{"id":782,"date":"2016-08-01T02:19:51","date_gmt":"2016-08-01T07:19:51","guid":{"rendered":"https:\/\/trigonography.com\/?p=782"},"modified":"2022-06-25T01:27:02","modified_gmt":"2022-06-25T06:27:02","slug":"angles-and-tangents-in-arithmetic-progression","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=782","title":{"rendered":"Angles and Tangents in Arithmetic Progression"},"content":{"rendered":"<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-anglesandtangentsinarithmeticprogression.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-anglesandtangentsinarithmeticprogression.png\" alt=\"trigonograph-anglesandtangentsinarithmeticprogression\" width=\"540\" height=\"360\"\/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center wp-block-paragraph\" style=\"font-size:16px\">For First-Quadrant angles \\(\\alpha\\) and \\(\\gamma\\;\\) (and \\(\\beta\\) and \\(\\beta^\\prime\\)), such that<br>\\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\) and \\(\\tan\\alpha\\), \\(\\tan\\beta^\\prime\\), \\(\\tan\\gamma\\) are both arithmetic progressions:<\/p>\n\n\n\n<div class=\"eqn-box\" style=\"margin-bottom: 20pt; font-size:12pt\">$$\\begin{align}<br>2\\,\\beta\\; &amp;\\;=\\; \\alpha + \\gamma \\\\[4pt]<br>2\\tan\\beta^\\prime &amp;\\;=\\; \\tan\\alpha + \\tan\\gamma<br>\\end{align}\\quad\\Longrightarrow\\quad<br>\\begin{array}{c}<br>\\beta \\;\\leq\\; \\beta^\\prime \\\\<br>\\text{with equality when and only when} \\\\<br>\\alpha = \\beta = \\beta^\\prime = \\gamma<br>\\end{array}$$<\/div>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\" style=\"font-size:16px\"><em style=\"color: gray;\">Motivated by <a href=\"http:\/\/math.stackexchange.com\/q\/1876639\/409\">this question<\/a> on the Mathematics Stack Exchange.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>For First-Quadrant angles \\(\\alpha\\) and \\(\\gamma\\;\\) (and \\(\\beta\\) and \\(\\beta^\\prime\\)), such that\\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\) and \\(\\tan\\alpha\\), \\(\\tan\\beta^\\prime\\), \\(\\tan\\gamma\\) are both arithmetic progressions: $$\\begin{align}2\\,\\beta\\; &amp;\\;=\\; \\alpha + \\gamma \\\\[4pt]2\\tan\\beta^\\prime &amp;\\;=\\; \\tan\\alpha + \\tan\\gamma\\end{align}\\quad\\Longrightarrow\\quad\\begin{array}{c}\\beta \\;\\leq\\; \\beta^\\prime \\\\\\text{with equality when and only when} \\\\\\alpha = \\beta = \\beta^\\prime = \\gamma\\end{array}$$ Motivated by this question on the Mathematics Stack [&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-782","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/782","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=782"}],"version-history":[{"count":10,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions"}],"predecessor-version":[{"id":1193,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/782\/revisions\/1193"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=782"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=782"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=782"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}