{"id":6585,"date":"2026-07-13T13:56:14","date_gmt":"2026-07-13T18:56:14","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=6585"},"modified":"2026-07-13T13:56:33","modified_gmt":"2026-07-13T18:56:33","slug":"pythagorean-identities","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/pythagorean-identities\/","title":{"rendered":"Pythagorean Identities"},"content":{"rendered":"<p>*There are three so-called \u201cPythagorean Identities\u201d that can be used to simplify expressions containing trigonometric\u00a0functions. The three Pythagorean identities are:<\/p>\n<p style=\"text-align: center;\">sin<sup>2<\/sup>x + cos<sup>2<\/sup>x = 1<br \/><br \/>1 + cot<sup>2<\/sup>x = csc<sup>2<\/sup>x<br \/><br \/>tan<sup>2<\/sup>x + 1 = sec<sup>2<\/sup>x<\/p>\n<p style=\"text-align: left;\">*These identities can be used to determine function values. <br \/><strong>Example:<\/strong> If the cot x = <span style=\"display: inline-block; text-align: center; vertical-align: middle;\"> <span style=\"display: block; border-bottom: 1px solid #000; line-height: .9;\"> \u221a3 <\/span> <span style=\"display: block; line-height: .9;\"> 2 <\/span> <\/span>, then what is the value of csc x if the angle is in Quadrant 3?<\/p>\n<p style=\"text-align: left;\">Using the second Pythagorean identity, we substitute the given value for cot x.<\/p>\n\n\n<div style=\"text-align:center; font-size:22px; line-height:2.2;\">\n\n1 +\n\n(\n\n<span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n    <span style=\"display:block; border-bottom:1px solid #000; line-height:.9;\">\n        \u221a3\n    <\/span>\n    <span style=\"display:block; line-height:.9;\">\n        2\n    <\/span>\n<\/span>\n\n)\n\n<sup>2<\/sup>\n\n&nbsp;=&nbsp;\n\ncsc<sup>2<\/sup>x\n\n<br><br>\n\n1 +\n\n<span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n    <span style=\"display:block; border-bottom:1px solid #000; line-height:.9;\">\n        3\n    <\/span>\n    <span style=\"display:block; line-height:.9;\">\n        4\n    <\/span>\n<\/span>\n\n&nbsp;=&nbsp;\n\ncsc<sup>2<\/sup>x\n\n<br><br>\n\n<span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n    <span style=\"display:block; border-bottom:1px solid #000; line-height:.9;\">\n        7\n    <\/span>\n    <span style=\"display:block; line-height:.9;\">\n        4\n    <\/span>\n<\/span>\n\n&nbsp;=&nbsp;\n\ncsc<sup>2<\/sup>x\n\n<br><br>\n\n<span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n    <span style=\"display:block; border-bottom:1px solid #000; line-height:.9;\">\n        \u2212\u221a7\n    <\/span>\n    <span style=\"display:block; line-height:.9;\">\n        2\n    <\/span>\n<\/span>\n\n&nbsp;=&nbsp;\n\ncsc x\n\n&nbsp;&nbsp;&nbsp;&nbsp;<br>\n\n<span style=\"font-size:20px;\">\n(The csc function is negative in quadrant 3.)\n<\/span>\n\n<\/div>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"455\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\" alt=\"91ԭƬ Learning Center Logo\" class=\"wp-image-4893\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg 602w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image-300x227.jpeg 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/a><figcaption class=\"wp-element-caption\">For more help, contact The Learning Center<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>*There are three so-called \u201cPythagorean Identities\u201d that can be used to simplify expressions containing trigonometric\u00a0functions. The three Pythagorean identities are: sin2x + cos2x = 1 1 + cot2x = csc2x tan2x + 1 = sec2x *These identities can be used to determine function values. Example: If the cot x = \u221a3 2 , then what [&hellip;]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-6585","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=6585"}],"version-history":[{"count":1,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6585\/revisions"}],"predecessor-version":[{"id":6586,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6585\/revisions\/6586"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6585"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}