{"id":1548,"date":"2025-07-25T02:44:02","date_gmt":"2025-07-25T02:44:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1548"},"modified":"2026-03-12T06:09:25","modified_gmt":"2026-03-12T06:09:25","slug":"sum-to-product-and-product-to-sum-formulas-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sum-to-product-and-product-to-sum-formulas-fresh-take\/","title":{"raw":"Sum-to-Product and Product-to-Sum Formulas: Fresh Take","rendered":"Sum-to-Product and Product-to-Sum Formulas: Fresh Take"},"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>Product-to-Sum Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"52\" data-end=\"360\">The product-to-sum formulas allow us to rewrite products of sine and cosine as sums or differences. This is especially helpful in simplifying trig expressions, evaluating integrals, or solving equations. By converting products into sums, we reduce complexity and often expose familiar angles or identities.<\/p>\r\n<p data-start=\"362\" data-end=\"381\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"383\" data-end=\"628\">\r\n \t<li data-start=\"383\" data-end=\"464\">\r\n<p data-start=\"385\" data-end=\"464\">[latex]\\sin A \\cos B = \\dfrac{1}{2}\\left[\\sin(A+B) + \\sin(A-B)\\right][\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"465\" data-end=\"546\">\r\n<p data-start=\"467\" data-end=\"546\">[latex]\\cos A \\cos B = \\dfrac{1}{2}\\left[\\cos(A+B) + \\cos(A-B)\\right][\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"547\" data-end=\"628\">\r\n<p data-start=\"549\" data-end=\"628\">[latex]\\sin A \\sin B = \\dfrac{1}{2}\\left[\\cos(A-B) - \\cos(A+B)\\right][\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Product-to-Sum Formulas<\/strong>\r\n<ol>\r\n \t<li data-start=\"687\" data-end=\"715\"><strong data-start=\"687\" data-end=\"713\">Pick the Right Formula<\/strong>\r\n<ul data-start=\"719\" data-end=\"795\">\r\n \t<li data-start=\"719\" data-end=\"795\">\r\n<p data-start=\"721\" data-end=\"795\">Look at whether you have sine \u00d7 sine, cosine \u00d7 cosine, or sine \u00d7 cosine.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"800\" data-end=\"824\"><strong data-start=\"800\" data-end=\"822\">Apply the Identity<\/strong>\r\n<ul data-start=\"828\" data-end=\"899\">\r\n \t<li data-start=\"828\" data-end=\"899\">\r\n<p data-start=\"830\" data-end=\"899\">Replace the product with the sum or difference of sines or cosines.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<ul data-start=\"1246\" data-end=\"1509\">\r\n \t<li data-start=\"1339\" data-end=\"1415\">\r\n<p data-start=\"1341\" data-end=\"1415\">[latex]= \\dfrac{1}{2}\\left[\\cos(90^\\circ)+\\cos(50^\\circ)\\right][\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1419\" data-end=\"1509\">\r\n<p data-start=\"1421\" data-end=\"1509\">[latex]= \\dfrac{1}{2}\\left[0+\\cos(50^\\circ)\\right]=\\dfrac{1}{2}\\cos(50^\\circ)[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div><section class=\"textbox example\">Express [latex]\\sin(5x)\\cos(3x)[\/latex] as a sum.[reveal-answer q=\"product-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"product-001\"]\r\nUse the product-to-sum formula:[latex]\\sin A \\cos B = \\dfrac{1}{2}\\left[\\sin(A+B) + \\sin(A-B)\\right][\/latex]Let [latex]A = 5x[\/latex] and [latex]B = 3x[\/latex]:[latex]\\begin{align*}\r\n\\sin(5x)\\cos(3x) &amp;= \\dfrac{1}{2}\\left[\\sin(5x+3x) + \\sin(5x-3x)\\right] \\\\\r\n&amp;= \\dfrac{1}{2}\\left[\\sin(8x) + \\sin(2x)\\right]\r\n\\end{align*}[\/latex]\r\n[\/hidden-answer]<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fehgcbgb-9mgjrdgpc_M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/9mgjrdgpc_M?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fehgcbgb-9mgjrdgpc_M\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661382&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fehgcbgb-9mgjrdgpc_M&vembed=0&video_id=9mgjrdgpc_M&video_target=tpm-plugin-fehgcbgb-9mgjrdgpc_M'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/The+Product-to-Sum+Formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Product-to-Sum Formulas\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Sum-to-Product Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"52\" data-end=\"396\">The sum-to-product formulas let us rewrite sums or differences of sines or cosines as products. This is the reverse of the product-to-sum process. These formulas are especially helpful for simplifying trig expressions and solving equations because they turn a sum of two terms into a single product, often making factoring and solving easier.<\/p>\r\n<p data-start=\"398\" data-end=\"417\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"419\" data-end=\"839\">\r\n \t<li data-start=\"419\" data-end=\"523\">\r\n<p data-start=\"421\" data-end=\"523\">[latex]\\sin A + \\sin B = 2\\sin!\\left(\\dfrac{A+B}{2}\\right)\\cos!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"524\" data-end=\"628\">\r\n<p data-start=\"526\" data-end=\"628\">[latex]\\sin A - \\sin B = 2\\cos!\\left(\\dfrac{A+B}{2}\\right)\\sin!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"629\" data-end=\"733\">\r\n<p data-start=\"631\" data-end=\"733\">[latex]\\cos A + \\cos B = 2\\cos!\\left(\\dfrac{A+B}{2}\\right)\\cos!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"734\" data-end=\"839\">\r\n<p data-start=\"736\" data-end=\"839\">[latex]\\cos A - \\cos B = -2\\sin!\\left(\\dfrac{A+B}{2}\\right)\\sin!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"52\" data-end=\"360\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum-to-Product Formulas<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li data-start=\"1246\" data-end=\"1399\">\r\n<p data-start=\"1249\" data-end=\"1275\"><strong data-start=\"1249\" data-end=\"1273\">Identify the Pattern<\/strong><\/p>\r\n\r\n<ul data-start=\"1279\" data-end=\"1399\">\r\n \t<li data-start=\"1279\" data-end=\"1347\">\r\n<p data-start=\"1281\" data-end=\"1347\">Look for two sine terms or two cosine terms added or subtracted.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1351\" data-end=\"1399\">\r\n<p data-start=\"1353\" data-end=\"1399\">Match to the correct sum-to-product formula.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1351\" data-end=\"1399\">\r\n<p data-start=\"1353\" data-end=\"1399\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"2391\" data-end=\"2410\">Why It\u2019s Useful<\/strong><\/p>\r\n\r\n<ul data-start=\"2416\" data-end=\"2544\">\r\n \t<li data-start=\"2416\" data-end=\"2466\">\r\n<p data-start=\"2418\" data-end=\"2466\">Converts tricky sums into manageable products.<\/p>\r\n<\/li>\r\n \t<li data-start=\"2470\" data-end=\"2544\">\r\n<p data-start=\"2472\" data-end=\"2544\">Especially useful in solving trig equations and in calculus integrals.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Express [latex]\\sin(7x) + \\sin(3x)[\/latex] as a product.[reveal-answer q=\"sum-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sum-001\"]\r\nUse the sum-to-product formula:[latex]\\sin A + \\sin B = 2\\sin\\left(\\dfrac{A+B}{2}\\right)\\cos\\left(\\dfrac{A-B}{2}\\right)[\/latex]Let [latex]A = 7x[\/latex] and [latex]B = 3x[\/latex]:[latex]\\begin{align*}\r\n\\sin(7x) + \\sin(3x) &amp;= 2\\sin\\left(\\dfrac{7x+3x}{2}\\right)\\cos\\left(\\dfrac{7x-3x}{2}\\right) \\\\\r\n&amp;= 2\\sin\\left(\\dfrac{10x}{2}\\right)\\cos\\left(\\dfrac{4x}{2}\\right) \\\\\r\n&amp;= 2\\sin(5x)\\cos(2x)\r\n\\end{align*}[\/latex]\r\n[\/hidden-answer]<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdheagec-k7FmEejVKgM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k7FmEejVKgM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdheagec-k7FmEejVKgM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661383&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdheagec-k7FmEejVKgM&vembed=0&video_id=k7FmEejVKgM&video_target=tpm-plugin-fdheagec-k7FmEejVKgM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/trig+formula+example+-+sum-to-product_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201ctrig formula example: sum-to-product\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","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>Product-to-Sum Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"52\" data-end=\"360\">The product-to-sum formulas allow us to rewrite products of sine and cosine as sums or differences. This is especially helpful in simplifying trig expressions, evaluating integrals, or solving equations. By converting products into sums, we reduce complexity and often expose familiar angles or identities.<\/p>\n<p data-start=\"362\" data-end=\"381\">The formulas are:<\/p>\n<ul data-start=\"383\" data-end=\"628\">\n<li data-start=\"383\" data-end=\"464\">\n<p data-start=\"385\" data-end=\"464\">[latex]\\sin A \\cos B = \\dfrac{1}{2}\\left[\\sin(A+B) + \\sin(A-B)\\right][\/latex]<\/p>\n<\/li>\n<li data-start=\"465\" data-end=\"546\">\n<p data-start=\"467\" data-end=\"546\">[latex]\\cos A \\cos B = \\dfrac{1}{2}\\left[\\cos(A+B) + \\cos(A-B)\\right][\/latex]<\/p>\n<\/li>\n<li data-start=\"547\" data-end=\"628\">\n<p data-start=\"549\" data-end=\"628\">[latex]\\sin A \\sin B = \\dfrac{1}{2}\\left[\\cos(A-B) - \\cos(A+B)\\right][\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Product-to-Sum Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"687\" data-end=\"715\"><strong data-start=\"687\" data-end=\"713\">Pick the Right Formula<\/strong>\n<ul data-start=\"719\" data-end=\"795\">\n<li data-start=\"719\" data-end=\"795\">\n<p data-start=\"721\" data-end=\"795\">Look at whether you have sine \u00d7 sine, cosine \u00d7 cosine, or sine \u00d7 cosine.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"800\" data-end=\"824\"><strong data-start=\"800\" data-end=\"822\">Apply the Identity<\/strong>\n<ul data-start=\"828\" data-end=\"899\">\n<li data-start=\"828\" data-end=\"899\">\n<p data-start=\"830\" data-end=\"899\">Replace the product with the sum or difference of sines or cosines.<\/p>\n<\/li>\n<\/ul>\n<ul data-start=\"1246\" data-end=\"1509\">\n<li data-start=\"1339\" data-end=\"1415\">\n<p data-start=\"1341\" data-end=\"1415\">[latex]= \\dfrac{1}{2}\\left[\\cos(90^\\circ)+\\cos(50^\\circ)\\right][\/latex].<\/p>\n<\/li>\n<li data-start=\"1419\" data-end=\"1509\">\n<p data-start=\"1421\" data-end=\"1509\">[latex]= \\dfrac{1}{2}\\left[0+\\cos(50^\\circ)\\right]=\\dfrac{1}{2}\\cos(50^\\circ)[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<ul>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ul>\n<\/div>\n<div>\n<section class=\"textbox example\">Express [latex]\\sin(5x)\\cos(3x)[\/latex] as a sum.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qproduct-001\">Show Solution<\/button><\/p>\n<div id=\"qproduct-001\" class=\"hidden-answer\" style=\"display: none\">\nUse the product-to-sum formula:[latex]\\sin A \\cos B = \\dfrac{1}{2}\\left[\\sin(A+B) + \\sin(A-B)\\right][\/latex]Let [latex]A = 5x[\/latex] and [latex]B = 3x[\/latex]:[latex]\\begin{align*}  \\sin(5x)\\cos(3x) &= \\dfrac{1}{2}\\left[\\sin(5x+3x) + \\sin(5x-3x)\\right] \\\\  &= \\dfrac{1}{2}\\left[\\sin(8x) + \\sin(2x)\\right]  \\end{align*}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fehgcbgb-9mgjrdgpc_M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/9mgjrdgpc_M?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fehgcbgb-9mgjrdgpc_M\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661382&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fehgcbgb-9mgjrdgpc_M&#38;vembed=0&#38;video_id=9mgjrdgpc_M&#38;video_target=tpm-plugin-fehgcbgb-9mgjrdgpc_M\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/The+Product-to-Sum+Formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Product-to-Sum Formulas\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Sum-to-Product Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"52\" data-end=\"396\">The sum-to-product formulas let us rewrite sums or differences of sines or cosines as products. This is the reverse of the product-to-sum process. These formulas are especially helpful for simplifying trig expressions and solving equations because they turn a sum of two terms into a single product, often making factoring and solving easier.<\/p>\n<p data-start=\"398\" data-end=\"417\">The formulas are:<\/p>\n<ul data-start=\"419\" data-end=\"839\">\n<li data-start=\"419\" data-end=\"523\">\n<p data-start=\"421\" data-end=\"523\">[latex]\\sin A + \\sin B = 2\\sin!\\left(\\dfrac{A+B}{2}\\right)\\cos!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"524\" data-end=\"628\">\n<p data-start=\"526\" data-end=\"628\">[latex]\\sin A - \\sin B = 2\\cos!\\left(\\dfrac{A+B}{2}\\right)\\sin!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"629\" data-end=\"733\">\n<p data-start=\"631\" data-end=\"733\">[latex]\\cos A + \\cos B = 2\\cos!\\left(\\dfrac{A+B}{2}\\right)\\cos!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"734\" data-end=\"839\">\n<p data-start=\"736\" data-end=\"839\">[latex]\\cos A - \\cos B = -2\\sin!\\left(\\dfrac{A+B}{2}\\right)\\sin!\\left(\\dfrac{A-B}{2}\\right)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"52\" data-end=\"360\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Sum-to-Product Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"1246\" data-end=\"1399\">\n<p data-start=\"1249\" data-end=\"1275\"><strong data-start=\"1249\" data-end=\"1273\">Identify the Pattern<\/strong><\/p>\n<ul data-start=\"1279\" data-end=\"1399\">\n<li data-start=\"1279\" data-end=\"1347\">\n<p data-start=\"1281\" data-end=\"1347\">Look for two sine terms or two cosine terms added or subtracted.<\/p>\n<\/li>\n<li data-start=\"1351\" data-end=\"1399\">\n<p data-start=\"1353\" data-end=\"1399\">Match to the correct sum-to-product formula.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1351\" data-end=\"1399\">\n<p data-start=\"1353\" data-end=\"1399\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"2391\" data-end=\"2410\">Why It\u2019s Useful<\/strong><\/p>\n<ul data-start=\"2416\" data-end=\"2544\">\n<li data-start=\"2416\" data-end=\"2466\">\n<p data-start=\"2418\" data-end=\"2466\">Converts tricky sums into manageable products.<\/p>\n<\/li>\n<li data-start=\"2470\" data-end=\"2544\">\n<p data-start=\"2472\" data-end=\"2544\">Especially useful in solving trig equations and in calculus integrals.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Express [latex]\\sin(7x) + \\sin(3x)[\/latex] as a product.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsum-001\">Show Solution<\/button><\/p>\n<div id=\"qsum-001\" class=\"hidden-answer\" style=\"display: none\">\nUse the sum-to-product formula:[latex]\\sin A + \\sin B = 2\\sin\\left(\\dfrac{A+B}{2}\\right)\\cos\\left(\\dfrac{A-B}{2}\\right)[\/latex]Let [latex]A = 7x[\/latex] and [latex]B = 3x[\/latex]:[latex]\\begin{align*}  \\sin(7x) + \\sin(3x) &= 2\\sin\\left(\\dfrac{7x+3x}{2}\\right)\\cos\\left(\\dfrac{7x-3x}{2}\\right) \\\\  &= 2\\sin\\left(\\dfrac{10x}{2}\\right)\\cos\\left(\\dfrac{4x}{2}\\right) \\\\  &= 2\\sin(5x)\\cos(2x)  \\end{align*}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdheagec-k7FmEejVKgM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k7FmEejVKgM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdheagec-k7FmEejVKgM\" class=\"p3sdk-target\"><\/div>\n<p 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