{"id":138,"date":"2021-02-03T21:52:34","date_gmt":"2021-02-03T21:52:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=138"},"modified":"2021-03-15T20:57:51","modified_gmt":"2021-03-15T20:57:51","slug":"summary-of-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-trigonometric-functions\/","title":{"raw":"Summary of Trigonometric Functions","rendered":"Summary of Trigonometric Functions"},"content":{"raw":"<div id=\"fs-id1170572167647\" class=\"learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572167654\">\r\n \t<li>Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180\u00b0 has a radian measure of [latex]\\pi [\/latex] rad.<\/li>\r\n \t<li>For acute angles [latex]\\theta[\/latex], the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is [latex]\\theta[\/latex].<\/li>\r\n \t<li>For a general angle [latex]\\theta[\/latex], let [latex](x,y)[\/latex] be a point on a circle of radius [latex]r[\/latex] corresponding to this angle [latex]\\theta[\/latex]. The trigonometric functions can be written as ratios involving [latex]x, \\, y[\/latex], and [latex]r[\/latex].<\/li>\r\n \t<li>The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period [latex]2\\pi[\/latex]. The tangent and cotangent functions have period [latex]\\pi[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572452321\">\r\n \t<li><strong>Generalized sine function<\/strong>\r\n[latex]f(x)=A\\sin(B(x-\\alpha))+C[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572481704\" class=\"definition\">\r\n \t<dt>periodic function<\/dt>\r\n \t<dd id=\"fs-id1170572481709\">a function is periodic if it has a repeating pattern as the values of [latex]x[\/latex] move from left to right<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572481719\" class=\"definition\">\r\n \t<dt>radians<\/dt>\r\n \t<dd id=\"fs-id1170572481724\">for a circular arc of length [latex]s[\/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\\theta [\/latex] is [latex]s[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572552272\" class=\"definition\">\r\n \t<dt>trigonometric functions<\/dt>\r\n \t<dd id=\"fs-id1170572552277\">functions of an angle defined as ratios of the lengths of the sides of a right triangle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572552282\" class=\"definition\">\r\n \t<dt>trigonometric identity<\/dt>\r\n \t<dd id=\"fs-id1170572552288\">an equation involving trigonometric functions that is true for all angles [latex]\\theta [\/latex] for which the functions in the equation are defined<\/dd>\r\n<\/dl>\r\n<div id=\"fs-id1170572455697\" class=\"key-equations\"><\/div>","rendered":"<div id=\"fs-id1170572167647\" class=\"learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572167654\">\n<li>Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180\u00b0 has a radian measure of [latex]\\pi[\/latex] rad.<\/li>\n<li>For acute angles [latex]\\theta[\/latex], the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is [latex]\\theta[\/latex].<\/li>\n<li>For a general angle [latex]\\theta[\/latex], let [latex](x,y)[\/latex] be a point on a circle of radius [latex]r[\/latex] corresponding to this angle [latex]\\theta[\/latex]. The trigonometric functions can be written as ratios involving [latex]x, \\, y[\/latex], and [latex]r[\/latex].<\/li>\n<li>The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period [latex]2\\pi[\/latex]. The tangent and cotangent functions have period [latex]\\pi[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572452321\">\n<li><strong>Generalized sine function<\/strong><br \/>\n[latex]f(x)=A\\sin(B(x-\\alpha))+C[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572481704\" class=\"definition\">\n<dt>periodic function<\/dt>\n<dd id=\"fs-id1170572481709\">a function is periodic if it has a repeating pattern as the values of [latex]x[\/latex] move from left to right<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572481719\" class=\"definition\">\n<dt>radians<\/dt>\n<dd id=\"fs-id1170572481724\">for a circular arc of length [latex]s[\/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\\theta[\/latex] is [latex]s[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572552272\" class=\"definition\">\n<dt>trigonometric functions<\/dt>\n<dd id=\"fs-id1170572552277\">functions of an angle defined as ratios of the lengths of the sides of a right triangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572552282\" class=\"definition\">\n<dt>trigonometric identity<\/dt>\n<dd id=\"fs-id1170572552288\">an equation involving trigonometric functions that is true for all angles [latex]\\theta[\/latex] for which the functions in the equation are defined<\/dd>\n<\/dl>\n<div id=\"fs-id1170572455697\" class=\"key-equations\"><\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-138\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-138","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions"}],"predecessor-version":[{"id":1255,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/138\/revisions\/1255"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}