{"id":243,"date":"2017-08-23T10:30:03","date_gmt":"2017-08-23T10:30:03","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=243"},"modified":"2017-09-22T16:18:03","modified_gmt":"2017-09-22T16:18:03","slug":"introduction-to-tangent-cotangent-secant-and-cosecant","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/introduction-to-tangent-cotangent-secant-and-cosecant\/","title":{"rendered":"Introduction to Tangent, Cotangent, Secant, and Cosecant"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/the-unit-circle\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/pythagorean-trigonometric-identities\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Introduction to Tangent, Cotangent, Secant, and Cosecant<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will cover what tangent, cotangent, secant, and cosecant are used for and how they are used. You\u00a0will also learn to use the basic trigonometric ratios that were mentioned in an earlier section to help you solve\u00a0for cotangent, secant, and cosecant.<\/p>\n<h4>Previously Covered:<\/h4>\n<p>There are six major <em><strong>trigonometric ratios <\/strong><\/em>that\u00a0can help you to solve for lengths of sides in right triangles. The\u00a0three that we have covered so far are as follows:<\/p>\n<ul>\n<li><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p1_clip_image003.gif\" width=\"127\" height=\"44\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p1_clip_image006.gif\" width=\"129\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p1_clip_image009.gif\" width=\"112\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<section>\n<h3>What are the tangent and cotangent?<\/h3>\n<p>The <abbr title=\"\">tangent<\/abbr> function was mentioned earlier when we first introduced\u00a0trigonometric ratios. The tangent function can be written as:<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image003.gif\" width=\"88\" height=\"41\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>This makes sense because, from the trigonometric ratios, we\u00a0know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image006.gif\" width=\"127\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image009.gif\" width=\"129\" height=\"44\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Since\u00a0<strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image011.gif\" width=\"88\" height=\"41\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><span style=\"text-decoration: none;\">,\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">w<\/span>e can\u00a0use substitution to find that:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image014.gif\" width=\"173\" height=\"72\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p align=\"LEFT\">which gives us:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image017.gif\" width=\"208\" height=\"48\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>And finally, after cancellation, we have:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image020.gif\" width=\"100\" height=\"44\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>.<\/p>\n<p>Since \u201c-toa,\u201d the trigonometric ratio from <em><span style=\"text-decoration: none;\">soh\u00a0cah toa,<\/span><\/em><span style=\"text-decoration: none;\"> is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image023.gif\" width=\"112\" height=\"44\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>,\u00a0we have just proven that<strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image025.gif\" width=\"88\" height=\"41\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0<\/strong>The\u00a0<abbr title=\"\">cotangent<\/abbr> is\u00a0<strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image028.gif\" width=\"87\" height=\"41\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/strong><\/p>\n<p>In terms of sine and cosine:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image031.gif\" width=\"120\" height=\"52\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image034.gif\" width=\"53\" height=\"41\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We mentioned earlier that there are six trigonometric ratios.\u00a0The cotangent is another one of the trigonometric ratios, and can\u00a0be written:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image037.gif\" width=\"88\" height=\"41\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image040.gif\" width=\"151\" height=\"75\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image043.gif\" width=\"184\" height=\"48\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image046.gif\" width=\"76\" height=\"44\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Tangent and cotangent functions are also referred to as <strong>ratio\u00a0identities<\/strong> since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image049.gif\" width=\"89\" height=\"41\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p2_clip_image052.gif\" width=\"89\" height=\"41\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>While sine and cosine can be considered the two most important\u00a0trigonometric functions (after all, every trigonometric function\u00a0can be written in terms of sine and cosine), it is still important\u00a0to recognize the tangent and cotangent. The tangent, in\u00a0particular, has some great uses that will be mentioned in a future\u00a0lesson.<\/p>\n<h3>Where are tangents and cotangents on a unit circle?<\/h3>\n<p>On the unit circle below, the tangent is located on segment\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s4_p3_html_m66540a80.gif\" width=\"27\" height=\"21\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Because you need to use an adjacent side, use triangle <em><span style=\"text-decoration: none;\">AFE.\u00a0<\/span><\/em>You know that, in triangle <em><span style=\"text-decoration: none;\">AFE,\u00a0<\/span><\/em>the side adjacent to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image003.gif\" width=\"15\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is 1.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image006.gif\" width=\"236\" height=\"44\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20003.JPG\" alt=\"Tan and cot on unit circle\" width=\"200\" height=\"173\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>On the unit circle above, the cotangent is segment\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s4_p3_html_534e007.gif\" width=\"29\" height=\"23\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Because you need to use an opposite side, use triangle <em><span style=\"text-decoration: none;\">ADG.\u00a0<\/span><\/em>You know that, in triangle <em><span style=\"text-decoration: none;\">ADG,\u00a0<\/span><\/em>the side opposite of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image008.gif\" width=\"15\" height=\"19\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>is 1.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image011.gif\" width=\"243\" height=\"44\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h4>What are the secant and cosecant?<\/h4>\n<p>The <abbr title=\"\">secant<\/abbr> is equal to one over the cosine, and can be written<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image014.gif\" width=\"88\" height=\"41\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>The trigonometric ratio for secant is<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image016.gif\" width=\"88\" height=\"41\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image019.gif\" width=\"151\" height=\"59\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image022.gif\" width=\"92\" height=\"44\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The <abbr title=\"\">cosecant<\/abbr> is similar to the secant, and can be written as<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image025.gif\" width=\"85\" height=\"41\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>The cosecant is the last important trigonometric ratio that you\u00a0will need to be familiar with. It can be expressed as<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image027.gif\" width=\"85\" height=\"41\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image030.gif\" width=\"149\" height=\"59\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image033.gif\" width=\"92\" height=\"44\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The cotangent, secant, and cosecant are also known as the\u00a0<abbr title=\"secant, cosecant, and cotangent\">reciprocal identities<\/abbr> because <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image036.gif\" width=\"93\" height=\"41\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image038.gif\" width=\"88\" height=\"41\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<\/strong>and \u00a0<strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p3_clip_image040.gif\" width=\"85\" height=\"41\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/strong><\/p>\n<h3>Where are the secant and cosecant on a unit circle?<\/h3>\n<p>On the unit circle below, the secant is the segment\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s4_p4_html_m2a45564b.gif\" width=\"27\" height=\"21\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20004.JPG\" alt=\" sec and cosec on unit circle\" width=\"200\" height=\"170\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Because you need to use an adjacent side, use triangle <em><span style=\"text-decoration: none;\">AEF.\u00a0<\/span><\/em>You know that, in triangle <em><span style=\"text-decoration: none;\">AEF,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">th<\/span>e side adjacent to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image003.gif\" width=\"15\" height=\"19\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is 1.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image006.gif\" width=\"252\" height=\"44\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>On the unit circle above, the cosecant is the segment\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s4_p4_html_23fb83cf.gif\" width=\"28\" height=\"23\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<blockquote><p>Because you need to use an opposite side, use triangle\u00a0<em><span style=\"text-decoration: none;\">ADG<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0<\/span>You know that, in triangle <em><u>ADG<\/u><\/em>, the side opposite\u00a0of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image008.gif\" width=\"15\" height=\"19\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is 1.<\/p><\/blockquote>\n<blockquote style=\"text-align: center;\"><p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image011.gif\" width=\"256\" height=\"44\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p><\/blockquote>\n<p>You should now have a working understanding of all six\u00a0trigonometric ratios.<\/p>\n<p>For a quick refresher, the trigonometric ratios are:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong><span style=\"text-decoration: none;\">1<\/span><\/strong><\/td>\n<td style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image014.gif\" width=\"127\" height=\"44\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><strong><span style=\"text-decoration: none;\">4<\/span><\/strong><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image017.gif\" width=\"112\" height=\"44\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong><span style=\"text-decoration: none;\">2<\/span><\/strong><\/td>\n<td style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image020.gif\" width=\"129\" height=\"44\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><strong><span style=\"text-decoration: none;\">5<\/span><\/strong><\/td>\n<td><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image023.gif\" width=\"128\" height=\"44\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong><span style=\"text-decoration: none;\">3<\/span><\/strong><\/td>\n<td style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image026.gif\" width=\"112\" height=\"44\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><strong><span style=\"text-decoration: none;\">6<\/span><\/strong><\/td>\n<td><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p4_clip_image029.gif\" width=\"128\" height=\"44\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<p>The <em><strong>ratio identities<\/strong><\/em> are:<\/p>\n<ul>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image003.gif\" width=\"88\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image006.gif\" width=\"88\" height=\"41\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<p>The\u00a0<em><strong>reciprocal\u00a0identities<\/strong><\/em> are:<\/p>\n<ul>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image009.gif\" width=\"88\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image012.gif\" width=\"85\" height=\"41\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image015.gif\" width=\"88\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<\/ul>\n<p>The <em><strong>new trigonometric ratios<\/strong><\/em> are:<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image018.gif\" width=\"112\" height=\"44\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image021.gif\" width=\"128\" height=\"44\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s4_p5_clip_image024.gif\" width=\"128\" height=\"44\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<\/ul>\n<p>All six trigonometric functions can be derived in terms of\u00a0sine and cosine.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/the-unit-circle\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/pythagorean-trigonometric-identities\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Introduction to Tangent, Cotangent, Secant, and Cosecant Objective In this lesson, you will cover what tangent, cotangent, secant, and cosecant are used for and how they are used. You\u00a0will also learn to use the basic trigonometric ratios that were mentioned in an earlier section to help you solve\u00a0for cotangent, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-243","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/243","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=243"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/243\/revisions"}],"predecessor-version":[{"id":801,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/243\/revisions\/801"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}