{"id":57946,"date":"2019-12-12T14:47:32","date_gmt":"2019-12-12T14:47:32","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=57946"},"modified":"2026-03-26T10:05:00","modified_gmt":"2026-03-26T15:05:00","slug":"law-of-cosines","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/law-of-cosines\/","title":{"rendered":"Law of Cosines"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_vhVnuxOeeRc\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_vhVnuxOeeRc\" data-source-videoID=\"vhVnuxOeeRc\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Law of Cosines Video\" height=\"720\" width=\"1280\" class=\"size-full\" data-matomo-title = \"Law of Cosines\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_vhVnuxOeeRc:hover {cursor:pointer;} img#videoThumbnailImage_vhVnuxOeeRc {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/thumb.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_vhVnuxOeeRc\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_vhVnuxOeeRc\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Law of Cosines\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_vhVnuxOeeRc\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_vhVnuxOeeRc\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_vhVnuxOeeRc\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction K27_Function() {\n  var x = document.getElementById(\"K27\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"K27_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"K27\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Pythagorean_Theorem\" class=\"smooth-scroll\">Pythagorean Theorem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Pythagorean_Theorem_Examples\" class=\"smooth-scroll\">Pythagorean Theorem Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Law_of_Cosines_Practice_Questions\" class=\"smooth-scroll\">Law of Cosines Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>How often do you break laws? Hopefully not too often, but you may actually be breaking more laws than you think!<\/p>\n<p>I\u2019m not talking about just missing a speed limit sign; I\u2019m talking about mathematical laws that help us take the most optimal route toward a correct answer.<\/p>\n<p>In this video, we\u2019re going to take a look specifically at the law of cosines and how it simplifies finding angles.<\/p>\n<h2><span id=\"Pythagorean_Theorem\" class=\"m-toc-anchor\"><\/span>Pythagorean Theorem<\/h2>\n<p>\nLet\u2019s start by taking a look at the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/pythagorean-theorem\/\">Pythagorean theorem<\/a>.<\/p>\n<p>It\u2019s that easy-to-remember formula for finding the longest side of a right triangle when we know the length of the other two sides.<\/p>\n<p>Since we know that the Pythagorean theorem only works when there is a 90\u00b0 angle, what happens when you\u2019re working with a triangle that isn\u2019t a right triangle? That\u2019s where the law of cosines comes in\u2014it allows us to find the third length of <em>any<\/em> triangle.<\/p>\n<h3><span id=\"Pythagorean_Theorem_Formula\" class=\"m-toc-anchor\"><\/span>Pythagorean Theorem Formula<\/h3>\n<p>\nHere\u2019s the formula:<\/p>\n<div class=\"examplesentence\">\\(c^2=a^2+b^2-2ab\\cdot\\text{cos}(C)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAs you can see from the end of the formula, we need one more piece of information than the Pythagorean theorem requires\u2014we need the measure of an angle. To be specific, we need the measure of the angle between the two sides we know (sides \\(a\\) and \\(b\\)).<\/p>\n<p>Here\u2019s triangle ABC with the sides labeled lowercase a, b, and c:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles.webp\" alt=\"Diagram of a triangle with sides labeled a, b, and c and angles labeled A, B, and C.\" width=\"2124\" height=\"1281\" class=\"aligncenter size-full wp-image-229369\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles.webp 2124w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles-300x181.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles-1024x618.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles-768x463.webp 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles-1536x926.webp 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Labeled-Triangles-2048x1235.webp 2048w\" sizes=\"auto, (max-width: 2124px) 100vw, 2124px\" \/><\/p>\n<p>It\u2019s important to know that, when naming a triangle, we use capital letters. And, to indicate sides, lowercase letters are used that match the opposite <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/angles\/\">angle<\/a> label. For example, side \\(a\\) is opposite angle \\(A\\).<\/p>\n<p>So in order to use the law of cosines to find side \\(c\\), we would need to know side \\(b\\), side \\(a\\), and angle \\(C\\), which is opposite side \\(c\\).<\/p>\n<h2><span id=\"Pythagorean_Theorem_Examples\" class=\"m-toc-anchor\"><\/span>Pythagorean Theorem Examples<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example 1<\/h3>\n<p>\nLet\u2019s try it out.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-1.png\" alt=\"Isosceles triangle with angles labeled: 45\u00b0 at vertex C. Sides AB is labeled c, AC is labeled b (10 cm), and BC is labeled a (10 cm).\" width=\"521\" height=\"532\" class=\"aligncenter size-full wp-image-229372\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-1.png 1042w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-1-294x300.png 294w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-1-1003x1024.png 1003w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-1-768x784.png 768w\" sizes=\"auto, (max-width: 521px) 100vw, 521px\" \/><\/p>\n<p>First, we need to check to see if we have everything we need to use the law of cosines. We have the measure of two sides (\\(a\\) and \\(b\\), both 10 centimeters), and the angle between those two sides is angle \\(C\\), which measures 45\u00b0. That\u2019s all we need to know, so let\u2019s plug sides \\(a\\) and \\(b\\) and angle \\(C\\) into the formula:<\/p>\n<div class=\"examplesentence\">\\(c^2=a^2+b^2-2ab\\text{ cos}(C)\\)<br \/>\n\\(c^2=10^2+10^2-2(10)(10)\\text{ cos}(45)\\)<\/div>\n<p>\n&nbsp;<br \/>\nBe careful with the order of operations here. Evaluate the exponents first, then take the cosine of 45. Since 10 squared is equal to 100, substitute 100 for 10 squared in both places. Then, multiply 2 times 10 times 10, which is equal to 200. Then, find the cosine of 45 degrees. <\/p>\n<div class=\"examplesentence\">\\(c^2=100+100-200(0.70710678)\\)<br \/>\n\\(c^2=200-144.42\\)<\/div>\n<p>\n&nbsp;<br \/>\nFind the difference between 200 and 141.42.<\/p>\n<div class=\"examplesentence\">\\(c^2=55.58\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhen we get to the point where we have \\(c^2\\) equal to a number, we take the square root of each side to find that \\(c\\) is approximately 7.65 centimeters.<\/p>\n<div class=\"examplesentence\">\\(c\u22457.65\\text{ cm}\\)<\/div>\n<p>\n&nbsp;<br \/>\nA common sense check of that answer tells us that it\u2019s reasonable since sides \\(a\\) and \\(b\\) both measure 10 centimeters and their opposite angles are larger than angle C, so our answer should be under 10 (which it is).<\/p>\n<p>That\u2019s all there is to using the law of cosines for finding the missing side of a triangle. In fact, it\u2019s a bit more versatile than the Pythagorean theorem because we can label any of the sides \\(a\\), \\(b\\) and \\(c\\), as long as we label the angle opposite that side \\(A\\), \\(B\\), and \\(C\\).<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example 2<\/h3>\n<p>\nLet\u2019s look at another problem.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.webp\" alt=\"A blue triangle with sides measuring 8 cm and 12 cm and an interior angle of 120 degrees.\" width=\"547.47\" height=\"332.64\" class=\"aligncenter size-full wp-image-229375\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.webp 1659w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2-300x182.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2-1024x622.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2-768x467.webp 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2-1536x933.webp 1536w\" sizes=\"(max-width: 1659px) 100vw, 1659px\" \/><\/p>\n<p>We haven\u2019t named any of the sides or angles yet, but we can see that we have been given the measure of two of the sides and the angle between them. So we should label the angle \\(C\\) and the sides \\(a\\) and \\(b\\), like this:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2.webp\" alt=\"Illustration of triangle ABC with angle C at 120 degrees, side a measuring 8 cm, and side b measuring 12 cm.\" width=\"595.65\" height=\"400.29\" class=\"aligncenter size-full wp-image-229378\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2.webp 1805w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2-300x202.webp 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2-1024x688.webp 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2-768x516.webp 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/Triangle-example-2.2-1536x1032.webp 1536w\" sizes=\"(max-width: 1805px) 100vw, 1805px\" \/><\/p>\n<p>Now we can go ahead and substitute the values into the law of cosines formula that we used previously.<\/p>\n<div class=\"examplesentence\">\\(c^2=a^2+b^2-2ab\\text{ cos}(C)\\)<br \/>\n\\(c^2=(8)^2+(12)^2-2(8)(12)\\text{ cos}(120)\\)<\/div>\n<p>\n&nbsp;<br \/>\nSince \\(8^2=64\\), substitute that value in. Since \\(12^2=144\\), substitute that value in as well. Then, we know that \\(2\\times 8 \\times 12=192\\). Finally, the cosine of 120\u00b0 is equal to -0.5.<\/p>\n<div class=\"examplesentence\">\\(c^2=64+144-192(-0.5)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAdd 64 and 144 together to compose a sum of 208. Then, multiply -192 times -0.5 to get a product of 96.<\/p>\n<div class=\"examplesentence\">\\(c^2=208+96\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, add 208 and 96 to get a sum of 304.<\/p>\n<div class=\"examplesentence\">\\(c^2=304\\)<\/div>\n<p>\n&nbsp;<br \/>\nTake the square root of 304.<\/p>\n<div class=\"examplesentence\">\\(\\sqrt{c^2}=\\sqrt{304}\\) <br \/>\n\\(c\u224517.44\\text{ cm}\\)<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s try our common sense check again. Side \\(c\\) is opposite the largest angle, so it should be larger than the other sides (and it is). It\u2019s also less than the sum of the other two sides, which it must be in order for this shape to be a triangle.<\/p>\n<p>Okay, that\u2019s all for this review. Don\u2019t break the law of cosines or else the math police will come after you!<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<ul class=\"citelist\">\n<li><a href=\"https:\/\/www.mathsisfun.com\/algebra\/trig-cosine-law.html\"target=\"_blank\">\u201cThe Law of Cosines.\u201d 2017. Mathsisfun.com<\/a><\/li>\n<\/ul>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Law_of_Cosines_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Law of Cosines Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nIn order to solve for the missing third side length of a triangle using the law of cosines, what two things must be provided?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">Hypotenuse and the measure of one angle <\/div><div class=\"PQ\"  id=\"PQ-1-2\">Measure of all angles and the longest side length <\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">Measure of two sides and the measure of the angle between those two sides <\/div><div class=\"PQ\"  id=\"PQ-1-4\">Measure of one side and the measure of one angle <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>In order to determine the missing third side length of a triangle, you need to know the measure of two sides and the angle measure between them.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich equation correctly shows the law of cosines?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">\\(a^2+b^2-2ab\\cos(C)=c^2\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(c^2+b^2+2ab\\cos(C)=a^2\\)<\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\(a^2-b^2-2ab\\sin(C)=c^2\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(a^3+b^3-2ab\\cos(C)=c^3\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The formula resembles the Pythagorean theorem, except the law of cosines works for any triangle, not just right triangles.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nSolve for the missing side length \\(c\\).<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Law-of-Cosine-Triangle-example-1.svg\" alt=\"Triangle ABC with angle C marked as 37 degrees, side AC measuring 11 cm, side BC measuring 8 cm, and side AB labeled as c.\" width=\"242\" height=\"228.8\" class=\"aligncenter size-full wp-image-274780\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">8.67 cm<\/div><div class=\"PQ\"  id=\"PQ-3-2\">7.67 cm<\/div><div class=\"PQ\"  id=\"PQ-3-3\">9.67 cm<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">6.67 cm<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The missing side length \\(c\\) can be calculated using the law of cosines.<\/p>\n<p>The formula \\(c^2=a^2+b^2-2ab\\cos(C)\\) becomes:<\/p>\n<p style=\"text-align: center\">\\(c^2=8^2+11^2-(2\\times8\\times11)\\cos(37)\\)<\/p>\n<p>\\(\\cos(37)\\) is approximately 0.7986, so this can also be plugged into the formula.<\/p>\n<p>When the order of operations is carefully followed:<\/p>\n<p style=\"text-align: center\">\\(c^2=8^2+11^2-(2\\times8\\times11)\\cos(37)\\)<\/p>\n<p>This simplifies to \\(c^2\\approx44.45\\).<\/p>\n<p>Now the square root of both sides can be calculated, which means that \\(c\\approx6.67\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDan wants to measure the distance across the pond in his backyard. The pond is too shallow to use a boat to measure the distance, so Dan decides to apply his knowledge of the law of cosines to make the measurement. He stands back from the pond, and measures 800 feet to one end of the pond, and 900 feet to the other end. The angle between these distances is 70\u00b0. What will the distance across the pond be?<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-92152 \" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/Triangle-with-side-lengths-of-800-and-900-ft-with-70-degrees-between-the-sides.png\" alt=\"Triangle with side lengths of 800 and 900 ft with 70 degrees between the sides\" width=\"424.8\" height=\"310.4\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/Triangle-with-side-lengths-of-800-and-900-ft-with-70-degrees-between-the-sides.png 972w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/Triangle-with-side-lengths-of-800-and-900-ft-with-70-degrees-between-the-sides-300x219.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/Triangle-with-side-lengths-of-800-and-900-ft-with-70-degrees-between-the-sides-768x562.png 768w\" sizes=\"(max-width: 972px) 100vw, 972px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">900.5 feet<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">978.5 feet<\/div><div class=\"PQ\"  id=\"PQ-4-3\">998.5 feet<\/div><div class=\"PQ\"  id=\"PQ-4-4\">1,978.5 feet<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The law of cosines can be used to determine the distance across the pond because two side lengths are known, as well as the angle between them.<\/p>\n<p>The formula \\(c^2=a^2+b^2-2ab\\cos(C)\\) becomes:<\/p>\n<p style=\"text-align: center\">\\(c^2=800^2+900^2-(2\\times800\\times900)\\cos(70)\\)<\/p>\n<p>This simplifies to \\(c^2\\approx3{,}420.2\\). When the square root of both sides of the equation is calculated, \\(c\\approx978.5\\text{ ft}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMegan is designing a triangular park for her city. She needs one length of the park to be 1 mile, and another length of the park to be two miles. The angle between these two sides needs to be 54\u00b0. How long will the third side of the triangular park be?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Law-of-Cosine-Triangle-example-2.svg\" alt=\"A triangle with sides labeled 1 mi, 2 mi, and c, and an included angle of 54\u00b0 between the 1 mi and 2 mi sides.\" width=\"320.1\" height=\"141.9\" class=\"aligncenter size-full wp-image-274783\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">1.63 mi<\/div><div class=\"PQ\"  id=\"PQ-5-2\">2.63 mi<\/div><div class=\"PQ\"  id=\"PQ-5-3\">3.63 mi<\/div><div class=\"PQ\"  id=\"PQ-5-4\">4.63 mi<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Two side lengths of the park are provided, as well as the angle in between them. This means that the third side of the triangular park can be calculated using the law of cosines.<\/p>\n<p>The formula \\(c^2=a^2+b^2-2ab\\cos(C)\\) becomes:<\/p>\n<p style=\"text-align: center\">\\(c^2=1^2+2^2-(2\\times1\\times2)\\cos(54)\\)<\/p>\n<p>When the order of operations are carefully applied, this simplifies to \\(c^2\\approx2.65\\). When the square roots of both sides are calculated, \\(c\\approx1.63\\text{ mi}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/trigonometry\/\">Return to Trigonometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Trigonometry Videos<\/p>\n","protected":false},"author":1,"featured_media":229363,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-57946","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-cosine-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57946","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=57946"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57946\/revisions"}],"predecessor-version":[{"id":263719,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/57946\/revisions\/263719"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/229363"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=57946"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}