{"id":3247,"date":"2025-08-15T23:44:36","date_gmt":"2025-08-15T23:44:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3247"},"modified":"2025-10-16T17:50:04","modified_gmt":"2025-10-16T17:50:04","slug":"sum-to-product-and-product-to-sum-formulas-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas-apply-it\/","title":{"raw":"Sum-to-Product and Product-to-Sum Formulas: Apply It","rendered":"Sum-to-Product and Product-to-Sum Formulas: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Express products as sums.<\/li>\r\n \t<li>Express sums as products.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Sound Wave Interference<\/h2>\r\n<p class=\"whitespace-normal break-words\">When two sound waves of different frequencies travel through the same space, they interfere with each other, creating patterns of constructive and destructive interference. These interference patterns can be analyzed using product-to-sum and sum-to-product formulas.<\/p>\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How To: Converting Sum to Product<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Identify which sum-to-product formula to use based on the functions involved<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] from your expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]\\frac{\\alpha + \\beta}{2}[\/latex] and [latex]\\frac{\\alpha - \\beta}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute into the formula and simplify<\/li>\r\n<\/ol>\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Two tuning forks produce sound waves with frequencies that can be modeled by [latex]\\sin(440t)[\/latex] and [latex]\\sin(446t)[\/latex], where [latex]t[\/latex] is time in seconds. When both forks sound simultaneously, the combined signal is [latex]\\sin(440t) + \\sin(446t)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Express this sum as a product to analyze the interference pattern.<\/p>\r\n<p class=\"whitespace-normal break-words\">\r\n[reveal-answer q=\"802128\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"802128\"]We have a sum of sines, so we use: [latex]\\sin\\alpha + \\sin\\beta = 2\\sin\\left(\\frac{\\alpha + \\beta}{2}\\right)\\cos\\left(\\frac{\\alpha - \\beta}{2}\\right)[\/latex] Here, [latex]\\alpha = 440t[\/latex] and [latex]\\beta = 446t[\/latex] Calculate the averages: [latex]\\begin{aligned} \\frac{\\alpha + \\beta}{2} &amp;= \\frac{440t + 446t}{2} = \\frac{886t}{2} = 443t \\ \\frac{\\alpha - \\beta}{2} &amp;= \\frac{440t - 446t}{2} = \\frac{-6t}{2} = -3t \\end{aligned}[\/latex] Substitute into the formula: [latex]\\begin{aligned} \\sin(440t) + \\sin(446t) &amp;= 2\\sin(443t)\\cos(-3t) \\ &amp;= 2\\sin(443t)\\cos(3t) \\end{aligned}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">Musicians use this beating phenomenon to tune instruments. When two strings are perfectly in tune, the beats disappear!\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Two flutes produce tones modeled by [latex]\\sin(523t)[\/latex] and [latex]\\sin(529t)[\/latex]. Express the sum as a product and identify the beat frequency.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">An audio engineer is working with two modulated signals: [latex]\\cos(1200t)[\/latex] and [latex]\\cos(800t)[\/latex]. The product of these signals is [latex]\\cos(1200t)\\cos(800t)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Express this product as a sum to understand the frequency components.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">\r\n[reveal-answer q=\"59397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59397\"]Use the product-to-sum formula for cosines: [latex]\\cos\\alpha\\cos\\beta = \\frac{1}{2}[\\cos(\\alpha - \\beta) + \\cos(\\alpha + \\beta)][\/latex] Here, [latex]\\alpha = 1200t[\/latex] and [latex]\\beta = 800t[\/latex] [latex]\\begin{aligned} \\cos(1200t)\\cos(800t) &amp;= \\frac{1}{2}[\\cos(1200t - 800t) + \\cos(1200t + 800t)] \\ &amp;= \\frac{1}{2}[\\cos(400t) + \\cos(2000t)] \\end{aligned}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Express the product [latex]\\sin(3000t)\\cos(500t)[\/latex] as a sum and identify the resulting frequency components.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Express products as sums.<\/li>\n<li>Express sums as products.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Sound Wave Interference<\/h2>\n<p class=\"whitespace-normal break-words\">When two sound waves of different frequencies travel through the same space, they interfere with each other, creating patterns of constructive and destructive interference. These interference patterns can be analyzed using product-to-sum and sum-to-product formulas.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How To: Converting Sum to Product<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Identify which sum-to-product formula to use based on the functions involved<\/li>\n<li class=\"whitespace-normal break-words\">Identify [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] from your expression<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]\\frac{\\alpha + \\beta}{2}[\/latex] and [latex]\\frac{\\alpha - \\beta}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute into the formula and simplify<\/li>\n<\/ol>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Two tuning forks produce sound waves with frequencies that can be modeled by [latex]\\sin(440t)[\/latex] and [latex]\\sin(446t)[\/latex], where [latex]t[\/latex] is time in seconds. When both forks sound simultaneously, the combined signal is [latex]\\sin(440t) + \\sin(446t)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Express this sum as a product to analyze the interference pattern.<\/p>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q802128\">Show Solution<\/button><\/p>\n<div id=\"q802128\" class=\"hidden-answer\" style=\"display: none\">We have a sum of sines, so we use: [latex]\\sin\\alpha + \\sin\\beta = 2\\sin\\left(\\frac{\\alpha + \\beta}{2}\\right)\\cos\\left(\\frac{\\alpha - \\beta}{2}\\right)[\/latex] Here, [latex]\\alpha = 440t[\/latex] and [latex]\\beta = 446t[\/latex] Calculate the averages: [latex]\\begin{aligned} \\frac{\\alpha + \\beta}{2} &= \\frac{440t + 446t}{2} = \\frac{886t}{2} = 443t \\ \\frac{\\alpha - \\beta}{2} &= \\frac{440t - 446t}{2} = \\frac{-6t}{2} = -3t \\end{aligned}[\/latex] Substitute into the formula: [latex]\\begin{aligned} \\sin(440t) + \\sin(446t) &= 2\\sin(443t)\\cos(-3t) \\ &= 2\\sin(443t)\\cos(3t) \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Musicians use this beating phenomenon to tune instruments. When two strings are perfectly in tune, the beats disappear!<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Two flutes produce tones modeled by [latex]\\sin(523t)[\/latex] and [latex]\\sin(529t)[\/latex]. Express the sum as a product and identify the beat frequency.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">An audio engineer is working with two modulated signals: [latex]\\cos(1200t)[\/latex] and [latex]\\cos(800t)[\/latex]. The product of these signals is [latex]\\cos(1200t)\\cos(800t)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Express this product as a sum to understand the frequency components.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q59397\">Show Solution<\/button><\/p>\n<div id=\"q59397\" class=\"hidden-answer\" style=\"display: none\">Use the product-to-sum formula for cosines: [latex]\\cos\\alpha\\cos\\beta = \\frac{1}{2}[\\cos(\\alpha - \\beta) + \\cos(\\alpha + \\beta)][\/latex] Here, [latex]\\alpha = 1200t[\/latex] and [latex]\\beta = 800t[\/latex] [latex]\\begin{aligned} \\cos(1200t)\\cos(800t) &= \\frac{1}{2}[\\cos(1200t - 800t) + \\cos(1200t + 800t)] \\ &= \\frac{1}{2}[\\cos(400t) + \\cos(2000t)] \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Express the product [latex]\\sin(3000t)\\cos(500t)[\/latex] as a sum and identify the resulting frequency components.<\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":24,"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":"apply_it","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\/3247"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3247\/revisions"}],"predecessor-version":[{"id":4688,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3247\/revisions\/4688"}],"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\/3247\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3247"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3247"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3247"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}