{"id":3231,"date":"2025-08-15T23:38:17","date_gmt":"2025-08-15T23:38:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3231"},"modified":"2026-03-24T17:11:06","modified_gmt":"2026-03-24T17:11:06","slug":"the-other-trigonometric-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions-apply-it\/","title":{"raw":"The Other Trigonometric Functions: Apply It","rendered":"The Other Trigonometric Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\r\n \t<li>Use properties of even and odd trigonometric functions.<\/li>\r\n \t<li>Recognize and use fundamental identities.<\/li>\r\n \t<li>Evaluate trigonometric functions with a calculator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Radio Tower Guy Wire Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\"><img class=\"alignleft\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/db\/KBRC_antenna_tower_guy_wires.JPG\/330px-KBRC_antenna_tower_guy_wires.JPG?20120729131325\" alt=\"File:KBRC antenna tower guy wires.JPG\" width=\"196\" height=\"451\" \/>A radio transmission tower stands vertically with guy wires attached at various points to provide stability. We'll use all six trigonometric functions to analyze the angles, distances, and tensions in these support structures.<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">A guy wire creates an angle whose terminal side passes through the point [latex]\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)[\/latex] on the unit circle. Find all six trigonometric functions for this angle.\r\n\r\n[reveal-answer q=\"629045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"629045\"]From the coordinates on the unit circle: [latex]\\cos t = x = \\frac{1}{2}[\/latex] [latex]\\sin t = y = \\frac{\\sqrt{3}}{2}[\/latex] Now find the remaining four functions: [latex]\\begin{aligned} \\tan t &amp;= \\frac{\\sin t}{\\cos t} = \\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} = \\frac{\\sqrt{3}}{2} \\cdot \\frac{2}{1} = \\sqrt{3} \\end{aligned}[\/latex] [latex]\\begin{aligned} \\sec t &amp;= \\frac{1}{\\cos t} = \\frac{1}{\\frac{1}{2}} = 2 \\end{aligned}[\/latex] [latex]\\begin{aligned} \\csc t &amp;= \\frac{1}{\\sin t} = \\frac{1}{\\frac{\\sqrt{3}}{2}} = \\frac{2}{\\sqrt{3}} = \\frac{2\\sqrt{3}}{3} \\end{aligned}[\/latex] [latex]\\begin{aligned} \\cot t &amp;= \\frac{\\cos t}{\\sin t} = \\frac{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}} = \\frac{1}{2} \\cdot \\frac{2}{\\sqrt{3}} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3} \\end{aligned}[\/latex]\r\n\r\n&nbsp;\r\n<p class=\"whitespace-normal break-words\">[latex]\\sin t = \\frac{\\sqrt{3}}{2}, \\cos t = \\frac{1}{2}, \\tan t = \\sqrt{3}, \\sec t = 2, \\csc t = \\frac{2\\sqrt{3}}{3}, \\cot t = \\frac{\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">A guy wire is attached to the tower at an angle of [latex]\\frac{5\\pi}{4}[\/latex] radians from the positive x-axis. Find all six trigonometric functions for this angle.[reveal-answer q=\"633463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"633463\"]First, determine the quadrant. Since [latex]\\frac{5\\pi}{4}[\/latex] is between [latex]\\pi[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex], the angle is in Quadrant III. Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &amp;= \\frac{5\\pi}{4} - \\pi \\ &amp;= \\frac{5\\pi}{4} - \\frac{4\\pi}{4} \\ &amp;= \\frac{\\pi}{4} \\end{aligned}[\/latex] For the reference angle [latex]\\frac{\\pi}{4}[\/latex], we know: [latex]\\cos\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}, \\sin\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}[\/latex] In Quadrant III, both x and y coordinates are negative, so both sine and cosine are negative. However, when both are negative, tangent and cotangent become positive (negative divided by negative).\r\n<p class=\"whitespace-normal break-words\">[latex]\\cos\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}, \\sin\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]\\tan\\left(\\frac{5\\pi}{4}\\right) = \\frac{-\\frac{\\sqrt{2}}{2}}{-\\frac{\\sqrt{2}}{2}} = 1[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]\\sec\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{-\\frac{\\sqrt{2}}{2}} = -\\frac{2}{\\sqrt{2}} = -\\sqrt{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]\\csc\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{-\\frac{\\sqrt{2}}{2}} = -\\sqrt{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]\\cot\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{1} = 1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A measurement device records an angle of [latex]-\\frac{\\pi}{3}[\/latex] radians. If [latex]\\tan\\left(\\frac{\\pi}{3}\\right) = \\sqrt{3}[\/latex] and [latex]\\sec\\left(\\frac{\\pi}{3}\\right) = 2[\/latex], find [latex]\\tan\\left(-\\frac{\\pi}{3}\\right)[\/latex] and [latex]\\sec\\left(-\\frac{\\pi}{3}\\right)[\/latex].[reveal-answer q=\"636739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"636739\"]Use the properties of even and odd functions: Even functions: [latex]\\cos(-t) = \\cos t[\/latex] and [latex]\\sec(-t) = \\sec t[\/latex] Odd functions: [latex]\\sin(-t) = -\\sin t[\/latex], [latex]\\tan(-t) = -\\tan t[\/latex], [latex]\\csc(-t) = -\\csc t[\/latex], [latex]\\cot(-t) = -\\cot t[\/latex] Since tangent is an odd function: [latex]\\tan\\left(-\\frac{\\pi}{3}\\right) = -\\tan\\left(\\frac{\\pi}{3}\\right) = -\\sqrt{3}[\/latex] Since secant is an even function: [latex]\\sec\\left(-\\frac{\\pi}{3}\\right) = \\sec\\left(\\frac{\\pi}{3}\\right) = 2[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">If [latex]\\sin t = -\\frac{3}{5}[\/latex] and [latex]t[\/latex] is in Quadrant III, find the values of the other five trigonometric functions.<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\n<li>Use properties of even and odd trigonometric functions.<\/li>\n<li>Recognize and use fundamental identities.<\/li>\n<li>Evaluate trigonometric functions with a calculator.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Radio Tower Guy Wire Analysis<\/h2>\n<p class=\"whitespace-normal break-words\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/d\/db\/KBRC_antenna_tower_guy_wires.JPG\/330px-KBRC_antenna_tower_guy_wires.JPG?20120729131325\" alt=\"File:KBRC antenna tower guy wires.JPG\" width=\"196\" height=\"451\" \/>A radio transmission tower stands vertically with guy wires attached at various points to provide stability. We&#8217;ll use all six trigonometric functions to analyze the angles, distances, and tensions in these support structures.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">A guy wire creates an angle whose terminal side passes through the point [latex]\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)[\/latex] on the unit circle. Find all six trigonometric functions for this angle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q629045\">Show Solution<\/button><\/p>\n<div id=\"q629045\" class=\"hidden-answer\" style=\"display: none\">From the coordinates on the unit circle: [latex]\\cos t = x = \\frac{1}{2}[\/latex] [latex]\\sin t = y = \\frac{\\sqrt{3}}{2}[\/latex] Now find the remaining four functions: [latex]\\begin{aligned} \\tan t &= \\frac{\\sin t}{\\cos t} = \\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} = \\frac{\\sqrt{3}}{2} \\cdot \\frac{2}{1} = \\sqrt{3} \\end{aligned}[\/latex] [latex]\\begin{aligned} \\sec t &= \\frac{1}{\\cos t} = \\frac{1}{\\frac{1}{2}} = 2 \\end{aligned}[\/latex] [latex]\\begin{aligned} \\csc t &= \\frac{1}{\\sin t} = \\frac{1}{\\frac{\\sqrt{3}}{2}} = \\frac{2}{\\sqrt{3}} = \\frac{2\\sqrt{3}}{3} \\end{aligned}[\/latex] [latex]\\begin{aligned} \\cot t &= \\frac{\\cos t}{\\sin t} = \\frac{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}} = \\frac{1}{2} \\cdot \\frac{2}{\\sqrt{3}} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3} \\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\sin t = \\frac{\\sqrt{3}}{2}, \\cos t = \\frac{1}{2}, \\tan t = \\sqrt{3}, \\sec t = 2, \\csc t = \\frac{2\\sqrt{3}}{3}, \\cot t = \\frac{\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">A guy wire is attached to the tower at an angle of [latex]\\frac{5\\pi}{4}[\/latex] radians from the positive x-axis. Find all six trigonometric functions for this angle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q633463\">Show Solution<\/button><\/p>\n<div id=\"q633463\" class=\"hidden-answer\" style=\"display: none\">First, determine the quadrant. Since [latex]\\frac{5\\pi}{4}[\/latex] is between [latex]\\pi[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex], the angle is in Quadrant III. Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &= \\frac{5\\pi}{4} - \\pi \\ &= \\frac{5\\pi}{4} - \\frac{4\\pi}{4} \\ &= \\frac{\\pi}{4} \\end{aligned}[\/latex] For the reference angle [latex]\\frac{\\pi}{4}[\/latex], we know: [latex]\\cos\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}, \\sin\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}[\/latex] In Quadrant III, both x and y coordinates are negative, so both sine and cosine are negative. However, when both are negative, tangent and cotangent become positive (negative divided by negative).<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\cos\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}, \\sin\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\tan\\left(\\frac{5\\pi}{4}\\right) = \\frac{-\\frac{\\sqrt{2}}{2}}{-\\frac{\\sqrt{2}}{2}} = 1[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\sec\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{-\\frac{\\sqrt{2}}{2}} = -\\frac{2}{\\sqrt{2}} = -\\sqrt{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\csc\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{-\\frac{\\sqrt{2}}{2}} = -\\sqrt{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\cot\\left(\\frac{5\\pi}{4}\\right) = \\frac{1}{1} = 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A measurement device records an angle of [latex]-\\frac{\\pi}{3}[\/latex] radians. If [latex]\\tan\\left(\\frac{\\pi}{3}\\right) = \\sqrt{3}[\/latex] and [latex]\\sec\\left(\\frac{\\pi}{3}\\right) = 2[\/latex], find [latex]\\tan\\left(-\\frac{\\pi}{3}\\right)[\/latex] and [latex]\\sec\\left(-\\frac{\\pi}{3}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q636739\">Show Solution<\/button><\/p>\n<div id=\"q636739\" class=\"hidden-answer\" style=\"display: none\">Use the properties of even and odd functions: Even functions: [latex]\\cos(-t) = \\cos t[\/latex] and [latex]\\sec(-t) = \\sec t[\/latex] Odd functions: [latex]\\sin(-t) = -\\sin t[\/latex], [latex]\\tan(-t) = -\\tan t[\/latex], [latex]\\csc(-t) = -\\csc t[\/latex], [latex]\\cot(-t) = -\\cot t[\/latex] Since tangent is an odd function: [latex]\\tan\\left(-\\frac{\\pi}{3}\\right) = -\\tan\\left(\\frac{\\pi}{3}\\right) = -\\sqrt{3}[\/latex] Since secant is an even function: [latex]\\sec\\left(-\\frac{\\pi}{3}\\right) = \\sec\\left(\\frac{\\pi}{3}\\right) = 2[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">If [latex]\\sin t = -\\frac{3}{5}[\/latex] and [latex]t[\/latex] is in Quadrant III, find the values of the other five trigonometric functions.<\/section>\n","protected":false},"author":67,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3231"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3231\/revisions"}],"predecessor-version":[{"id":5992,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3231\/revisions\/5992"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3231\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3231"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3231"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3231"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}