{"id":59,"date":"2005-01-08T21:42:36","date_gmt":"2005-01-08T21:42:36","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=59"},"modified":"2005-01-08T21:42:36","modified_gmt":"2005-01-08T21:42:36","slug":"spherical-trigonometry","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=59","title":{"rendered":"Spherical Trigonometry"},"content":{"rendered":"<p>The idea that the angles of a triangle add up to 180&deg; is so well engrained that it comes as a shock to some students that it isn&#8217;t true in other geometries. These geometries don&#8217;t have to be obscure or abstract since the angles of a spherical triangle drawn on the earth&#8217;s surface (assuming it is a sphere) always add up to more than 180&deg; by an amount proportional to its area.<\/p>\n<p>It&#8217;s a shame that like much else, spherical trigonometry has long since disappeared from the A level syllabus. The 2-dimensional sine rule <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/580ac6a2156cbc6c3feb43eaa0e50744.gif' title='\\displaystyle \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}' alt='\\displaystyle \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}' align=absmiddle><\/ul>\n<p> is (or ought to be 8-)) well-known but how many are aware of the 3-dimensional version <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/01301dfca5fec70c3190640c24f5d662.gif' title='\\displaystyle \\frac{\\sin a}{\\sin A}=\\frac{\\sin b}{\\sin B}=\\frac{\\sin c}{\\sin C}' alt='\\displaystyle \\frac{\\sin a}{\\sin A}=\\frac{\\sin b}{\\sin B}=\\frac{\\sin c}{\\sin C}' align=absmiddle> <\/ul>\n<p>or the fact that the great circle distance (the shortest distance) <img src='\/maths\/latexrender\/pictures\/7b8b965ad4bca0e41ab51de7b31363a1.gif' title='n' alt='n' align=absmiddle> in nautical miles between points with latitude &#038; longitude <img src='\/maths\/latexrender\/pictures\/3c998fd9a68da396f4eb54672773b8bc.gif' title='\\left(\\alpha_1,\\beta_1\\right),\\left(\\alpha_2,\\beta_2\\right)' alt='\\left(\\alpha_1,\\beta_1\\right),\\left(\\alpha_2,\\beta_2\\right)' align=absmiddle> is given by <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/ee652e4d3c2ab55af2a0c41814900c65.gif' title='\\cos n = \\sin \\alpha_1 \\sin \\alpha_2 + \\cos \\alpha_1 \\cos \\alpha_2 \\cos \\left(\\beta_1-\\beta_2\\right)' alt='\\cos n = \\sin \\alpha_1 \\sin \\alpha_2 + \\cos \\alpha_1 \\cos \\alpha_2 \\cos \\left(\\beta_1-\\beta_2\\right)' align=absmiddle><\/ul>\n<p>where <img src='\/maths\/latexrender\/pictures\/7b8b965ad4bca0e41ab51de7b31363a1.gif' title='n' alt='n' align=absmiddle> is measured in minutes? You could use sites such as <a href=\"http:\/\/www.wcrl.ars.usda.gov\/cec\/java\/lat-long.htm\" target=\"_blank\">Surface Distance Between Two Points of Latitude and Longitude<\/a> but it&#8217;s not the same as sitting down and proving the general result.<\/p>\n<p>Without this introduction to three dimensions how is one going to start visualising geometry in four or more dimensions? Of course, reading Flatland would be an excellent start. <a href=\"http:\/\/www.phoenix-library.org\" target=\"_blank\">Phoenix-Library<\/a> has some excellent versions of this book in a number of online formats.<\/p>\n<p>This is perhaps a suitable place to publicise my all time favourite puzzle:<\/p>\n<ul>A hunter leaves his house one morning and walks one mile due south. He then walks one mile due west and shoots a bear, before walking a mile due north back to his house. What colour is the bear?<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The idea that the angles of a triangle add up to 180&deg; is so well engrained that it comes as a shock to some students that it isn&#8217;t true in other geometries. These geometries don&#8217;t have to be obscure or abstract since the angles of a spherical triangle drawn on the earth&#8217;s surface (assuming it [&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-59","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\/59","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=59"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/59\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=59"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=59"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=59"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}