{"id":3220,"date":"2025-08-15T23:32:13","date_gmt":"2025-08-15T23:32:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3220"},"modified":"2025-10-09T16:12:40","modified_gmt":"2025-10-09T16:12:40","slug":"angles-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles-apply-it\/","title":{"raw":"Angles: Apply It","rendered":"Angles: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Draw angles in standard position.<\/li>\r\n \t<li>Convert between degrees and radians.<\/li>\r\n \t<li>Convert angles between decimal form and degree, minutes, and seconds form<\/li>\r\n \t<li>Find coterminal angles.<\/li>\r\n \t<li>Find complementary and supplementary angles<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div data-test-render-count=\"1\">\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\r\n<p class=\"whitespace-normal break-words\">Angles are measured differently in navigation. While mathematicians measure angles counterclockwise from the positive x-axis (east), navigators measure angles clockwise from north. Understanding how to convert between these two systems is essential for pilots, sailors, surveyors, and anyone working with maps and GPS systems. In this page, you'll learn to work with both systems and convert between them.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">In mathematics, angles in <strong>standard position<\/strong> have:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Vertex at the origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Initial side along the positive x-axis (pointing east)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Positive angles measured counterclockwise<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Angles measured from 0\u00b0 to 360\u00b0<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Aircraft Navigation and Heading<\/h2>\r\n<p class=\"whitespace-normal break-words\">Pilots use [pb_glossary id=\"4573\"]heading[\/pb_glossary] to describe the direction an aircraft is pointed, measured as an angle in standard position from north (0\u00b0). A heading of 90\u00b0 means the aircraft points due east, 180\u00b0 means due south, and 270\u00b0 means due west.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>bearing from North<\/h3>\r\nA navigation angle measured clockwise from north, ranging from 0\u00b0 to 360\u00b0, where 0\u00b0 is north, 90\u00b0 is east, 180\u00b0 is south, and 270\u00b0 is west.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">An aircraft departs from an airport with a heading of 135\u00b0. Draw this angle in standard position.[reveal-answer q=\"749633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"749633\"]135\u00b0 is between 90\u00b0 and 180\u00b0, so the angle lands in Quadrant II. Since 135\u00b0 = 90\u00b0 + 45\u00b0, we rotate 90\u00b0 to point north, then an additional 45\u00b0 toward the west. The aircraft heading of 135\u00b0 points southeast (halfway between south and east).[\/hidden-answer]<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How To: Drawing Angles for Navigation Heading<\/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-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Determine which quadrant the angle belongs in (Quadrant I: 0\u00b0-90\u00b0, Quadrant II: 90\u00b0-180\u00b0, Quadrant III: 180\u00b0-270\u00b0, Quadrant IV: 270\u00b0-360\u00b0)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Place the vertex at the origin with the initial side along the positive x-axis (pointing east)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rotate counterclockwise the appropriate amount<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw the terminal side proportionately within the correct quadrant<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<ol>\r\n \t<li>An aircraft's heading is 30\u00b0 in mathematical standard position. Find the bearing from north.<\/li>\r\n \t<li>A ship travels on a bearing of 120\u00b0 from north. What is this angle in mathematical standard position?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"660176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660176\"]\r\n<ol>\r\n \t<li>A math angle of 30\u00b0 is in Quadrant I (between 0\u00b0 and 90\u00b0). In math standard position, 30\u00b0 points northeast (30\u00b0 counterclockwise from east). To find the bearing from north: [latex]\\begin{aligned} \\text{Bearing from North} &amp;= 90\u00b0 - 30\u00b0 \\ &amp;= 60\u00b0 \\end{aligned}[\/latex] The bearing from north is 60\u00b0 (which means 60\u00b0 clockwise from north, pointing northeast).<\/li>\r\n \t<li>A bearing of 120\u00b0 from north means 120\u00b0 clockwise from north, which points southeast.\r\nTo convert to mathematical standard position: [latex]\\begin{aligned} \\text{Math angle} &amp;= 90\u00b0 - \\text{Bearing} \\ &amp;= 90\u00b0 - 120\u00b0 \\ &amp;= -30\u00b0 \\end{aligned}[\/latex]\r\nSince we get a negative result, we need to add 360\u00b0: [latex]-30\u00b0 + 360\u00b0 = 330\u00b0[\/latex]\r\nThe mathematical standard position angle is 330\u00b0.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">When converting from bearing to math standard position, if you get a negative angle, add 360\u00b0.<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"h-px w-full pointer-events-none\" aria-hidden=\"true\"><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>NS bearing<\/h3>\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Describes direction as an angle from either north or south toward east or west<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Written as: N [angle] E, N [angle] W, S [angle] E, or S [angle] W<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The angle is always between 0\u00b0 and 90\u00b0<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A mathematical standard position angle of 40\u00b0 needs to be expressed as an NS bearing. Find the NS bearing.[reveal-answer q=\"949994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"949994\"]A math angle of 40\u00b0 is in Quadrant I, pointing northeast. The angle from the north-south line (north) is: [latex]90\u00b0 - 40\u00b0 = 50\u00b0[\/latex] Since the angle is in Quadrant I (northeast), the NS bearing is N 50\u00b0 E. This means: starting from north, rotate 50\u00b0 toward the east.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313439[\/ohm_question]<\/section>\r\n<p class=\"whitespace-normal break-words\">Here's how the same directions appear in each system:<\/p>\r\n\r\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\" style=\"width: 75.8404%;\">\r\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\"><strong>Direction<\/strong><\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\"><strong>Math Standard Position<\/strong><\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\"><strong>Bearing from North<\/strong><\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\"><strong>NS Bearing<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">East<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">0\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">90\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 90\u00b0 E or S 90\u00b0 E<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">North<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">90\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">0\u00b0 (or 360\u00b0)<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 0\u00b0 E<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">West<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">180\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">270\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 90\u00b0 W or S 90\u00b0 W<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">South<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">270\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">180\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 0\u00b0 E<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Northeast<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">45\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">45\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 45\u00b0 E<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Southeast<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">315\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">135\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 45\u00b0 E<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Southwest<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">225\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">225\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 45\u00b0 W<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Northwest<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">135\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">315\u00b0<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 45\u00b0 W<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Draw angles in standard position.<\/li>\n<li>Convert between degrees and radians.<\/li>\n<li>Convert angles between decimal form and degree, minutes, and seconds form<\/li>\n<li>Find coterminal angles.<\/li>\n<li>Find complementary and supplementary angles<\/li>\n<\/ul>\n<\/section>\n<div data-test-render-count=\"1\">\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<p class=\"whitespace-normal break-words\">Angles are measured differently in navigation. While mathematicians measure angles counterclockwise from the positive x-axis (east), navigators measure angles clockwise from north. Understanding how to convert between these two systems is essential for pilots, sailors, surveyors, and anyone working with maps and GPS systems. In this page, you&#8217;ll learn to work with both systems and convert between them.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">In mathematics, angles in <strong>standard position<\/strong> have:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Vertex at the origin<\/li>\n<li class=\"whitespace-normal break-words\">Initial side along the positive x-axis (pointing east)<\/li>\n<li class=\"whitespace-normal break-words\">Positive angles measured counterclockwise<\/li>\n<li class=\"whitespace-normal break-words\">Angles measured from 0\u00b0 to 360\u00b0<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Aircraft Navigation and Heading<\/h2>\n<p class=\"whitespace-normal break-words\">Pilots use <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_3220_4573\">heading<\/a> to describe the direction an aircraft is pointed, measured as an angle in standard position from north (0\u00b0). A heading of 90\u00b0 means the aircraft points due east, 180\u00b0 means due south, and 270\u00b0 means due west.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>bearing from North<\/h3>\n<p>A navigation angle measured clockwise from north, ranging from 0\u00b0 to 360\u00b0, where 0\u00b0 is north, 90\u00b0 is east, 180\u00b0 is south, and 270\u00b0 is west.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">An aircraft departs from an airport with a heading of 135\u00b0. Draw this angle in standard position.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q749633\">Show Solution<\/button><\/p>\n<div id=\"q749633\" class=\"hidden-answer\" style=\"display: none\">135\u00b0 is between 90\u00b0 and 180\u00b0, so the angle lands in Quadrant II. Since 135\u00b0 = 90\u00b0 + 45\u00b0, we rotate 90\u00b0 to point north, then an additional 45\u00b0 toward the west. The aircraft heading of 135\u00b0 points southeast (halfway between south and east).<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How To: Drawing Angles for Navigation Heading<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Determine which quadrant the angle belongs in (Quadrant I: 0\u00b0-90\u00b0, Quadrant II: 90\u00b0-180\u00b0, Quadrant III: 180\u00b0-270\u00b0, Quadrant IV: 270\u00b0-360\u00b0)<\/li>\n<li class=\"whitespace-normal break-words\">Place the vertex at the origin with the initial side along the positive x-axis (pointing east)<\/li>\n<li class=\"whitespace-normal break-words\">Rotate counterclockwise the appropriate amount<\/li>\n<li class=\"whitespace-normal break-words\">Draw the terminal side proportionately within the correct quadrant<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ol>\n<li>An aircraft&#8217;s heading is 30\u00b0 in mathematical standard position. Find the bearing from north.<\/li>\n<li>A ship travels on a bearing of 120\u00b0 from north. What is this angle in mathematical standard position?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q660176\">Show Solution<\/button><\/p>\n<div id=\"q660176\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>A math angle of 30\u00b0 is in Quadrant I (between 0\u00b0 and 90\u00b0). In math standard position, 30\u00b0 points northeast (30\u00b0 counterclockwise from east). To find the bearing from north: [latex]\\begin{aligned} \\text{Bearing from North} &= 90\u00b0 - 30\u00b0 \\ &= 60\u00b0 \\end{aligned}[\/latex] The bearing from north is 60\u00b0 (which means 60\u00b0 clockwise from north, pointing northeast).<\/li>\n<li>A bearing of 120\u00b0 from north means 120\u00b0 clockwise from north, which points southeast.<br \/>\nTo convert to mathematical standard position: [latex]\\begin{aligned} \\text{Math angle} &= 90\u00b0 - \\text{Bearing} \\ &= 90\u00b0 - 120\u00b0 \\ &= -30\u00b0 \\end{aligned}[\/latex]<br \/>\nSince we get a negative result, we need to add 360\u00b0: [latex]-30\u00b0 + 360\u00b0 = 330\u00b0[\/latex]<br \/>\nThe mathematical standard position angle is 330\u00b0.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">When converting from bearing to math standard position, if you get a negative angle, add 360\u00b0.<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"h-px w-full pointer-events-none\" aria-hidden=\"true\">\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>NS bearing<\/h3>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Describes direction as an angle from either north or south toward east or west<\/li>\n<li class=\"whitespace-normal break-words\">Written as: N [angle] E, N [angle] W, S [angle] E, or S [angle] W<\/li>\n<li class=\"whitespace-normal break-words\">The angle is always between 0\u00b0 and 90\u00b0<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A mathematical standard position angle of 40\u00b0 needs to be expressed as an NS bearing. Find the NS bearing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q949994\">Show Solution<\/button><\/p>\n<div id=\"q949994\" class=\"hidden-answer\" style=\"display: none\">A math angle of 40\u00b0 is in Quadrant I, pointing northeast. The angle from the north-south line (north) is: [latex]90\u00b0 - 40\u00b0 = 50\u00b0[\/latex] Since the angle is in Quadrant I (northeast), the NS bearing is N 50\u00b0 E. This means: starting from north, rotate 50\u00b0 toward the east.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313439\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313439&theme=lumen&iframe_resize_id=ohm313439&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"whitespace-normal break-words\">Here&#8217;s how the same directions appear in each system:<\/p>\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\" style=\"width: 75.8404%;\">\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\"><strong>Direction<\/strong><\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\"><strong>Math Standard Position<\/strong><\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\"><strong>Bearing from North<\/strong><\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\"><strong>NS Bearing<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">East<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">0\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">90\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 90\u00b0 E or S 90\u00b0 E<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">North<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">90\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">0\u00b0 (or 360\u00b0)<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 0\u00b0 E<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">West<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">180\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">270\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 90\u00b0 W or S 90\u00b0 W<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">South<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">270\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">180\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 0\u00b0 E<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Northeast<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">45\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">45\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 45\u00b0 E<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Southeast<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">315\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">135\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 45\u00b0 E<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Southwest<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">225\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">225\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">S 45\u00b0 W<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 17.3038%;\">Northwest<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 27.8639%;\">135\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 26.4487%;\">315\u00b0<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"width: 70.292%;\">N 45\u00b0 W<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_3220_4573\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_3220_4573\"><div tabindex=\"-1\"><p>The direction an aircraft is pointed, expressed as an angle measured clockwise from north in standard position.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":67,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"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\/3220"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3220\/revisions"}],"predecessor-version":[{"id":4576,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3220\/revisions\/4576"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3220\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3220"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3220"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3220"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3220"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}