{"id":3262,"date":"2025-08-15T23:52:29","date_gmt":"2025-08-15T23:52:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3262"},"modified":"2025-10-17T21:23:18","modified_gmt":"2025-10-17T21:23:18","slug":"inverse-trigonometric-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-trigonometric-functions-apply-it\/","title":{"raw":"Inverse Trigonometric Functions: Apply It","rendered":"Inverse Trigonometric Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Understand the domain restrictions on inverse sine, cosine, and tangent<\/li>\r\n \t<li style=\"font-weight: 400;\">Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use a calculator to evaluate inverse trigonometric functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use inverse trigonometric functions to solve right triangles.<\/li>\r\n \t<li style=\"font-weight: 400;\">Find exact values of composite functions with inverse trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Angles and Surveying<\/h2>\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\">Inverse trigonometric functions are essential tools in navigation and surveying, where angles must be determined from measured distances. Surveyors measure side lengths and use inverse trig functions to find the angles needed to complete their maps and calculations.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">Each inverse trigonometric function has a restricted range to ensure it returns only one angle value:<\/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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(x)[\/latex]: domain [\u22121, 1], range [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\cos^{-1}(x)[\/latex]: domain [\u22121, 1], range [latex][0, \\pi][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan^{-1}(x)[\/latex]: domain (\u2212\u221e, \u221e), range [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A surveyor stands 50 feet from the base of a cell tower. The top of the tower is 120 feet above eye level. What is the angle of elevation to the top of the tower?<\/p>\r\n<p class=\"whitespace-normal break-words\">\r\n[reveal-answer q=\"681360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"681360\"][latex]\\begin{align} \\tan(\\theta) &amp;= \\frac{120}{50} \\[0.5em] \\theta &amp;= \\tan^{-1}\\left(\\frac{120}{50}\\right) &amp;&amp; \\text{Apply inverse tangent} \\[0.5em] \\theta &amp;= \\tan^{-1}(2.4) \\[0.5em] \\theta &amp;\\approx 1.176 \\text{ radians or } 67.4\u00b0 \\end{align}[\/latex] The angle of elevation is approximately 67.4\u00b0.[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A surveyor measures a horizontal distance of 85 feet from a point to the base of a building. The vertical height to a window is 45 feet. Find the angle of elevation to the window.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">A surveyor measures a slope distance of 340 feet with a horizontal distance of 320 feet. What angle does the slope make with the horizontal?\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">A surveyor calculates that for a particular sight line, [latex]\\cos(\\theta) = \\frac{3}{5}[\/latex]. She needs to find [latex]\\sin(\\theta)[\/latex] for additional calculations. Find [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right)[\/latex].\r\n\r\n[reveal-answer q=\"227715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"227715\"]Let [latex]\\theta = \\cos^{-1}\\left(\\frac{3}{5}\\right)[\/latex], so [latex]\\cos(\\theta) = \\frac{3}{5}[\/latex]. Using the Pythagorean identity: [latex]\\begin{align} \\sin^2(\\theta) + \\cos^2(\\theta) &amp;= 1 \\[0.5em] \\sin^2(\\theta) + \\left(\\frac{3}{5}\\right)^2 &amp;= 1 \\[0.5em] \\sin^2(\\theta) &amp;= 1 - \\frac{9}{25} = \\frac{16}{25} \\[0.5em] \\sin(\\theta) &amp;= \\frac{4}{5} &amp;&amp; \\text{Positive since } \\theta \\in [0, \\pi] \\end{align}[\/latex] Therefore, [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right) = \\frac{4}{5}[\/latex]. Try It: Find the exact value of [latex]\\tan\\left(\\sin^{-1}\\left(\\frac{5}{13}\\right)\\right)[\/latex].[\/hidden-answer]\r\n\r\n<\/section><\/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;\">Understand the domain restrictions on inverse sine, cosine, and tangent<\/li>\n<li style=\"font-weight: 400;\">Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.<\/li>\n<li style=\"font-weight: 400;\">Use a calculator to evaluate inverse trigonometric functions.<\/li>\n<li style=\"font-weight: 400;\">Use inverse trigonometric functions to solve right triangles.<\/li>\n<li style=\"font-weight: 400;\">Find exact values of composite functions with inverse trigonometric functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Angles and Surveying<\/h2>\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\">Inverse trigonometric functions are essential tools in navigation and surveying, where angles must be determined from measured distances. Surveyors measure side lengths and use inverse trig functions to find the angles needed to complete their maps and calculations.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">Each inverse trigonometric function has a restricted range to ensure it returns only one angle value:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\sin^{-1}(x)[\/latex]: domain [\u22121, 1], range [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cos^{-1}(x)[\/latex]: domain [\u22121, 1], range [latex][0, \\pi][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan^{-1}(x)[\/latex]: domain (\u2212\u221e, \u221e), range [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A surveyor stands 50 feet from the base of a cell tower. The top of the tower is 120 feet above eye level. What is the angle of elevation to the top of the tower?<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q681360\">Show Solution<\/button><\/p>\n<div id=\"q681360\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align} \\tan(\\theta) &= \\frac{120}{50} \\[0.5em] \\theta &= \\tan^{-1}\\left(\\frac{120}{50}\\right) && \\text{Apply inverse tangent} \\[0.5em] \\theta &= \\tan^{-1}(2.4) \\[0.5em] \\theta &\\approx 1.176 \\text{ radians or } 67.4\u00b0 \\end{align}[\/latex] The angle of elevation is approximately 67.4\u00b0.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A surveyor measures a horizontal distance of 85 feet from a point to the base of a building. The vertical height to a window is 45 feet. Find the angle of elevation to the window.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">A surveyor measures a slope distance of 340 feet with a horizontal distance of 320 feet. What angle does the slope make with the horizontal?<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">A surveyor calculates that for a particular sight line, [latex]\\cos(\\theta) = \\frac{3}{5}[\/latex]. She needs to find [latex]\\sin(\\theta)[\/latex] for additional calculations. Find [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q227715\">Show Solution<\/button><\/p>\n<div id=\"q227715\" class=\"hidden-answer\" style=\"display: none\">Let [latex]\\theta = \\cos^{-1}\\left(\\frac{3}{5}\\right)[\/latex], so [latex]\\cos(\\theta) = \\frac{3}{5}[\/latex]. Using the Pythagorean identity: [latex]\\begin{align} \\sin^2(\\theta) + \\cos^2(\\theta) &= 1 \\[0.5em] \\sin^2(\\theta) + \\left(\\frac{3}{5}\\right)^2 &= 1 \\[0.5em] \\sin^2(\\theta) &= 1 - \\frac{9}{25} = \\frac{16}{25} \\[0.5em] \\sin(\\theta) &= \\frac{4}{5} && \\text{Positive since } \\theta \\in [0, \\pi] \\end{align}[\/latex] Therefore, [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right) = \\frac{4}{5}[\/latex]. Try It: Find the exact value of [latex]\\tan\\left(\\sin^{-1}\\left(\\frac{5}{13}\\right)\\right)[\/latex].<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":221,"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\/3262"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3262\/revisions"}],"predecessor-version":[{"id":4734,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3262\/revisions\/4734"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/221"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3262\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3262"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3262"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3262"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}