{"id":25946,"date":"2022-02-06T12:00:41","date_gmt":"2022-02-06T06:30:41","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=25946"},"modified":"2024-05-26T13:58:13","modified_gmt":"2024-05-26T08:28:13","slug":"inverse-trigonometric-functions-excercise","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=25946","title":{"rendered":"INVERSE TRIGONOMETRIC FUNCTIONS (Excercise)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<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} {Prove\\ that}\\  Sin\\ 50^{0}\\ -\\  Sin\\ 70^{0}\\ +\\ Sin\\ 10^{0}\\ =\\ 0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ W.\\ K.\\ T.\\  Sin\\ A\\ -\\ Sin\\ B =\\ 2\\ Cos\\ (\\frac{A\\ +\\ B}{2})\\ Sin\\ (\\frac{A\\ -\\ B}{2})\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ Sin\\ 50^{0}\\ -\\  Sin\\ 70^{0}\\ +\\ Sin\\ 10^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\ Cos\\ (\\frac{50^{0}\\ +\\ 70^{0}}{2})\\ Sin\\ (\\frac{50^{0}\\ -\\ 70^{0}}{2})\\ +\\ Sin\\ 10^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2\\ Cos\\ 60^{0}\\ Sin\\ (-10^{0})\\  +\\  Sin\\ 10^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2\\ \u00d7\\  \\frac{1}{2}\\ (-\\ Sin\\ 10^{0}\\  +\\  Sin\\ 10^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2}.\\ \\color {red} {Prove\\ that}\\  Cos\\ 80^{0}\\ +\\  Cos\\ 40^{0}\\ -\\ Cos\\ 20^{0}\\ =\\ 0\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ W.\\ K.\\ T.\\  Cos\\ A\\ +\\ Cos\\ B =\\ 2\\ Cos\\ (\\frac{A\\ +\\ B}{2})\\ Cos\\ (\\frac{A\\ -\\ B}{2})\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ Cos\\ 80^{0}\\ +\\  Cos\\ 40^{0}\\ -\\ Cos\\ 20^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\ Cos\\ (\\frac{80^{0}\\ +\\ 40^{0}}{2})\\ Cos\\ (\\frac{80^{0}\\ -\\ 40^{0}}{2})\\ -\\ Cos\\ 20^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2\\ Cos\\ 60^{0}\\ Cos\\ (-20^{0})\\  -\\  Cos\\ 20^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2\\ \u00d7\\  \\frac{1}{2}\\ (Cos\\ 20^{0}\\  -\\  Cos\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 0\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3}.\\ \\color {red} {Prove\\ that}\\  Sin\\ 20^{0}\\   Sin\\ 40^{0}\\ Sin\\ 60^{0}\\ Sin\\ 80^{0}\\ =\\ \\frac{3}{16}\\ \\hspace{15cm}\\] <\/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\">\\[W.\\ K.\\ T\\ Sin\\ (A\\ +\\ B)Sin\\ (A\\ -\\ B)\\ =\\ Sin^2\\ A\\ -\\  Sin^2\\ B\\ and\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Sin\\ 3A\\  =\\ 3\\ Sin\\ A\\  -\\ 4\\ Sin^3\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ Sin\\ 20^{0}\\   Sin\\ 40^{0}\\ Sin\\ 60^{0}\\ Sin\\ 80^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ \\frac{\\sqrt{3}}{2}\\ Sin\\ 20^{0}\\   Sin\\ 40^{0}\\ Sin\\ 80^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{2}\\ Sin\\ 20^{0}\\   Sin\\ (60^{0}\\ -\\ 20^{0}) Sin\\ ((60^{0}\\ +\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{2}\\ Sin\\ 20^{0}\\   (Sin^2\\ 60^{0}\\ -\\ Sin^2\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{2}\\ Sin\\ 20^{0}\\   ((\\frac{\\sqrt{3}}{2})^2\\ -\\ Sin^2\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{2}\\ Sin\\ 20^{0}\\   (\\frac{3}{4}\\ -\\ Sin^2\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{8}\\ Sin\\ 20^{0}\\   (3\\ -\\ 4\\ Sin^2\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{8}(3\\ Sin\\ 20^{0}\\ -\\ 4\\ Sin^3\\ 20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{8}\\ Sin\\ 3(20^{0})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{8}\\ Sin\\ 60^{0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{\\sqrt{3}}{8}\\ (\\frac{\\sqrt{3}}{2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3}{16}\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4.}\\ \\color {red} {Prove:}\\ Tan^{-1}\\ (\\frac{3x\\ -\\ x^3}{1\\ -\\ 3\\ x^2})\\ =\\ 3\\ Tan^{-1}\\ x\\ \\hspace{15cm}\\] <\/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\">\\[Put\\ x\\ =\\ Tan\\ \u03b8,\\ \\implies\\ \u03b8\\ =\\ Tan^{-1}\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[L.\\ H.\\ S\\ =\\ Tan^{-1}\\ (\\frac{3x\\ -\\ x^3}{1\\ -\\ 3\\ x^2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ Tan^{-1}\\ (\\frac{3\\ Tan\\ \u03b8\\ -\\ Tan^3\\ \u03b8}{1\\ -\\ 3\\ Tan^2\\ \u03b8})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ Tan^{-1}\\ (Tan\\ 3\u03b8)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ \u03b8\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 3\\ Tan^{-1}\\ x\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5.}\\ \\color {red} {Show\\ that}\\ Tan^{-1}\\ (\\frac{4}{3})\\ &#8211; Tan^{-1}\\ (\\frac{1}{7})\\ =\\ \\frac{\\pi}{4}\\ \\hspace{15cm}\\] <\/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\">\\[L.\\ H.\\ S\\ =\\ Tan^{-1}\\ (\\frac{4}{3})\\ &#8211; Tan^{-1}\\ (\\frac{1}{7})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{4}{3}\\ -\\ \\frac{1}{7}}{1\\ +\\ \\frac{4}{3}\\ \u00d7\\ \\frac{1}{7}})\\ \\hspace{2cm}\\ \\because\\ Tan^{-1}\\ x\\ &#8211; Tan^{-1}\\ y\\ =\\ Tan^{-1}\\ (\\frac{x\\ -\\ y}{1\\ +\\ x\\ y})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{28\\ -\\ 3}{21}}{\\frac{21\\ +\\ 4}{21}})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{25}{21}}{\\frac{25}{21}})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ Tan^{-1}\\ (1)\\ =\\ \\frac{\\pi}{4}\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6.}\\ \\color {red} {Show\\ that}\\  2\\ Tan^{-1}\\ (\\frac{2}{3})\\ =\\ Tan^{-1}\\ (\\frac{12}{5})\\ \\hspace{15cm}\\] <\/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\">\\[L.\\ H.\\ S\\ =\\ 2\\ Tan^{-1}\\ (\\frac{2}{3})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{2}{3})\\ +\\ Tan^{-1}\\ (\\frac{2}{3})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{2}{3}\\ +\\ \\frac{2}{3}}{1\\ -\\ \\frac{2}{3}\\ \u00d7\\ \\frac{2}{3}})\\ \\hspace{2cm}\\ \\because\\ Tan^{-1}\\ x\\ + Tan^{-1}\\ y\\ =\\ Tan^{-1}\\ (\\frac{x\\ +\\ y}{1\\ -\\ x\\ y})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{2\\ +\\ 2}{3}}{\\frac{9\\ -\\ 4}{9}})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  Tan^{-1}\\ (\\frac{\\frac{4}{3}}{\\frac{5}{9}})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ Tan^{-1}\\ (\\frac{12}{5})\\ =\\ R.H.S\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"1366\" height=\"768\" src=\"https:\/\/www.youtube.com\/embed\/NPdKUbwIckA\" title=\"INVERSE TRIGONOMETRIC FUNCTIONS - PART - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" 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