{"id":275,"date":"2016-06-04T05:42:59","date_gmt":"2016-06-04T10:42:59","guid":{"rendered":"https:\/\/trigonography.com\/?p=275"},"modified":"2022-06-25T01:54:30","modified_gmt":"2022-06-25T06:54:30","slug":"compound-hypotenuse","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=275","title":{"rendered":"Compound Hypotenuse"},"content":{"rendered":"<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/06\/trigonograph-compoundhypotenuse.png\"><img decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/06\/trigonograph-compoundhypotenuse.png\" alt=\"trigonograph-compoundhypotenuse\"\/><\/a><\/figure>\n<\/div>\n\n\n<div class=\"eqn-box\" style=\"width:640px; margin:auto; margin-bottom: 20pt; font-size:12pt;\">$$\\left(\\;\\cos 2\\alpha + \\cos 2\\beta + 2 \\cos( \\alpha+\\beta ) \\;\\right)^2 \\;\\;+\\;\\; \\left(\\;\\sin 2\\alpha + \\sin 2\\beta + 2 \\sin( \\alpha+\\beta )\\; \\right)^2 \\\\[6pt]<br>= \\;\\;\\left(\\;2\\;\\left(\\;1 + \\cos(\\alpha-\\beta)\\;\\right)\\;\\right)^2<br>$$<\/div>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\" style=\"font-size:16px\"><b>Note:<\/b> The simplest form of the hypotenuse length is, of course, \\( 4 \\cos^2\\left(\\frac{\\alpha-\\beta}{2}\\right) \\).<br>Augmenting this trigonograph to illustrate introduces too much clutter for my taste.<br>I&#8217;d rather find a more-direct alternative construction.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\" style=\"font-size:16px\"><em style=\"color: gray;\"><br><br><br>Motivated by <a href=\"http:\/\/math.stackexchange.com\/q\/1811867\/409\">this question<\/a> on the Mathematics Stack Exchange.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>$$\\left(\\;\\cos 2\\alpha + \\cos 2\\beta + 2 \\cos( \\alpha+\\beta ) \\;\\right)^2 \\;\\;+\\;\\; \\left(\\;\\sin 2\\alpha + \\sin 2\\beta + 2 \\sin( \\alpha+\\beta )\\; \\right)^2 \\\\[6pt]= \\;\\;\\left(\\;2\\;\\left(\\;1 + \\cos(\\alpha-\\beta)\\;\\right)\\;\\right)^2$$ Note: The simplest form of the hypotenuse length is, of course, \\( 4 \\cos^2\\left(\\frac{\\alpha-\\beta}{2}\\right) \\).Augmenting this trigonograph to illustrate introduces too much clutter for my taste.I&#8217;d rather find [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-275","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/275","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=275"}],"version-history":[{"count":9,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/275\/revisions"}],"predecessor-version":[{"id":1201,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/275\/revisions\/1201"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=275"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=275"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}