{"id":81,"date":"2005-05-23T21:42:29","date_gmt":"2005-05-23T21:42:29","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=81"},"modified":"2005-05-23T21:42:29","modified_gmt":"2005-05-23T21:42:29","slug":"more-trig-ratios","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=81","title":{"rendered":"More Trig Ratios"},"content":{"rendered":"<p>Students studying A level mathematics are expected to know exact values of a few trig ratios such as <img src='\/maths\/latexrender\/pictures\/506f0ee7aefa230bcebd8e931a6d88ef.gif' title='\\sin\\left(\\frac{\\pi}{3}\\right)=\\frac{\\sqrt{3}}{2}' alt='\\sin\\left(\\frac{\\pi}{3}\\right)=\\frac{\\sqrt{3}}{2}' align=absmiddle> (see <a href=\"http:\/\/sixthform.info\/maths\/index.php?p=18\" target=\"_blank\">Trig Ratios<\/a> posting). But a visit to <a href=\"http:\/\/mathworld.wolfram.com\" target=\"_blank\">Mathworld<\/a> reveals a whole world of fascinating values.<\/p>\n<p>Here are just a few of them (you may wish to try proving them)<\/p>\n<p><img src='\/maths\/latexrender\/pictures\/582104b66f6cfb8d988b27f2b7d58ea5.gif' title='\\cos\\left(\\dfrac{\\pi}{9}\\right)=2^{-\\frac{4}{3}}\\left(\\sqrt[3]{1+i\\sqrt{3}}+\\sqrt[3]{1-i\\sqrt{3}}\\right)' alt='\\cos\\left(\\dfrac{\\pi}{9}\\right)=2^{-\\frac{4}{3}}\\left(\\sqrt[3]{1+i\\sqrt{3}}+\\sqrt[3]{1-i\\sqrt{3}}\\right)' align=absmiddle><\/p>\n<p><img src='\/maths\/latexrender\/pictures\/bbb15599833cad833d5b64e2b441f5fa.gif' title='\\tan\\left(\\dfrac{3\\pi}{10}\\right)=\\frac{1}{5}\\sqrt{25+10\\sqrt{5}}' alt='\\tan\\left(\\dfrac{3\\pi}{10}\\right)=\\frac{1}{5}\\sqrt{25+10\\sqrt{5}}' align=absmiddle><\/p>\n<p><img src='\/maths\/latexrender\/pictures\/1177c922f9e2f36e2eaeb3f0b9ce069c.gif' title='\\tan\\left(\\dfrac{\\pi}{16}\\right)=\\sqrt{\\dfrac{2-\\sqrt{2+\\sqrt{2}}}{2+\\sqrt{2+\\sqrt{2}}}}' alt='\\tan\\left(\\dfrac{\\pi}{16}\\right)=\\sqrt{\\dfrac{2-\\sqrt{2+\\sqrt{2}}}{2+\\sqrt{2+\\sqrt{2}}}}' align=absmiddle><\/p>\n<p><img src='\/maths\/latexrender\/pictures\/d96c38bb044f64eab33ac3dacf4b476a.gif' title='\\sin\\left(\\dfrac{\\pi}{17}\\right)=\\frac{1}{8}\\left[34-2\\sqrt{17}-2\\sqrt{2}\\epsilon^*-2\\sqrt{68+12\\sqrt{17}+2\\sqrt{2}(\\sqrt{17}-1)\\epsilon^*-16\\sqrt{2}\\epsilon}\\:\\right]^{\\frac{1}{2}}' alt='\\sin\\left(\\dfrac{\\pi}{17}\\right)=\\frac{1}{8}\\left[34-2\\sqrt{17}-2\\sqrt{2}\\epsilon^*-2\\sqrt{68+12\\sqrt{17}+2\\sqrt{2}(\\sqrt{17}-1)\\epsilon^*-16\\sqrt{2}\\epsilon}\\:\\right]^{\\frac{1}{2}}' align=absmiddle> <\/p>\n<p>where <img src='\/maths\/latexrender\/pictures\/bfba5faacdbb7cd53cc53ce2bacfae64.gif' title='\\epsilon=\\sqrt{17+\\sqrt{17}},\\ \\epsilon^*=\\sqrt{17-\\sqrt{17}}' alt='\\epsilon=\\sqrt{17+\\sqrt{17}},\\ \\epsilon^*=\\sqrt{17-\\sqrt{17}}' align=absmiddle><\/p>\n<p><img src='\/maths\/latexrender\/pictures\/1dbea9a4d28105934ffd6a5f62668964.gif' title='\\sin\\left(\\dfrac{\\pi}{18}\\right)=\\frac{1}{2}\\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{2-\\dots}}}}' alt='\\sin\\left(\\dfrac{\\pi}{18}\\right)=\\frac{1}{2}\\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{2-\\dots}}}}' align=absmiddle> where the sequence of signs <img src='\/maths\/latexrender\/pictures\/920676ac308552b045a84e839ce67f2c.gif' title='+,\\;+,\\;-' alt='+,\\;+,\\;-' align=absmiddle> repeats with period 3<\/p>\n<p>As I said, fascinating! <\/p>\n<p><small>Thank goodness <img src='\/maths\/latexrender\/pictures\/c51d7e23458ca0e7373a8ed6ab56b2b9.gif' title='\\LaTeX' alt='\\LaTeX' align=absmiddle> can show these values easily!<\/small><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Students studying A level mathematics are expected to know exact values of a few trig ratios such as (see Trig Ratios posting). But a visit to Mathworld reveals a whole world of fascinating values. Here are just a few of them (you may wish to try proving them) where where the sequence of signs repeats [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-81","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/81","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=81"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/81\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=81"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=81"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=81"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}