{"id":2742,"date":"2025-08-13T18:34:08","date_gmt":"2025-08-13T18:34:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2742"},"modified":"2025-10-17T21:02:41","modified_gmt":"2025-10-17T21:02:41","slug":"trigonometric-identities-and-equations-background-youll-need-4-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/trigonometric-identities-and-equations-background-youll-need-4-2\/","title":{"raw":"Trigonometric Identities and Equations: Background You'll Need 4","rendered":"Trigonometric Identities and Equations: Background You&#8217;ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Identify the period, amplitude and midline of a sinusoidal function<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p data-start=\"3994\" data-end=\"4058\">A sinusoidal function is a sine or cosine graph of the form:<\/p>\r\n<p data-start=\"4060\" data-end=\"4134\">[latex]\r\ny = A\\sin(Bx) + D \\quad \\text{or} \\quad y = A\\cos(Bx) + D\r\n[\/latex]<\/p>\r\n\r\n<ul data-start=\"4136\" data-end=\"4374\">\r\n \t<li data-start=\"4136\" data-end=\"4217\">\r\n<p data-start=\"4138\" data-end=\"4217\">Amplitude [latex]|A|[\/latex]: height from the midline to a peak or trough<\/p>\r\n<\/li>\r\n \t<li data-start=\"4218\" data-end=\"4301\">\r\n<p data-start=\"4220\" data-end=\"4301\">Period: [latex] \\frac{2\\pi}{B}[\/latex]: horizontal length of one full cycle<\/p>\r\n<\/li>\r\n \t<li data-start=\"4302\" data-end=\"4374\">\r\n<p data-start=\"4304\" data-end=\"4374\">Midline: [latex]y = D[\/latex]: vertical center line of the wave<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"4396\" data-end=\"4474\">Find the amplitude, period, and midline of<br data-start=\"4438\" data-end=\"4441\" \/>[latex]y = 3\\sin(2x) - 1[\/latex].<\/p>\r\n[reveal-answer q=\"955831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"955831\"]\r\n\r\n[latex]\r\n\\begin{align}\r\nA &amp;= 3 &amp;&amp; \\text{Amplitude: } 3 \\\\[6pt]\r\nB &amp;= 2 &amp;&amp; \\text{Period: } \\frac{2\\pi}{B} = \\frac{2\\pi}{2} = \\pi \\\\[6pt]\r\nD &amp;= -1 &amp;&amp; \\text{Midline: } y = -1\r\n\\end{align}\r\n[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313775[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313776[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Identify the period, amplitude and midline of a sinusoidal function<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p data-start=\"3994\" data-end=\"4058\">A sinusoidal function is a sine or cosine graph of the form:<\/p>\n<p data-start=\"4060\" data-end=\"4134\">[latex]y = A\\sin(Bx) + D \\quad \\text{or} \\quad y = A\\cos(Bx) + D[\/latex]<\/p>\n<ul data-start=\"4136\" data-end=\"4374\">\n<li data-start=\"4136\" data-end=\"4217\">\n<p data-start=\"4138\" data-end=\"4217\">Amplitude [latex]|A|[\/latex]: height from the midline to a peak or trough<\/p>\n<\/li>\n<li data-start=\"4218\" data-end=\"4301\">\n<p data-start=\"4220\" data-end=\"4301\">Period: [latex]\\frac{2\\pi}{B}[\/latex]: horizontal length of one full cycle<\/p>\n<\/li>\n<li data-start=\"4302\" data-end=\"4374\">\n<p data-start=\"4304\" data-end=\"4374\">Midline: [latex]y = D[\/latex]: vertical center line of the wave<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"4396\" data-end=\"4474\">Find the amplitude, period, and midline of<br data-start=\"4438\" data-end=\"4441\" \/>[latex]y = 3\\sin(2x) - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q955831\">Show Solution<\/button><\/p>\n<div id=\"q955831\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}  A &= 3 && \\text{Amplitude: } 3 \\\\[6pt]  B &= 2 && \\text{Period: } \\frac{2\\pi}{B} = \\frac{2\\pi}{2} = \\pi \\\\[6pt]  D &= -1 && \\text{Midline: } y = -1  \\end{align}[\/latex]\n<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313775\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313775&theme=lumen&iframe_resize_id=ohm313775&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313776\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313776&theme=lumen&iframe_resize_id=ohm313776&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":201,"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\/2742"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2742\/revisions"}],"predecessor-version":[{"id":4724,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2742\/revisions\/4724"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2742\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2742"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2742"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2742"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}