{"id":2710,"date":"2025-08-13T18:29:49","date_gmt":"2025-08-13T18:29:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2710"},"modified":"2025-10-13T16:24:04","modified_gmt":"2025-10-13T16:24:04","slug":"trigonometric-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/trigonometric-functions-background-youll-need-2\/","title":{"raw":"Trigonometric Functions: Background You'll Need 2","rendered":"Trigonometric Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Rationalize radical denominators<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3><strong data-start=\"1220\" data-end=\"1249\">rationalize a denominator<\/strong><\/h3>\r\nTo <strong data-start=\"1220\" data-end=\"1249\">rationalize a denominator<\/strong> means to eliminate radicals (square roots, cube roots, etc.) from the denominator of a fraction.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Rationalize the denominator.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\frac{3}{\\sqrt{5}}[\/latex].<\/li>\r\n \t<li>\r\n<p data-start=\"1622\" data-end=\"1669\">[latex]\\frac{2}{3 + \\sqrt{2}}[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"666705\"]Show Solutions[\/reveal-answer]\r\n[hidden-answer a=\"666705\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex] \\frac{3}{\\sqrt{5}} \\times \\frac{\\sqrt{5}}{\\sqrt{5}} = \\frac{3\\sqrt{5}}{5} [\/latex] Result: [latex]\\frac{3\\sqrt{5}}{5}[\/latex] Simplify Multiply by the conjugate of the denominator, [latex]3 - \\sqrt{2}[\/latex]: [latex] \\frac{2}{3 + \\sqrt{2}} \\times \\frac{3 - \\sqrt{2}}{3 - \\sqrt{2}} = \\frac{2(3 - \\sqrt{2})}{(3 + \\sqrt{2})(3 - \\sqrt{2})} [\/latex] [latex] = \\frac{6 - 2\\sqrt{2}}{9 - 2} = \\frac{6 - 2\\sqrt{2}}{7} [\/latex] Result: [latex]\\frac{6 - 2\\sqrt{2}}{7}[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313565[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313566[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">When there\u2019s more than one term in the denominator, use its conjugate (change the sign between terms).\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Rationalize radical denominators<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3><strong data-start=\"1220\" data-end=\"1249\">rationalize a denominator<\/strong><\/h3>\n<p>To <strong data-start=\"1220\" data-end=\"1249\">rationalize a denominator<\/strong> means to eliminate radicals (square roots, cube roots, etc.) from the denominator of a fraction.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rationalize the denominator.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{3}{\\sqrt{5}}[\/latex].<\/li>\n<li>\n<p data-start=\"1622\" data-end=\"1669\">[latex]\\frac{2}{3 + \\sqrt{2}}[\/latex].<\/p>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q666705\">Show Solutions<\/button><\/p>\n<div id=\"q666705\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{3}{\\sqrt{5}} \\times \\frac{\\sqrt{5}}{\\sqrt{5}} = \\frac{3\\sqrt{5}}{5}[\/latex] Result: [latex]\\frac{3\\sqrt{5}}{5}[\/latex] Simplify Multiply by the conjugate of the denominator, [latex]3 - \\sqrt{2}[\/latex]: [latex]\\frac{2}{3 + \\sqrt{2}} \\times \\frac{3 - \\sqrt{2}}{3 - \\sqrt{2}} = \\frac{2(3 - \\sqrt{2})}{(3 + \\sqrt{2})(3 - \\sqrt{2})}[\/latex] [latex]= \\frac{6 - 2\\sqrt{2}}{9 - 2} = \\frac{6 - 2\\sqrt{2}}{7}[\/latex] Result: [latex]\\frac{6 - 2\\sqrt{2}}{7}[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313565\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313565&theme=lumen&iframe_resize_id=ohm313565&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313566\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313566&theme=lumen&iframe_resize_id=ohm313566&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">When there\u2019s more than one term in the denominator, use its conjugate (change the sign between terms).<\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"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\/2710"}],"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\/2710\/revisions"}],"predecessor-version":[{"id":4608,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2710\/revisions\/4608"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2710\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2710"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2710"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2710"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}