{"id":194,"date":"2025-02-13T22:44:53","date_gmt":"2025-02-13T22:44:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-trigonometric-functions\/"},"modified":"2025-12-02T22:42:34","modified_gmt":"2025-12-02T22:42:34","slug":"inverse-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-trigonometric-functions\/","title":{"raw":"Inverse Trigonometric Functions: Learn It 1","rendered":"Inverse Trigonometric Functions: Learn It 1"},"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>Understanding and Using the Inverse Sine, Cosine, and Tangent Functions<\/h2>\r\nIn order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function \u201cundoes\u201d what the original trigonometric function \u201cdoes,\u201d as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa.\r\n\r\n<img class=\" wp-image-4965 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-300x126.png\" alt=\"A diagram comparing the sine function and the inverse sine function. On the left, the equation y = sin x is labeled \u2018sine,\u2019 with arrows showing that its domain is angle measure and its range is a ratio. On the right, the equation y = sin\u207b\u00b9 x is labeled \u2018inverse sine,\u2019 with arrows showing that its domain is a ratio and its range is angle measure. The diagram illustrates how the domain and range switch for inverse functions.\" width=\"443\" height=\"186\" \/>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Be aware that [latex]{\\sin}^{-1}x[\/latex] does not mean [latex]\\frac{1}{\\sin{x}}[\/latex].<\/section>For example, if [latex]f(x)=\\sin x[\/latex], then we would write [latex]f^{1}(x)={\\sin}^{-1}{x}[\/latex]. The following examples illustrate the inverse trigonometric functions:\r\n<ul>\r\n \t<li>Since [latex]\\sin\\left(\\frac{\\pi}{6}\\right)=\\frac{1}{2}[\/latex], then [latex]\\frac{\\pi}{6}=\\sin^{\u22121}(\\frac{1}{2})[\/latex].<\/li>\r\n \t<li>Since [latex]\\cos(\\pi)=\u22121[\/latex], then [latex]\\pi=\\cos^{\u22121}(\u22121)[\/latex].<\/li>\r\n \t<li>Since [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], then [latex]\\frac{\\pi}{4}=\\tan^{\u22121}(1)[\/latex].<\/li>\r\n<\/ul>\r\nIn previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">For a <strong>one-to-one function<\/strong>, if [latex]f(a)=b[\/latex], then an inverse function would satisfy [latex]f^{\u22121}(b)=a[\/latex].<\/section>The sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the <strong>domain<\/strong> of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>inverse trig functions<\/h3>\r\n<ul>\r\n \t<li>The <strong>inverse sine function<\/strong>\u00a0[latex]y=\\sin^{\u22121}x[\/latex] means [latex]x=\\sin y[\/latex]. The inverse sine function is sometimes called the <strong>arcsine<\/strong> function, and notated arcsin <em>x<\/em>.\r\n<div>\r\n<div style=\"text-align: center;\">[latex]y=\\sin^{\u22121}x[\/latex] has\u00a0domain [\u22121, 1] and\u00a0range [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]<\/div>\r\n<div><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164005\/CNX_Precalc_Figure_06_03_004n.jpg\" alt=\"A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions\" width=\"508\" height=\"301\" \/><\/div>\r\n<\/div><\/li>\r\n \t<li>The <strong>inverse cosine function<\/strong>\u00a0[latex]y=\\cos^{\u22121}x[\/latex] means [latex]x=\\cos y[\/latex]. The inverse cosine function is sometimes called the <strong>arccosine<\/strong> function, and notated arccos <em>x<\/em>.\r\n<div>\r\n<div style=\"text-align: center;\">[latex]y=\\cos^{\u22121}x[\/latex] has\u00a0domain [\u22121, 1] and\u00a0range [0, \u03c0]<\/div>\r\n<div><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164008\/CNX_Precalc_Figure_06_03_005n.jpg\" alt=\"A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions.\" width=\"487\" height=\"343\" \/><\/div>\r\n<\/div><\/li>\r\n \t<li>The <strong>inverse tangent function<\/strong>\u00a0[latex]y=\\tan^{\u22121}x[\/latex] means [latex]x=\\tan y[\/latex]. The inverse tangent function is sometimes called the <strong>arctangent<\/strong> function, and notated arctan <em>x<\/em>.\r\n<div>\r\n<div style=\"text-align: center;\">[latex]y=\\tan^{\u22121}x[\/latex] has\u00a0domain (\u2212\u221e, \u221e) and\u00a0range [latex]\\left(\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right)[\/latex]<\/div>\r\n<div><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164010\/CNX_Precalc_Figure_06_03_006n.jpg\" alt=\"A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions.\" width=\"487\" height=\"433\" \/><\/div>\r\n<\/div><\/li>\r\n<\/ul>\r\n<\/section>To find the <strong>domain<\/strong> and <strong>range<\/strong> of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line [latex]y=x[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]\\sin\\left(\\frac{5\\pi}{12}\\right)\\approx 0.96593[\/latex], write a relation involving the inverse sine.[reveal-answer q=\"641490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641490\"]Use the relation for the inverse sine. If [latex]\\sin y=x[\/latex], then [latex]\\sin^{\u22121}x=y[\/latex].In this problem, [latex]x=0.96593[\/latex], and [latex]y=\\frac{5\\pi}{12}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\sin^{\u22121}(0.96593)\\approx \\frac{5\\pi}{12}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Given [latex]\\cos(0.5)\\approx 0.8776[\/latex], write a relation involving the inverse cosine.[reveal-answer q=\"359839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359839\"][latex]\\arccos(0.8776)\\approx0.5[\/latex][\/hidden-answer]<\/section>","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>Understanding and Using the Inverse Sine, Cosine, and Tangent Functions<\/h2>\n<p>In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function \u201cundoes\u201d what the original trigonometric function \u201cdoes,\u201d as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4965 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-300x126.png\" alt=\"A diagram comparing the sine function and the inverse sine function. On the left, the equation y = sin x is labeled \u2018sine,\u2019 with arrows showing that its domain is angle measure and its range is a ratio. On the right, the equation y = sin\u207b\u00b9 x is labeled \u2018inverse sine,\u2019 with arrows showing that its domain is a ratio and its range is angle measure. The diagram illustrates how the domain and range switch for inverse functions.\" width=\"443\" height=\"186\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-300x126.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-65x27.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-225x94.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1-350x147.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/02\/02224139\/16.2.L.1.Diagram1.png 605w\" sizes=\"(max-width: 443px) 100vw, 443px\" \/><\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Be aware that [latex]{\\sin}^{-1}x[\/latex] does not mean [latex]\\frac{1}{\\sin{x}}[\/latex].<\/section>\n<p>For example, if [latex]f(x)=\\sin x[\/latex], then we would write [latex]f^{1}(x)={\\sin}^{-1}{x}[\/latex]. The following examples illustrate the inverse trigonometric functions:<\/p>\n<ul>\n<li>Since [latex]\\sin\\left(\\frac{\\pi}{6}\\right)=\\frac{1}{2}[\/latex], then [latex]\\frac{\\pi}{6}=\\sin^{\u22121}(\\frac{1}{2})[\/latex].<\/li>\n<li>Since [latex]\\cos(\\pi)=\u22121[\/latex], then [latex]\\pi=\\cos^{\u22121}(\u22121)[\/latex].<\/li>\n<li>Since [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], then [latex]\\frac{\\pi}{4}=\\tan^{\u22121}(1)[\/latex].<\/li>\n<\/ul>\n<p>In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">For a <strong>one-to-one function<\/strong>, if [latex]f(a)=b[\/latex], then an inverse function would satisfy [latex]f^{\u22121}(b)=a[\/latex].<\/section>\n<p>The sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the <strong>domain<\/strong> of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>inverse trig functions<\/h3>\n<ul>\n<li>The <strong>inverse sine function<\/strong>\u00a0[latex]y=\\sin^{\u22121}x[\/latex] means [latex]x=\\sin y[\/latex]. The inverse sine function is sometimes called the <strong>arcsine<\/strong> function, and notated arcsin <em>x<\/em>.\n<div>\n<div style=\"text-align: center;\">[latex]y=\\sin^{\u22121}x[\/latex] has\u00a0domain [\u22121, 1] and\u00a0range [latex]\\left[\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right][\/latex]<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164005\/CNX_Precalc_Figure_06_03_004n.jpg\" alt=\"A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions\" width=\"508\" height=\"301\" \/><\/div>\n<\/div>\n<\/li>\n<li>The <strong>inverse cosine function<\/strong>\u00a0[latex]y=\\cos^{\u22121}x[\/latex] means [latex]x=\\cos y[\/latex]. The inverse cosine function is sometimes called the <strong>arccosine<\/strong> function, and notated arccos <em>x<\/em>.\n<div>\n<div style=\"text-align: center;\">[latex]y=\\cos^{\u22121}x[\/latex] has\u00a0domain [\u22121, 1] and\u00a0range [0, \u03c0]<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164008\/CNX_Precalc_Figure_06_03_005n.jpg\" alt=\"A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions.\" width=\"487\" height=\"343\" \/><\/div>\n<\/div>\n<\/li>\n<li>The <strong>inverse tangent function<\/strong>\u00a0[latex]y=\\tan^{\u22121}x[\/latex] means [latex]x=\\tan y[\/latex]. The inverse tangent function is sometimes called the <strong>arctangent<\/strong> function, and notated arctan <em>x<\/em>.\n<div>\n<div style=\"text-align: center;\">[latex]y=\\tan^{\u22121}x[\/latex] has\u00a0domain (\u2212\u221e, \u221e) and\u00a0range [latex]\\left(\u2212\\frac{\\pi}{2}\\text{, }\\frac{\\pi}{2}\\right)[\/latex]<\/div>\n<div><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164010\/CNX_Precalc_Figure_06_03_006n.jpg\" alt=\"A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions.\" width=\"487\" height=\"433\" \/><\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<p>To find the <strong>domain<\/strong> and <strong>range<\/strong> of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line [latex]y=x[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]\\sin\\left(\\frac{5\\pi}{12}\\right)\\approx 0.96593[\/latex], write a relation involving the inverse sine.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q641490\">Show Solution<\/button><\/p>\n<div id=\"q641490\" class=\"hidden-answer\" style=\"display: none\">Use the relation for the inverse sine. If [latex]\\sin y=x[\/latex], then [latex]\\sin^{\u22121}x=y[\/latex].In this problem, [latex]x=0.96593[\/latex], and [latex]y=\\frac{5\\pi}{12}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sin^{\u22121}(0.96593)\\approx \\frac{5\\pi}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Given [latex]\\cos(0.5)\\approx 0.8776[\/latex], write a relation involving the inverse cosine.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q359839\">Show Solution<\/button><\/p>\n<div id=\"q359839\" class=\"hidden-answer\" style=\"display: none\">[latex]\\arccos(0.8776)\\approx0.5[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":221,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"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\/194"}],"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\/6"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions"}],"predecessor-version":[{"id":4966,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions\/4966"}],"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\/194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}