{"id":1564,"date":"2025-07-25T03:44:16","date_gmt":"2025-07-25T03:44:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1564"},"modified":"2026-03-12T06:50:13","modified_gmt":"2026-03-12T06:50:13","slug":"non-right-triangles-with-law-of-cosines-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/non-right-triangles-with-law-of-cosines-fresh-take\/","title":{"raw":"Non-right Triangles with Law of Cosines: Fresh Take","rendered":"Non-right Triangles with Law of Cosines: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the Law of Cosines to solve oblique triangles.<\/li>\r\n \t<li>Solve applied problems using the Law of Cosines.<\/li>\r\n \t<li>Use Heron\u2019s formula to \ufb01nd the area of a triangle.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Oblique Triangles with the Law of Cosines<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"78\" data-end=\"437\">The Law of Cosines extends the Pythagorean Theorem to all triangles, making it possible to solve oblique (non-right) triangles when the Law of Sines is not convenient. It is especially useful for <strong data-start=\"274\" data-end=\"281\">SAS<\/strong> (two sides and the included angle) and <strong data-start=\"321\" data-end=\"328\">SSS<\/strong> (all three sides) cases. With it, we can find unknown sides or angles even when no right angle is present.<\/p>\r\n<p data-start=\"439\" data-end=\"460\">The Law of Cosines:<\/p>\r\n\r\n<ul data-start=\"462\" data-end=\"620\">\r\n \t<li data-start=\"462\" data-end=\"514\">\r\n<p data-start=\"464\" data-end=\"514\">[latex]a^{2} = b^{2} + c^{2} - 2bc\\cos A[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"515\" data-end=\"567\">\r\n<p data-start=\"517\" data-end=\"567\">[latex]b^{2} = a^{2} + c^{2} - 2ac\\cos B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"568\" data-end=\"620\">\r\n<p data-start=\"570\" data-end=\"620\">[latex]c^{2} = a^{2} + b^{2} - 2ab\\cos C[\/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 the Law of Cosines<\/strong>\r\n<ol>\r\n \t<li data-start=\"671\" data-end=\"886\">\r\n<p data-start=\"674\" data-end=\"699\"><strong data-start=\"674\" data-end=\"697\">Know When to Use It<\/strong><\/p>\r\n\r\n<ul data-start=\"703\" data-end=\"886\">\r\n \t<li data-start=\"703\" data-end=\"804\">\r\n<p data-start=\"705\" data-end=\"804\"><strong data-start=\"705\" data-end=\"712\">SAS<\/strong>: when two sides and the included angle are given, use the formula to find the third side.<\/p>\r\n<\/li>\r\n \t<li data-start=\"808\" data-end=\"886\">\r\n<p data-start=\"810\" data-end=\"886\"><strong data-start=\"810\" data-end=\"817\">SSS<\/strong>: when all three sides are given, use the formula to find an angle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"888\" data-end=\"1206\">\r\n<p data-start=\"891\" data-end=\"915\"><strong data-start=\"891\" data-end=\"913\">Solving for a Side<\/strong><\/p>\r\n\r\n<ul data-start=\"919\" data-end=\"1206\">\r\n \t<li data-start=\"919\" data-end=\"966\">\r\n<p data-start=\"921\" data-end=\"966\">Plug in known sides and the included angle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1208\" data-end=\"1598\">\r\n<p data-start=\"1211\" data-end=\"1237\"><strong data-start=\"1211\" data-end=\"1235\">Solving for an Angle<\/strong><\/p>\r\n\r\n<ul data-start=\"1241\" data-end=\"1598\">\r\n \t<li data-start=\"1241\" data-end=\"1334\">\r\n<p data-start=\"1243\" data-end=\"1334\">Rearrange the formula:<br data-start=\"1265\" data-end=\"1268\" \/>[latex]\\cos A = \\dfrac{b^{2} + c^{2} - a^{2}}{2bc}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1241\" data-end=\"1334\">Plug in the known sides<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1600\" data-end=\"1779\">\r\n<p data-start=\"1603\" data-end=\"1638\"><strong data-start=\"1603\" data-end=\"1636\">Combine with the Law of Sines<\/strong><\/p>\r\n\r\n<ul data-start=\"1642\" data-end=\"1779\">\r\n \t<li data-start=\"1642\" data-end=\"1779\">\r\n<p data-start=\"1644\" data-end=\"1779\">After using the Law of Cosines to find one side or angle, switch to the Law of Sines for quicker calculations of the remaining parts.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1781\" data-end=\"1945\">\r\n<p data-start=\"1784\" data-end=\"1808\"><strong data-start=\"1784\" data-end=\"1806\">Check for Validity<\/strong><\/p>\r\n\r\n<ul data-start=\"1812\" data-end=\"1945\">\r\n \t<li data-start=\"1812\" data-end=\"1887\">\r\n<p data-start=\"1814\" data-end=\"1887\">Angles must add up to [latex]180^\\circ[\/latex] (or [latex]\\pi[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1891\" data-end=\"1945\">\r\n<p data-start=\"1893\" data-end=\"1945\">Side lengths must satisfy the triangle inequality.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In triangle ABC, [latex]b = 12[\/latex], [latex]c = 15[\/latex], and [latex]A = 65\u00b0[\/latex]. Find side [latex]a[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"326789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"326789\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">We have two sides and the included angle, so use the Law of Cosines: [latex] a^2 = b^2 + c^2 - 2bc\\cos A [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Substitute the known values: [latex] \\begin{aligned} a^2 &amp;= 12^2 + 15^2 - 2(12)(15)\\cos 65\u00b0 \\\\ a^2 &amp;= 144 + 225 - 360\\cos 65\u00b0 \\ a^2 &amp;= 369 - 360(0.4226) \\\\ a^2 &amp;= 369 - 152.14 \\\\ a^2 &amp;= 216.86 \\\\ a &amp;= \\sqrt{216.86} \\\\ a &amp;\\approx 14.7 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Side [latex]a[\/latex] is approximately [latex]14.7[\/latex] units.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/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-adghhgaa-66VuRaNQlVM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/66VuRaNQlVM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-adghhgaa-66VuRaNQlVM\" 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=14661411&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-adghhgaa-66VuRaNQlVM&vembed=0&video_id=66VuRaNQlVM&video_target=tpm-plugin-adghhgaa-66VuRaNQlVM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Law+of+Cosines+(SSS+AND+SAS)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLaw of Cosines (SSS AND SAS)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Applied Problems with the Law of Cosines<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"69\" data-end=\"461\">The Law of Cosines is especially useful for real-world problems where no right triangle is present. It allows us to calculate unknown distances, heights, or angles when two sides and the included angle (<strong data-start=\"272\" data-end=\"279\">SAS<\/strong>) or all three sides (<strong data-start=\"301\" data-end=\"308\">SSS<\/strong>) are known. This makes it a powerful tool in navigation, surveying, construction, and physics, where many situations naturally form oblique triangles.<\/p>\r\n<p data-start=\"463\" data-end=\"484\">The Law of Cosines:<\/p>\r\n\r\n<ul data-start=\"486\" data-end=\"644\">\r\n \t<li data-start=\"486\" data-end=\"538\">\r\n<p data-start=\"488\" data-end=\"538\">[latex]a^{2} = b^{2} + c^{2} - 2bc\\cos A[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"539\" data-end=\"591\">\r\n<p data-start=\"541\" data-end=\"591\">[latex]b^{2} = a^{2} + c^{2} - 2ac\\cos B[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"592\" data-end=\"644\">\r\n<p data-start=\"594\" data-end=\"644\">[latex]c^{2} = a^{2} + b^{2} - 2ab\\cos C[\/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: Solving Applied Problems<\/strong>\r\n<ol data-start=\"695\" data-end=\"1863\">\r\n \t<li data-start=\"695\" data-end=\"878\">\r\n<p data-start=\"698\" data-end=\"741\"><strong data-start=\"698\" data-end=\"739\">Translate the Problem into a Triangle<\/strong><\/p>\r\n\r\n<ul data-start=\"745\" data-end=\"878\">\r\n \t<li data-start=\"745\" data-end=\"820\">\r\n<p data-start=\"747\" data-end=\"820\">Draw a diagram of the situation (survey lines, navigation paths, etc.).<\/p>\r\n<\/li>\r\n \t<li data-start=\"824\" data-end=\"878\">\r\n<p data-start=\"826\" data-end=\"878\">Label sides and angles with the given information.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"880\" data-end=\"1094\">\r\n<p data-start=\"883\" data-end=\"915\"><strong data-start=\"883\" data-end=\"913\">Use SAS or SSS Information<\/strong><\/p>\r\n\r\n<ul data-start=\"919\" data-end=\"1094\">\r\n \t<li data-start=\"919\" data-end=\"1015\">\r\n<p data-start=\"921\" data-end=\"1015\"><strong data-start=\"921\" data-end=\"928\">SAS<\/strong>: two sides and the included angle \u2192 use the formula directly to find the third side.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1019\" data-end=\"1094\">\r\n<p data-start=\"1021\" data-end=\"1094\"><strong data-start=\"1021\" data-end=\"1028\">SSS<\/strong>: all three sides \u2192 rearrange the formula to solve for an angle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two hikers leave a campsite at the same time. One hikes 6 miles in a direction [latex]35\u00b0[\/latex] north of east. The other hikes 8 miles in a direction [latex]50\u00b0[\/latex] south of east. How far apart are the hikers?<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"911707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"911707\"]Draw a diagram showing both paths from the campsite.<\/p>\r\n<img class=\"alignnone wp-image-5634\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM.png\" alt=\"The diagram shows a point labeled Campsite at the origin of a dashed horizontal and vertical reference line. The upper segment from the Campsite to point C makes an angle of 35 degrees above the horizontal dashed line. The lower segment from the Campsite to point D makes an angle of 50 degrees below the horizontal dashed line.\" width=\"335\" height=\"319\" \/>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The angle between the two paths is: [latex] 35\u00b0 + 50\u00b0 = 85\u00b0 [\/latex]<\/p>\r\nThis is an SAS situation with sides 6 and 8, and included angle [latex]85\u00b0[\/latex].\r\n\r\nLet [latex]d[\/latex] be the distance between the hikers.\r\nUse the Law of Cosines:\r\n\r\n[latex] \\begin{aligned} d^2 &amp;= 6^2 + 8^2 - 2(6)(8)\\cos 85\u00b0 \\\\ d^2 &amp;= 36 + 64 - 96\\cos 85\u00b0 \\\\ d^2 &amp;= 100 - 96(0.0872) \\\\ d^2 &amp;= 100 - 8.37 \\\\ d^2 &amp;= 91.63 \\\\ d &amp;= \\sqrt{91.63} \\\\ d &amp;\\approx 9.6 \\text{ miles} \\end{aligned} [\/latex]\r\n\r\nThe hikers are approximately [latex]9.6[\/latex] miles apart.[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<div><section><\/section><section><\/section><section><\/section><section><\/section><section><\/section><section><\/section><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-bhfhfcce-mMLkdjPc5gQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mMLkdjPc5gQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bhfhfcce-mMLkdjPc5gQ\" 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=14661411&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bhfhfcce-mMLkdjPc5gQ&vembed=0&video_id=mMLkdjPc5gQ&video_target=tpm-plugin-bhfhfcce-mMLkdjPc5gQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Bearing+Word+Problem+Find+Distance+and+Angle+Using+Law+of+Sines+and+Law+of+Cosines_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBearing Word Problem Find Distance and Angle Using Law of Sines and Law of Cosines\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Area of a Triangle with Heron's Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"68\" data-end=\"368\">Heron\u2019s Formula is a method for finding the area of a triangle when <strong data-start=\"136\" data-end=\"165\">all three sides are known<\/strong>. Unlike the sine-based formula, it doesn\u2019t require knowing or calculating any angles. This makes it especially useful in surveying, geometry, or applied problems where side lengths are given directly.<\/p>\r\n<p data-start=\"370\" data-end=\"388\">Heron\u2019s Formula:<\/p>\r\n\r\n<ul data-start=\"390\" data-end=\"549\">\r\n \t<li data-start=\"390\" data-end=\"470\">\r\n<p data-start=\"392\" data-end=\"470\">First compute the <strong data-start=\"410\" data-end=\"427\">semiperimeter<\/strong>:<br data-start=\"428\" data-end=\"431\" \/>[latex]s = \\dfrac{a+b+c}{2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"471\" data-end=\"549\">\r\n<p data-start=\"473\" data-end=\"549\">Then the area is:<br data-start=\"490\" data-end=\"493\" \/>[latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/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 Heron's Formula<\/strong>\r\n<ol>\r\n \t<li data-start=\"597\" data-end=\"702\">\r\n<p data-start=\"600\" data-end=\"638\"><strong data-start=\"600\" data-end=\"636\">Check That the Triangle is Valid<\/strong><\/p>\r\n\r\n<ul data-start=\"642\" data-end=\"702\">\r\n \t<li data-start=\"642\" data-end=\"702\">\r\n<p data-start=\"644\" data-end=\"702\">The sum of any two sides must be greater than the third.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"704\" data-end=\"784\">\r\n<p data-start=\"707\" data-end=\"740\"><strong data-start=\"707\" data-end=\"738\">Calculate the Semiperimeter<\/strong><\/p>\r\n\r\n<ul data-start=\"744\" data-end=\"784\">\r\n \t<li data-start=\"744\" data-end=\"784\">\r\n<p data-start=\"746\" data-end=\"784\">Add all three sides and divide by 2.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"786\" data-end=\"900\">\r\n<p data-start=\"789\" data-end=\"812\"><strong data-start=\"789\" data-end=\"810\">Apply the Formula<\/strong><\/p>\r\n\r\n<ul data-start=\"1263\" data-end=\"1391\">\r\n \t<li data-start=\"816\" data-end=\"900\">\r\n<p data-start=\"818\" data-end=\"900\">Plug [latex]s[\/latex] and each side into [latex]\\sqrt{s(s-a)(s-b)(s-c)}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/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-cfbgdhba--YI6UC4qVEY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-YI6UC4qVEY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfbgdhba--YI6UC4qVEY\" 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=14661413&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfbgdhba--YI6UC4qVEY&vembed=0&video_id=-YI6UC4qVEY&video_target=tpm-plugin-cfbgdhba--YI6UC4qVEY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Heron's+formula+%7C+Perimeter%2C+area%2C+and+volume+%7C+Geometry+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHeron's formula | Perimeter, area, and volume | Geometry | Khan Academy\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>Use the Law of Cosines to solve oblique triangles.<\/li>\n<li>Solve applied problems using the Law of Cosines.<\/li>\n<li>Use Heron\u2019s formula to \ufb01nd the area of a triangle.<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Oblique Triangles with the Law of Cosines<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"78\" data-end=\"437\">The Law of Cosines extends the Pythagorean Theorem to all triangles, making it possible to solve oblique (non-right) triangles when the Law of Sines is not convenient. It is especially useful for <strong data-start=\"274\" data-end=\"281\">SAS<\/strong> (two sides and the included angle) and <strong data-start=\"321\" data-end=\"328\">SSS<\/strong> (all three sides) cases. With it, we can find unknown sides or angles even when no right angle is present.<\/p>\n<p data-start=\"439\" data-end=\"460\">The Law of Cosines:<\/p>\n<ul data-start=\"462\" data-end=\"620\">\n<li data-start=\"462\" data-end=\"514\">\n<p data-start=\"464\" data-end=\"514\">[latex]a^{2} = b^{2} + c^{2} - 2bc\\cos A[\/latex]<\/p>\n<\/li>\n<li data-start=\"515\" data-end=\"567\">\n<p data-start=\"517\" data-end=\"567\">[latex]b^{2} = a^{2} + c^{2} - 2ac\\cos B[\/latex]<\/p>\n<\/li>\n<li data-start=\"568\" data-end=\"620\">\n<p data-start=\"570\" data-end=\"620\">[latex]c^{2} = a^{2} + b^{2} - 2ab\\cos C[\/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 the Law of Cosines<\/strong><\/p>\n<ol>\n<li data-start=\"671\" data-end=\"886\">\n<p data-start=\"674\" data-end=\"699\"><strong data-start=\"674\" data-end=\"697\">Know When to Use It<\/strong><\/p>\n<ul data-start=\"703\" data-end=\"886\">\n<li data-start=\"703\" data-end=\"804\">\n<p data-start=\"705\" data-end=\"804\"><strong data-start=\"705\" data-end=\"712\">SAS<\/strong>: when two sides and the included angle are given, use the formula to find the third side.<\/p>\n<\/li>\n<li data-start=\"808\" data-end=\"886\">\n<p data-start=\"810\" data-end=\"886\"><strong data-start=\"810\" data-end=\"817\">SSS<\/strong>: when all three sides are given, use the formula to find an angle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"888\" data-end=\"1206\">\n<p data-start=\"891\" data-end=\"915\"><strong data-start=\"891\" data-end=\"913\">Solving for a Side<\/strong><\/p>\n<ul data-start=\"919\" data-end=\"1206\">\n<li data-start=\"919\" data-end=\"966\">\n<p data-start=\"921\" data-end=\"966\">Plug in known sides and the included angle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1208\" data-end=\"1598\">\n<p data-start=\"1211\" data-end=\"1237\"><strong data-start=\"1211\" data-end=\"1235\">Solving for an Angle<\/strong><\/p>\n<ul data-start=\"1241\" data-end=\"1598\">\n<li data-start=\"1241\" data-end=\"1334\">\n<p data-start=\"1243\" data-end=\"1334\">Rearrange the formula:<br data-start=\"1265\" data-end=\"1268\" \/>[latex]\\cos A = \\dfrac{b^{2} + c^{2} - a^{2}}{2bc}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1241\" data-end=\"1334\">Plug in the known sides<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1600\" data-end=\"1779\">\n<p data-start=\"1603\" data-end=\"1638\"><strong data-start=\"1603\" data-end=\"1636\">Combine with the Law of Sines<\/strong><\/p>\n<ul data-start=\"1642\" data-end=\"1779\">\n<li data-start=\"1642\" data-end=\"1779\">\n<p data-start=\"1644\" data-end=\"1779\">After using the Law of Cosines to find one side or angle, switch to the Law of Sines for quicker calculations of the remaining parts.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1781\" data-end=\"1945\">\n<p data-start=\"1784\" data-end=\"1808\"><strong data-start=\"1784\" data-end=\"1806\">Check for Validity<\/strong><\/p>\n<ul data-start=\"1812\" data-end=\"1945\">\n<li data-start=\"1812\" data-end=\"1887\">\n<p data-start=\"1814\" data-end=\"1887\">Angles must add up to [latex]180^\\circ[\/latex] (or [latex]\\pi[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1891\" data-end=\"1945\">\n<p data-start=\"1893\" data-end=\"1945\">Side lengths must satisfy the triangle inequality.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In triangle ABC, [latex]b = 12[\/latex], [latex]c = 15[\/latex], and [latex]A = 65\u00b0[\/latex]. Find side [latex]a[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q326789\">Show Solution<\/button><\/p>\n<div id=\"q326789\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">We have two sides and the included angle, so use the Law of Cosines: [latex]a^2 = b^2 + c^2 - 2bc\\cos A[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Substitute the known values: [latex]\\begin{aligned} a^2 &= 12^2 + 15^2 - 2(12)(15)\\cos 65\u00b0 \\\\ a^2 &= 144 + 225 - 360\\cos 65\u00b0 \\ a^2 &= 369 - 360(0.4226) \\\\ a^2 &= 369 - 152.14 \\\\ a^2 &= 216.86 \\\\ a &= \\sqrt{216.86} \\\\ a &\\approx 14.7 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Side [latex]a[\/latex] is approximately [latex]14.7[\/latex] units.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/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-adghhgaa-66VuRaNQlVM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/66VuRaNQlVM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-adghhgaa-66VuRaNQlVM\" 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=14661411&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-adghhgaa-66VuRaNQlVM&#38;vembed=0&#38;video_id=66VuRaNQlVM&#38;video_target=tpm-plugin-adghhgaa-66VuRaNQlVM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Law+of+Cosines+(SSS+AND+SAS)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLaw of Cosines (SSS AND SAS)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Applied Problems with the Law of Cosines<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"69\" data-end=\"461\">The Law of Cosines is especially useful for real-world problems where no right triangle is present. It allows us to calculate unknown distances, heights, or angles when two sides and the included angle (<strong data-start=\"272\" data-end=\"279\">SAS<\/strong>) or all three sides (<strong data-start=\"301\" data-end=\"308\">SSS<\/strong>) are known. This makes it a powerful tool in navigation, surveying, construction, and physics, where many situations naturally form oblique triangles.<\/p>\n<p data-start=\"463\" data-end=\"484\">The Law of Cosines:<\/p>\n<ul data-start=\"486\" data-end=\"644\">\n<li data-start=\"486\" data-end=\"538\">\n<p data-start=\"488\" data-end=\"538\">[latex]a^{2} = b^{2} + c^{2} - 2bc\\cos A[\/latex]<\/p>\n<\/li>\n<li data-start=\"539\" data-end=\"591\">\n<p data-start=\"541\" data-end=\"591\">[latex]b^{2} = a^{2} + c^{2} - 2ac\\cos B[\/latex]<\/p>\n<\/li>\n<li data-start=\"592\" data-end=\"644\">\n<p data-start=\"594\" data-end=\"644\">[latex]c^{2} = a^{2} + b^{2} - 2ab\\cos C[\/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: Solving Applied Problems<\/strong><\/p>\n<ol data-start=\"695\" data-end=\"1863\">\n<li data-start=\"695\" data-end=\"878\">\n<p data-start=\"698\" data-end=\"741\"><strong data-start=\"698\" data-end=\"739\">Translate the Problem into a Triangle<\/strong><\/p>\n<ul data-start=\"745\" data-end=\"878\">\n<li data-start=\"745\" data-end=\"820\">\n<p data-start=\"747\" data-end=\"820\">Draw a diagram of the situation (survey lines, navigation paths, etc.).<\/p>\n<\/li>\n<li data-start=\"824\" data-end=\"878\">\n<p data-start=\"826\" data-end=\"878\">Label sides and angles with the given information.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"880\" data-end=\"1094\">\n<p data-start=\"883\" data-end=\"915\"><strong data-start=\"883\" data-end=\"913\">Use SAS or SSS Information<\/strong><\/p>\n<ul data-start=\"919\" data-end=\"1094\">\n<li data-start=\"919\" data-end=\"1015\">\n<p data-start=\"921\" data-end=\"1015\"><strong data-start=\"921\" data-end=\"928\">SAS<\/strong>: two sides and the included angle \u2192 use the formula directly to find the third side.<\/p>\n<\/li>\n<li data-start=\"1019\" data-end=\"1094\">\n<p data-start=\"1021\" data-end=\"1094\"><strong data-start=\"1021\" data-end=\"1028\">SSS<\/strong>: all three sides \u2192 rearrange the formula to solve for an angle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two hikers leave a campsite at the same time. One hikes 6 miles in a direction [latex]35\u00b0[\/latex] north of east. The other hikes 8 miles in a direction [latex]50\u00b0[\/latex] south of east. How far apart are the hikers?<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q911707\">Show Solution<\/button><\/p>\n<div id=\"q911707\" class=\"hidden-answer\" style=\"display: none\">Draw a diagram showing both paths from the campsite.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5634\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM.png\" alt=\"The diagram shows a point labeled Campsite at the origin of a dashed horizontal and vertical reference line. The upper segment from the Campsite to point C makes an angle of 35 degrees above the horizontal dashed line. The lower segment from the Campsite to point D makes an angle of 50 degrees below the horizontal dashed line.\" width=\"335\" height=\"319\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM.png 942w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM-300x285.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM-768x730.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM-65x62.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM-225x214.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/13173145\/Screenshot-2026-02-13-at-10.31.11%E2%80%AFAM-350x333.png 350w\" sizes=\"(max-width: 335px) 100vw, 335px\" \/><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The angle between the two paths is: [latex]35\u00b0 + 50\u00b0 = 85\u00b0[\/latex]<\/p>\n<p>This is an SAS situation with sides 6 and 8, and included angle [latex]85\u00b0[\/latex].<\/p>\n<p>Let [latex]d[\/latex] be the distance between the hikers.<br \/>\nUse the Law of Cosines:<\/p>\n<p>[latex]\\begin{aligned} d^2 &= 6^2 + 8^2 - 2(6)(8)\\cos 85\u00b0 \\\\ d^2 &= 36 + 64 - 96\\cos 85\u00b0 \\\\ d^2 &= 100 - 96(0.0872) \\\\ d^2 &= 100 - 8.37 \\\\ d^2 &= 91.63 \\\\ d &= \\sqrt{91.63} \\\\ d &\\approx 9.6 \\text{ miles} \\end{aligned}[\/latex]<\/p>\n<p>The hikers are approximately [latex]9.6[\/latex] miles apart.<\/p><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section><\/section>\n<section><\/section>\n<section><\/section>\n<section><\/section>\n<section><\/section>\n<section><\/section>\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-bhfhfcce-mMLkdjPc5gQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/mMLkdjPc5gQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bhfhfcce-mMLkdjPc5gQ\" 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=14661411&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bhfhfcce-mMLkdjPc5gQ&#38;vembed=0&#38;video_id=mMLkdjPc5gQ&#38;video_target=tpm-plugin-bhfhfcce-mMLkdjPc5gQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Bearing+Word+Problem+Find+Distance+and+Angle+Using+Law+of+Sines+and+Law+of+Cosines_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBearing Word Problem Find Distance and Angle Using Law of Sines and Law of Cosines\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Area of a Triangle with Heron&#8217;s Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"68\" data-end=\"368\">Heron\u2019s Formula is a method for finding the area of a triangle when <strong data-start=\"136\" data-end=\"165\">all three sides are known<\/strong>. Unlike the sine-based formula, it doesn\u2019t require knowing or calculating any angles. This makes it especially useful in surveying, geometry, or applied problems where side lengths are given directly.<\/p>\n<p data-start=\"370\" data-end=\"388\">Heron\u2019s Formula:<\/p>\n<ul data-start=\"390\" data-end=\"549\">\n<li data-start=\"390\" data-end=\"470\">\n<p data-start=\"392\" data-end=\"470\">First compute the <strong data-start=\"410\" data-end=\"427\">semiperimeter<\/strong>:<br data-start=\"428\" data-end=\"431\" \/>[latex]s = \\dfrac{a+b+c}{2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"471\" data-end=\"549\">\n<p data-start=\"473\" data-end=\"549\">Then the area is:<br data-start=\"490\" data-end=\"493\" \/>[latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/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 Heron&#8217;s Formula<\/strong><\/p>\n<ol>\n<li data-start=\"597\" data-end=\"702\">\n<p data-start=\"600\" data-end=\"638\"><strong data-start=\"600\" data-end=\"636\">Check That the Triangle is Valid<\/strong><\/p>\n<ul data-start=\"642\" data-end=\"702\">\n<li data-start=\"642\" data-end=\"702\">\n<p data-start=\"644\" data-end=\"702\">The sum of any two sides must be greater than the third.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"704\" data-end=\"784\">\n<p data-start=\"707\" data-end=\"740\"><strong data-start=\"707\" data-end=\"738\">Calculate the Semiperimeter<\/strong><\/p>\n<ul data-start=\"744\" data-end=\"784\">\n<li data-start=\"744\" data-end=\"784\">\n<p data-start=\"746\" data-end=\"784\">Add all three sides and divide by 2.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"786\" data-end=\"900\">\n<p data-start=\"789\" data-end=\"812\"><strong data-start=\"789\" data-end=\"810\">Apply the Formula<\/strong><\/p>\n<ul data-start=\"1263\" data-end=\"1391\">\n<li data-start=\"816\" data-end=\"900\">\n<p data-start=\"818\" data-end=\"900\">Plug [latex]s[\/latex] and each side into [latex]\\sqrt{s(s-a)(s-b)(s-c)}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\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-cfbgdhba--YI6UC4qVEY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-YI6UC4qVEY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfbgdhba--YI6UC4qVEY\" 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=14661413&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cfbgdhba--YI6UC4qVEY&#38;vembed=0&#38;video_id=-YI6UC4qVEY&#38;video_target=tpm-plugin-cfbgdhba--YI6UC4qVEY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Heron's+formula+%7C+Perimeter%2C+area%2C+and+volume+%7C+Geometry+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHeron&#8217;s formula | Perimeter, area, and volume | Geometry | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Law of Cosines 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