{"id":1914,"date":"2025-07-30T21:25:20","date_gmt":"2025-07-30T21:25:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1914"},"modified":"2025-10-13T20:53:47","modified_gmt":"2025-10-13T20:53:47","slug":"graphs-of-the-other-trigonometric-functions-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions-apply-it-1\/","title":{"raw":"Graphs of the Other Trigonometric Functions: Apply It 1","rendered":"Graphs of the Other Trigonometric Functions: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Graph transformations of y=tan x and y=cot x.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine a function formula from a tangent or cotangent graph.<\/li>\r\n \t<li style=\"font-weight: 400;\">Graph transformations of y=sec x and y=csc x.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine a function formula from a secant or cosecant graph.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\r\nMany real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Suppose the function [latex]y=5\\tan\\left(\\frac{\\pi}{4}t\\right)[\/latex] marks the distance in the movement of a light beam from the top of a police car across a wall where <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.\r\n<ol>\r\n \t<li>Find and interpret the stretching factor and period.<\/li>\r\n \t<li>Graph on the interval [0, 5].<\/li>\r\n \t<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"351813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"351813\"]\r\n<ol>\r\n \t<li>We know from the general form of \u00a0[latex]y=A\\tan(Bt)\\\\[\/latex] \u00a0that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.\r\n<figure id=\"Image_06_02_022\" class=\"small\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163819\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/><\/figure>\r\nWe see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.\r\n\r\nThe period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\frac{4}{\\pi}=4[\/latex]. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/li>\r\n \t<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period.\r\n<figure id=\"Image_06_02_021\" class=\"small\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163821\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/><\/figure>\r\n<\/li>\r\n \t<li>period: [latex]f(1)=5\\tan \\left(\\frac{\\pi}{4}\\left(1\\right)\\right)=5\\left(1\\right)=5[\/latex]; after 1 second, the beam of has moved 5 ft from the spot across from the police car.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\r\n<div data-test-render-count=\"1\">\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\r\n<p class=\"whitespace-normal break-words\">While sine and cosine create smooth wave patterns, the graphs of tangent, cotangent, secant, and cosecant have unique characteristics including vertical asymptotes and periodic gaps. These functions model real-world phenomena such as rotating light beams, shadows cast by the sun, and signal strength in telecommunications. In this page, you'll apply these functions to analyze a lighthouse beacon and a solar panel tracking system.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A periodic signal has the following characteristics:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Period: 8 seconds<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Minimum absolute value: 5<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Maximum absolute value: approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical asymptotes at [latex]t = 2, 6, 10, ...[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Write a function in the form [latex]y = A\\sec(Bt)[\/latex].<\/p>\r\n[reveal-answer q=\"284198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"284198\"]Step 1: Find [latex]B[\/latex] using the period. [latex]\\begin{aligned} \\frac{2\\pi}{B} &amp;= 8 \\ B &amp;= \\frac{2\\pi}{8} = \\frac{\\pi}{4} \\end{aligned}[\/latex] Step 2: Verify asymptote location. Asymptotes occur when [latex]\\frac{\\pi}{4}t = \\frac{\\pi}{2} + n\\pi[\/latex] [latex]t = 2 + 4n[\/latex], giving [latex]t = 2, 6, 10, ...[\/latex] \u2713 Step 3: Find [latex]A[\/latex]. The minimum absolute value equals [latex]|A|[\/latex], so [latex]|A| = 5[\/latex]. We'll use [latex]A = 5[\/latex] (positive). Function: [latex]y = 5\\sec\\left(\\frac{\\pi}{4}t\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">The height (in feet) a crane cable makes with the ground is modeled by [latex]h = 50\\cot\\left(\\frac{\\pi}{8}t\\right)[\/latex] where [latex]t[\/latex] is time in seconds. Find the period and evaluate [latex]h(2)[\/latex].<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Graph transformations of y=tan x and y=cot x.<\/li>\n<li style=\"font-weight: 400;\">Determine a function formula from a tangent or cotangent graph.<\/li>\n<li style=\"font-weight: 400;\">Graph transformations of y=sec x and y=csc x.<\/li>\n<li style=\"font-weight: 400;\">Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/section>\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\n<p>Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose the function [latex]y=5\\tan\\left(\\frac{\\pi}{4}t\\right)[\/latex] marks the distance in the movement of a light beam from the top of a police car across a wall where <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.<\/p>\n<ol>\n<li>Find and interpret the stretching factor and period.<\/li>\n<li>Graph on the interval [0, 5].<\/li>\n<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q351813\">Show Solution<\/button><\/p>\n<div id=\"q351813\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We know from the general form of \u00a0[latex]y=A\\tan(Bt)\\\\[\/latex] \u00a0that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.<br \/>\n<figure id=\"Image_06_02_022\" class=\"small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163819\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/><\/figure>\n<p>We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.<\/p>\n<p>The period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\frac{4}{\\pi}=4[\/latex]. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/li>\n<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period.<br \/>\n<figure id=\"Image_06_02_021\" class=\"small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163821\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/><\/figure>\n<\/li>\n<li>period: [latex]f(1)=5\\tan \\left(\\frac{\\pi}{4}\\left(1\\right)\\right)=5\\left(1\\right)=5[\/latex]; after 1 second, the beam of has moved 5 ft from the spot across from the police car.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\n<div data-test-render-count=\"1\">\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<p class=\"whitespace-normal break-words\">While sine and cosine create smooth wave patterns, the graphs of tangent, cotangent, secant, and cosecant have unique characteristics including vertical asymptotes and periodic gaps. These functions model real-world phenomena such as rotating light beams, shadows cast by the sun, and signal strength in telecommunications. In this page, you&#8217;ll apply these functions to analyze a lighthouse beacon and a solar panel tracking system.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A periodic signal has the following characteristics:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Period: 8 seconds<\/li>\n<li class=\"whitespace-normal break-words\">Minimum absolute value: 5<\/li>\n<li class=\"whitespace-normal break-words\">Maximum absolute value: approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptotes at [latex]t = 2, 6, 10, ...[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Write a function in the form [latex]y = A\\sec(Bt)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q284198\">Show Solution<\/button><\/p>\n<div id=\"q284198\" class=\"hidden-answer\" style=\"display: none\">Step 1: Find [latex]B[\/latex] using the period. [latex]\\begin{aligned} \\frac{2\\pi}{B} &= 8 \\ B &= \\frac{2\\pi}{8} = \\frac{\\pi}{4} \\end{aligned}[\/latex] Step 2: Verify asymptote location. Asymptotes occur when [latex]\\frac{\\pi}{4}t = \\frac{\\pi}{2} + n\\pi[\/latex] [latex]t = 2 + 4n[\/latex], giving [latex]t = 2, 6, 10, ...[\/latex] \u2713 Step 3: Find [latex]A[\/latex]. The minimum absolute value equals [latex]|A|[\/latex], so [latex]|A| = 5[\/latex]. We&#8217;ll use [latex]A = 5[\/latex] (positive). Function: [latex]y = 5\\sec\\left(\\frac{\\pi}{4}t\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">The height (in feet) a crane cable makes with the ground is modeled by [latex]h = 50\\cot\\left(\\frac{\\pi}{8}t\\right)[\/latex] where [latex]t[\/latex] is time in seconds. Find the period and evaluate [latex]h(2)[\/latex].<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":13,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":191,"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\/1914"}],"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\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1914\/revisions"}],"predecessor-version":[{"id":4639,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1914\/revisions\/4639"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1914\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1914"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1914"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1914"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}