sin^2\\ (\\frac{x\\ +\\ y}{2})\\ Cos^2\\ (\\frac{x\\ -\\ y}{2})\\ +\\ 4\\ Cos^2\\ (\\frac{x\\ +\\ y}{2})\\ Cos^2\\ (\\frac{x\\ -\\ y}{2}) \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ Cos^2\\ (\\frac{x\\ -\\ y}{2})[Sin^2\\ (\\frac{x\\ +\\ y}{2})\\ +\\  Cos^2\\ (\\frac{x\\ +\\ y}{2})]\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ Cos^2\\ (\\frac{x\\ -\\ y}{2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group\\ 6:\\ Show\\ that\\  Tan^{-1}\\ (\\frac{x\\ -\\ y}{1\\ +\\ x\\ y})\\ =\\ Tan^{-1}\\ x\\ &#8211; Tan^{-1}\\ y\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ \\hspace{19cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Tan^{-1}\\ (x)\\ =\\ A,\\  \\implies\\ x\\ =\\ Tan\\ A\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ Tan^{-1}\\ (y)\\ =\\ B,\\  \\implies\\ y\\ =\\ Tan\\ B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\ Tan(A &#8211; B)\\ =\\ \\frac{Tan A\\ -\\ Tan B}{1\\ +\\ Tan A\\ Tan B}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Tan(A &#8211; B)\\ =\\ \\frac{x\\ -\\ y}{1\\ +\\ x\\ y}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A &#8211; B\\ =\\ Tan^{-1}(\\frac{x\\ -\\ y}{1\\ +\\ x\\ y})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Tan^{-1}\\ x\\ &#8211; Tan^{-1}\\ y\\ =\\ Tan^{-1}\\ (\\frac{x\\ -\\ y}{1\\ +\\ x\\ y})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Group\\ 7:\\  Evaluate:\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{x^5\\ -\\ 3^5}{x^4\\ -\\ 3^4}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\  W.\\ K.\\ T\\ \\lim\\ _{x\\ \\to\\ a}\\ \\frac{x^n\\ -\\ a^n}{x\\ -\\ a}\\ =\\  n\\ a^{n\\ -\\ 1}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Multiply\\ and\\ divide\\ by\\ \u2018x &#8211; 3&#8217;\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\lim\\ _{x\\ \\to\\ 3}\\ \\frac{\\frac{x^5\\ -\\ 3^5}{x\\ -\\ 3}}{\\frac{x^4\\ -\\ 3^4}{x\\ -\\ 3}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 3^{5\\ -\\ 1}}{4\\ 3^{4\\ -\\ 1}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ 3^4}{4\\ (3^3)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (3^{4\\ -\\ 3})}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ (3)}{4}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{15}{4}\\ \\hspace{10cm}\\]<\/div>\n","protected":false},"excerpt":{"rendered":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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