{"id":13050,"date":"2014-02-07T15:52:18","date_gmt":"2014-02-07T15:52:18","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=13050"},"modified":"2026-03-28T10:47:42","modified_gmt":"2026-03-28T15:47:42","slug":"defined-and-reciprocal-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/","title":{"rendered":"Defined and Reciprocal Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_c0aVQ8SNTYw\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_c0aVQ8SNTYw\" data-source-videoID=\"c0aVQ8SNTYw\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Defined and Reciprocal Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Defined and Reciprocal Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_c0aVQ8SNTYw:hover {cursor:pointer;} img#videoThumbnailImage_c0aVQ8SNTYw {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-perfect-squares-and-square-roots-64c14790d341a.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_c0aVQ8SNTYw\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_c0aVQ8SNTYw\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Defined and Reciprocal Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_c0aVQ8SNTYw\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_c0aVQ8SNTYw\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_c0aVQ8SNTYw\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Y1Q_Function() {\n  var x = document.getElementById(\"Y1Q\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"Y1Q_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"Y1Q\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Common_Trig_Functions\" class=\"smooth-scroll\">Common Trig Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Reciprocal_Functions\" class=\"smooth-scroll\">Reciprocal Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Importance_of_Knowing_Reciprocals\" class=\"smooth-scroll\">Importance of Knowing Reciprocals<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_Problem\" class=\"smooth-scroll\">Example Problem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Defined_and_Reciprocal_Function_Practice_Questions\" class=\"smooth-scroll\">Defined and Reciprocal Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on reciprocal trig functions!<\/p>\n<p>Before we dive into what reciprocal trig functions are, let\u2019s quickly review what normal trig functions are. <\/p>\n<h2><span id=\"Common_Trig_Functions\" class=\"m-toc-anchor\"><\/span>Common Trig Functions<\/h2>\n<p>\nThe three most common trig functions are <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/sine\/\">sine<\/a>, <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/cosine\/\">cosine<\/a>, and <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/finding-tangent\/\">tangent<\/a>. When given a right triangle, sine is the ratio of the triangle\u2019s opposite to its hypotenuse, cosine is the ratio of its adjacent to its hypotenuse, and tangent is the ratio of its opposite to its adjacent. We remember these by using the phrase \u201cSOH-CAH-TOA\u201d.<\/p>\n<p>When you take the reciprocal of an expression, you divide one by the expression. A super simple way of doing this is to turn your expression into a fraction and then flip it. So if our expression is 2, we would turn that into 2 over 1, which makes it easy to see that the reciprocal is one half. The reciprocal of three fifths is five thirds. And the reciprocal of \\(2x\\) is \\(\\frac{1}{2x}\\).<\/p>\n<h2><span id=\"Reciprocal_Functions\" class=\"m-toc-anchor\"><\/span>Reciprocal Functions<\/h2>\n<p>\nWith reciprocal trig functions, we have the same thing, except the reciprocals are given new names.<\/p>\n<h3><span id=\"Cosecant\" class=\"m-toc-anchor\"><\/span>Cosecant<\/h3>\n<p>\nLet\u2019s start with the reciprocal of sine, cosecant. Cosecant is the same as 1 over sine, or when you are given a triangle, the ratio of the triangle\u2019s hypotenuse to its opposite. This is just the flipped version of SOH, sine equals opposite over hypotenuse.<\/p>\n<p>Take a look at this example.<\/p>\n<p>Given this triangle, what is the cosecant of theta?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-01-e1741868457687.webp\" alt=\"A right triangle with sides labeled 5, 12, and 13. The angle opposite the side labeled 5 has a theta symbol. The triangle is shaded blue.\" width=\"451\" height=\"272\" class=\"aligncenter size-full wp-image-248851\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-01-e1741868457687.webp 451w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-01-e1741868457687-300x181.webp 300w\" sizes=\"auto, (max-width: 451px) 100vw, 451px\" \/><\/p>\n<p>First, we need to remember that cosecant is the reciprocal of sine, so that means we need to figure out what sine is and flip it. Sine is opposite over hypotenuse, so in this case, \\(\\frac{5}{13}\\). Then we flip that to get cosecant of theta, which is \\(\\frac{13}{5}\\).<\/p>\n<h3><span id=\"Secant\" class=\"m-toc-anchor\"><\/span>Secant<\/h3>\n<p>\nThe second reciprocal function we are going to look at is secant, the reciprocal of cosine. Secant is the ratio of a triangle\u2019s hypotenuse to its adjacent, or \\(\\frac{H}{A}\\), which, if you think back to our CAH from SOH-CAH-TOA is the flipped version of cosine equals \\(\\frac{A}{H}\\).<\/p>\n<p>Let\u2019s try an example problem using secant.<\/p>\n<p>Find the missing side using secant.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-02-e1741868740555.webp\" alt=\"A right triangle with sides labeled 5 and 29, angle labeled 47 degrees, and the base labeled x.\" width=\"435\" height=\"273\" class=\"aligncenter size-full wp-image-248854\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-02-e1741868740555.webp 435w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-02-e1741868740555-300x188.webp 300w\" sizes=\"auto, (max-width: 435px) 100vw, 435px\" \/><\/p>\n<p>If we set up our equation, we get \\(\\text{sec}47 =\\frac{29}{x}\\) since our angle is measured 47 degrees and our hypotenuse is 29.<\/p>\n<p>This can be rearranged so that \\(x\\) is by itself, which looks like this: \\(x=\\frac{29}{\\text{sec}47}\\).<\/p>\n<p>When we plug this into our calculator, making sure our calculator is in degrees mode instead of radians, we get that \\(x\\) is approximately 19.78.<\/p>\n<h3><span id=\"Cotangent\" class=\"m-toc-anchor\"><\/span>Cotangent<\/h3>\n<p>\nWe have one more reciprocal function to cover and that is the reciprocal of tangent, which is cotangent. Cotangent is 1 over tangent, or the ratio of a triangle\u2019s adjacent over its hypotenuse.<\/p>\n<p>Using the things we have learned from our examples on cosecant and secant, I want to try another one, this time using cotangent.<\/p>\n<p>Given this triangle, what is the cotangent of theta?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-03-e1741868785402.webp\" alt=\"A right triangle with sides labeled 9, 12, and 15. Angle theta is marked at the bottom right.\" width=\"437\" height=\"277\" class=\"aligncenter size-full wp-image-248845\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-03-e1741868785402.webp 437w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-03-e1741868785402-300x190.webp 300w\" sizes=\"auto, (max-width: 437px) 100vw, 437px\" \/><\/p>\n<p>We know that tangent is opposite over adjacent, so since cotangent is the reciprocal of tangent, it must be the adjacent over opposite. This gives us the ratio \\(\\frac{12}{9}\\), so the cotangent of theta is \\(\\frac{12}{9}\\).<\/p>\n<h2><span id=\"Importance_of_Knowing_Reciprocals\" class=\"m-toc-anchor\"><\/span>Importance of Knowing Reciprocals<\/h2>\n<p>\nNow that we have covered all the reciprocal trig functions, I want to mention something important about them. Reciprocal trig functions pop up all the time in homework or test problems, so it is important to recognize the names and recall what function they go to. However, as you may have noticed from our examples, when solving triangles by yourself, you almost always are able to use the original, or defined, trig function instead of its reciprocal.<\/p>\n<p>Most commonly you will only use sine, cosine, and tangent, but as I said, the reciprocal trig functions will come up on homework and tests occasionally, so it is very important to recognize them and understand what they mean even if you do not use them in your own personal application of trigonometry.<\/p>\n<h2><span id=\"Example_Problem\" class=\"m-toc-anchor\"><\/span>Example Problem<\/h2>\n<p>\nBefore we end this video, I want to give you one more example to try on your own. Once I put it up, pause the video and see what answers you can come up with. Then you can check them with mine and see how you did.<\/p>\n<p>Given this triangle, find the cosecant of theta, the secant of theta, and the cotangent of theta.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-04-e1741868760291.webp\" alt=\"A right triangle with sides labeled 7, 24, and hypotenuse 25. An angle marked with a theta symbol is adjacent to the side labeled 24.\" width=\"435\" height=\"274\" class=\"aligncenter size-full wp-image-248848\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-04-e1741868760291.webp 435w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/03\/Defined-and-Reciprocal-Functions-04-e1741868760291-300x189.webp 300w\" sizes=\"auto, (max-width: 435px) 100vw, 435px\" \/><\/p>\n<p>Remember, cosecant is the reciprocal of sine so we are looking for the hypotenuse over the opposite. The cosecant of theta is \\(\\frac{25}{7}\\). Secant is the reciprocal of cosine, so we need the hypotenuse over the adjacent, which gives us that secant of theta is equal to \\(\\frac{25}{24}\\). And finally, cotangent is the reciprocal of tangent so we need the adjacent over the opposite, which leaves us with cotangent of theta is \\(\\frac{24}{7}\\).<\/p>\n<p>I hope this review of reciprocal trig functions was helpful. Thanks for watching and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Defined_and_Reciprocal_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Defined and Reciprocal Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following ratios describes the secant function \\(\\sec(\\theta)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(\\large{\\frac{\\text{opp}}{\\text{adj}}}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(\\large{\\frac{\\text{hyp}}{\\text{opp}}}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">\\(\\large{\\frac{\\text{hyp}}{\\text{adj}}}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-4\">\\(\\large{\\frac{\\text{opp}}{\\text{hyp}}}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The secant function is the reciprocal of the cosine function, and \\(\\cos=\\frac{\\text{adj}}{\\text{hyp}}\\), which we can recall easily from SOH-CAH-TOA. To take the reciprocal, we simply flip this fraction to get the solution:<\/p>\n<p style=\"text-align: center\">\\(\\sec=\\dfrac{\\text{hyp}}{\\text{adj}}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following is the correct ratio for the cosecant function, \\(\\csc(\u03b8)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(\\large{\\frac{\\text{opp}}{\\text{hyp}}}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">\\(\\large{\\frac{\\text{hyp}}{\\text{opp}}}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(\\large{\\frac{\\text{hyp}}{\\text{adj}}}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\large{\\frac{\\text{adj}}{\\text{opp}}}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Cosecant is the reciprocal of the sine function. Now, recall that \\(\\sin(\\theta)=\\frac{\\text{opp}}{\\text{hyp}}\\). To find the ratio for \\(\\csc(\\theta)\\), we simply need to flip the sine ratio, which will give us \\(\\frac{\\text{hyp}}{\\text{opp}}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nObserve the triangle below and calculate the value of \\(\\cot(\\theta)\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Defined-and-reciprocal-function-triangle-1.svg\" alt=\"A right triangle with side lengths of 6 cm, 11 cm, and 13 cm. The right angle is opposite the 13 cm side, and angle \u03b8 is adjacent to the 13 cm side.\" width=\"326\" height=\"167\" class=\"aligncenter size-full wp-image-275152\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">\\(\\large{\\frac{11}{6}}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-2\">\\(\\large{\\frac{6}{13}}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\(\\large{\\frac{13}{6}}\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\(\\large{\\frac{6}{11}}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The cotangent function is the reciprocal of the tangent, so instead of opposite over adjacent (\u201cTOA\u201d), we will take adjacent over opposite. The length of the triangle side next to (adjacent to) \\(\\theta\\) is 11&nbsp;cm, while the length across from \\(\\theta\\) (opposite) is 6 cm.<\/p>\n<p>Therefore, our solution is \\(\\large{\\frac{11}{6}}\\normalsize{ \\approx 1.833}\\). <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nObserve the following triangle, and use the secant function to determine side length \\(s\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Defined-and-reciprocal-function-triangle-2.svg\" alt=\"A right triangle with one angle labeled 58\u00b0, hypotenuse labeled 7.5 in, and one leg labeled s.\" width=\"318\" height=\"137\" class=\"aligncenter size-full wp-image-275149\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">2.75 in<\/div><div class=\"PQ\"  id=\"PQ-4-2\">3.6 in<\/div><div class=\"PQ\"  id=\"PQ-4-3\">4.25 in<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">3.97 in<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Once again, \\(\\sec(\\theta)=\\frac{\\text{hyp}}{\\text{adj}}\\). Even though we are only given the length of one side, we can solve for the adjacent side length using a little algebra. Simply plug in the values of \\(\\theta\\) and the hypotenuse as follows:<\/p>\n<p style=\"text-align:center;\">\\(\\sec(58\u00b0)=\\dfrac{7.5}{s}\\)<\/p>\n<p>Rearranging, we can get \\(s\\) by itself:<\/p>\n<p style=\"text-align:center;\">\\(s=\\dfrac{7.5}{\\sec(58\u00b0)}\\)<\/p>\n<p>Using a calculator, we can determine that \\(\\sec(58\u00b0) \\approx 1.887\\). From here:<\/p>\n<p style=\"text-align:center;\">\\(s=\\dfrac{7.5}{1.887\u2026} \\approx 3.97\\text{ in}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe value of the sine function evaluated at \\(\\frac{\\pi}{3}\\) radians is \\(\\sin(\\frac{\\pi}{3})=\\frac{\\sqrt{3}}{2}\\). Determine the value for \\(\\csc(\\frac{\\pi}{3})\\), in radians.<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">\\(\\large{\\frac{2\\sqrt{3}}{3}}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(\\large{\\frac{\\sqrt{3}}{2}}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(\\large{\\frac{\\sqrt{2}}{3}}\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(\\large{\\frac{3}{\\pi}}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Recall that throughout this lesson, we have been referring to \\(\\sec\\), \\(\\csc\\), and \\(\\cot\\) as reciprocal functions. Of course, we have been jumping straight to thinking of cosecant as the ratio \\(\\frac{\\text{hyp}}{\\text{opp}}\\).<\/p>\n<p>However, for this example we are not given side length information, so we need to examine the problem from a different perspective.<\/p>\n<p>Consider again the nature of reciprocals: In the same way that \\(\\frac{1}{x}\\) is the reciprocal of \\(\\frac{x}{1}\\), the reciprocal of \\(\\sin(\\theta)\\) is \\(\\frac{1}{\\sin(\\theta)}\\). This fraction is equivalent to cosecant!<\/p>\n<ul>\n<li style=\"margin-bottom: 12px;\">\\(\\large{\\frac{1}{x}}\\) is the reciprocal of \\(\\large{\\frac{x}{1}}\\)<\/li>\n<li>\\(\\large{\\frac{1}{\\sin(\\theta)}}\\) is the reciprocal of \\(\\sin(\\theta)\\)<\/li>\n<\/ul>\n<p>Since \\(\\csc\\) is the reciprocal of \\(\\sin\\), \\(\\csc(\\theta)=\\large{\\frac{1}{\\sin(\\theta)}}\\).<\/p>\n<p>Now, all we need to do to calculate \\(\\csc(\\frac{\\pi}{3})\\) is divide 1 by \\(\\sin(\\frac{\\pi}{3})\\).<\/p>\n<p>Since we were given the value of \\(\\sin(\\frac{\\pi}{3})\\), \\(\\frac{\\sqrt{3}}{2}\\), we can get the solution by flipping it, then rationalizing the denominator.<\/p>\n<p style=\"text-align:center;\">\\(\\dfrac{2}{\\sqrt{3}}\\times\\dfrac{\\sqrt{3}}{3}=\\dfrac{2\\sqrt{3}}{3}\\)<\/p>\n<p>The solution can also be attained by entering \\(1\\div\\sin(\\frac{\\pi}{3})\\) into your calculator.<\/p>\n<p style=\"text-align:center;\">\\(1\\div\\sin(\\dfrac{\\pi}{3})=\\dfrac{2\\sqrt{3}}{3}\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/trigonometry\/\">Return to Trigonometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Trigonometry Videos<\/p>\n","protected":false},"author":1,"featured_media":187115,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-13050","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-trigonometry-miscellaneous-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.9 - aioseo.com -->\n\t<meta name=\"description\" content=\"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" href=\"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO Pro (AIOSEO) 4.9.9\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"|\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"Defined and Reciprocal Functions (Video &amp; Practice Questions)\" \/>\n\t\t<meta property=\"og:description\" content=\"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/\" \/>\n\t\t<meta property=\"og:image\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp\" \/>\n\t\t<meta property=\"og:image:secure_url\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp\" \/>\n\t\t<meta property=\"og:image:width\" content=\"825\" \/>\n\t\t<meta property=\"og:image:height\" content=\"464\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2014-02-07T15:52:18+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2026-03-28T15:47:42+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n\t\t<meta name=\"twitter:title\" content=\"Defined and Reciprocal Functions (Video &amp; Practice Questions)\" \/>\n\t\t<meta name=\"twitter:description\" content=\"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!\" \/>\n\t\t<meta name=\"twitter:image\" content=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Mometrix-Academy-FI.png\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.mometrix.com\\\/academy\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#listItem\",\"name\":\"Defined and Reciprocal Functions\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#listItem\",\"position\":2,\"name\":\"Defined and Reciprocal Functions\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy#listItem\",\"name\":\"Home\"}}]},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#organization\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/wp-content\\\/uploads\\\/2022\\\/06\\\/Mometrix-Test-Prep-Logo-min.png\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#organizationLogo\",\"width\":557,\"height\":242,\"caption\":\"Mometrix Test Preparation logo\"},\"image\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#organizationLogo\"}},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#webpage\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/\",\"name\":\"Defined and Reciprocal Functions (Video & Practice Questions)\",\"description\":\"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!\",\"inLanguage\":\"en-US\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#website\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#breadcrumblist\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/wp-content\\\/uploads\\\/2023\\\/07\\\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#mainImage\",\"width\":825,\"height\":464,\"caption\":\"Thumbnail for the Defined and Reciprocal Functions video.\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/defined-and-reciprocal-functions\\\/#mainImage\"},\"datePublished\":\"2014-02-07T15:52:18-06:00\",\"dateModified\":\"2026-03-28T10:47:42-05:00\"},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#website\",\"url\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/\",\"inLanguage\":\"en-US\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.mometrix.com\\\/academy\\\/#organization\"}}]}\n\t\t<\/script>\n\t\t<!-- All in One SEO Pro -->\r\n\t\t<title>Defined and Reciprocal Functions (Video &amp; Practice Questions)<\/title>\n\n","aioseo_head_json":{"title":"Defined and Reciprocal Functions (Video & Practice Questions)","description":"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!","canonical_url":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"BreadcrumbList","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#breadcrumblist","itemListElement":[{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy#listItem","position":1,"name":"Home","item":"https:\/\/www.mometrix.com\/academy","nextItem":{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#listItem","name":"Defined and Reciprocal Functions"}},{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#listItem","position":2,"name":"Defined and Reciprocal Functions","previousItem":{"@type":"ListItem","@id":"https:\/\/www.mometrix.com\/academy#listItem","name":"Home"}}]},{"@type":"Organization","@id":"https:\/\/www.mometrix.com\/academy\/#organization","url":"https:\/\/www.mometrix.com\/academy\/","logo":{"@type":"ImageObject","url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/06\/Mometrix-Test-Prep-Logo-min.png","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#organizationLogo","width":557,"height":242,"caption":"Mometrix Test Preparation logo"},"image":{"@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#organizationLogo"}},{"@type":"WebPage","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#webpage","url":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/","name":"Defined and Reciprocal Functions (Video & Practice Questions)","description":"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!","inLanguage":"en-US","isPartOf":{"@id":"https:\/\/www.mometrix.com\/academy\/#website"},"breadcrumb":{"@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#breadcrumblist"},"image":{"@type":"ImageObject","url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp","@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#mainImage","width":825,"height":464,"caption":"Thumbnail for the Defined and Reciprocal Functions video."},"primaryImageOfPage":{"@id":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/#mainImage"},"datePublished":"2014-02-07T15:52:18-06:00","dateModified":"2026-03-28T10:47:42-05:00"},{"@type":"WebSite","@id":"https:\/\/www.mometrix.com\/academy\/#website","url":"https:\/\/www.mometrix.com\/academy\/","inLanguage":"en-US","publisher":{"@id":"https:\/\/www.mometrix.com\/academy\/#organization"}}]},"og:locale":"en_US","og:site_name":"|","og:type":"article","og:title":"Defined and Reciprocal Functions (Video &amp; Practice Questions)","og:description":"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!","og:url":"https:\/\/www.mometrix.com\/academy\/defined-and-reciprocal-functions\/","og:image":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp","og:image:secure_url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp","og:image:width":"825","og:image:height":"464","article:published_time":"2014-02-07T15:52:18+00:00","article:modified_time":"2026-03-28T15:47:42+00:00","twitter:card":"summary_large_image","twitter:title":"Defined and Reciprocal Functions (Video &amp; Practice Questions)","twitter:description":"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!","twitter:image":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Mometrix-Academy-FI.png"},"aioseo_meta_data":{"post_id":"13050","title":"Defined and Reciprocal Functions (Video &amp; Practice Questions)","description":"The reciprocal functions of sine, cosine, and tangent, are cosecant, secant, and cotangent, respectively. Learn how to work these in context with examples here!","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"featured","og_image_url":"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-defined-and-reciprocal-functions-64c150fcd5168.webp","og_image_width":"825","og_image_height":"464","og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"WebPage","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":"{\"article\":{\"articleType\":\"BlogPosting\"},\"course\":{\"name\":\"\",\"description\":\"\",\"provider\":\"\"},\"faq\":{\"pages\":[]},\"product\":{\"reviews\":[]},\"recipe\":{\"ingredients\":[],\"instructions\":[],\"keywords\":[]},\"software\":{\"reviews\":[],\"operatingSystems\":[]},\"webPage\":{\"webPageType\":\"WebPage\"}}","pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"seo_analyzer_scan_date":null,"breadcrumb_settings":null,"limit_modified_date":false,"open_ai":"{\"title\":{\"suggestions\":[],\"usage\":0},\"description\":{\"suggestions\":[],\"usage\":0}}","ai":{"faqs":[],"keyPoints":[],"titles":[],"descriptions":[],"socialPosts":{"email":[],"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"created":"2021-02-18 17:07:38","updated":"2026-03-28 16:32:25"},"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/13050","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=13050"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/13050\/revisions"}],"predecessor-version":[{"id":280598,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/13050\/revisions\/280598"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/187115"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=13050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}