{"id":2729,"date":"2025-08-13T18:31:52","date_gmt":"2025-08-13T18:31:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2729"},"modified":"2025-10-14T15:58:27","modified_gmt":"2025-10-14T15:58:27","slug":"periodic-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/periodic-functions-background-youll-need-3\/","title":{"raw":"Periodic Functions: Background You'll Need 3","rendered":"Periodic Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Identify values where trigonometric functions are undefined<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">Trigonometric functions are undefined when their denominators are zero or when they involve a ratio that cannot exist.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Examples of when trigonometric functions are undefined:\r\n<ul data-start=\"3260\" data-end=\"3703\">\r\n \t<li data-start=\"3260\" data-end=\"3376\">\r\n<p data-start=\"3262\" data-end=\"3376\">[latex]\\tan(\\theta) = \\frac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex] \u2192 undefined when [latex]\\cos(\\theta) = 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"3377\" data-end=\"3482\">\r\n<p data-start=\"3379\" data-end=\"3482\">[latex]\\sec(\\theta) = \\frac{1}{\\cos(\\theta)}[\/latex] \u2192 undefined when [latex]\\cos(\\theta) = 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"3483\" data-end=\"3588\">\r\n<p data-start=\"3485\" data-end=\"3588\">[latex]\\csc(\\theta) = \\frac{1}{\\sin(\\theta)}[\/latex] \u2192 undefined when [latex]\\sin(\\theta) = 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"3589\" data-end=\"3703\">\r\n<p data-start=\"3591\" data-end=\"3703\">[latex]\\cot(\\theta) = \\frac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex] \u2192 undefined when [latex]\\sin(\\theta) = 0[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">sine and cosine have infinite domains and will never be undefined, but secant, cosecant, tangent, and cotangent are all undefined for some angles in their domain.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313585[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Identify values where trigonometric functions are undefined<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">Trigonometric functions are undefined when their denominators are zero or when they involve a ratio that cannot exist.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Examples of when trigonometric functions are undefined:<\/p>\n<ul data-start=\"3260\" data-end=\"3703\">\n<li data-start=\"3260\" data-end=\"3376\">\n<p data-start=\"3262\" data-end=\"3376\">[latex]\\tan(\\theta) = \\frac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex] \u2192 undefined when [latex]\\cos(\\theta) = 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"3377\" data-end=\"3482\">\n<p data-start=\"3379\" data-end=\"3482\">[latex]\\sec(\\theta) = \\frac{1}{\\cos(\\theta)}[\/latex] \u2192 undefined when [latex]\\cos(\\theta) = 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"3483\" data-end=\"3588\">\n<p data-start=\"3485\" data-end=\"3588\">[latex]\\csc(\\theta) = \\frac{1}{\\sin(\\theta)}[\/latex] \u2192 undefined when [latex]\\sin(\\theta) = 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"3589\" data-end=\"3703\">\n<p data-start=\"3591\" data-end=\"3703\">[latex]\\cot(\\theta) = \\frac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex] \u2192 undefined when [latex]\\sin(\\theta) = 0[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">sine and cosine have infinite domains and will never be undefined, but secant, cosecant, tangent, and cotangent are all undefined for some angles in their domain.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313585\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313585&theme=lumen&iframe_resize_id=ohm313585&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":4,"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":"background_you_need","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\/2729"}],"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\/2729\/revisions"}],"predecessor-version":[{"id":4653,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2729\/revisions\/4653"}],"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\/2729\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2729"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2729"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2729"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2729"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}