{"id":1504,"date":"2025-07-25T02:17:36","date_gmt":"2025-07-25T02:17:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1504"},"modified":"2026-03-12T05:22:17","modified_gmt":"2026-03-12T05:22:17","slug":"angles-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles-fresh-take\/","title":{"raw":"Angles: Fresh Take","rendered":"Angles: Fresh Take"},"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>Find coterminal angles.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Angles in Standard Position<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nAn angle in standard position has its vertex at the origin and its initial side lying along the positive x-axis. The direction of rotation determines the sign of the angle:\r\n<ul>\r\n \t<li>counterclockwise rotations are positive<\/li>\r\n \t<li>clockwise rotations are negative<\/li>\r\n<\/ul>\r\nUnderstanding an angle in standard position is important because it provides a consistent way to classify angles, no matter their size or sign. By using standard position, you can easily determine which quadrant an angle's terminal side lands in and connect angles to trigonometric values.\r\n\r\n<strong>Quick Tips: Drawing Angles in Standard Position<\/strong>\r\n<ol>\r\n \t<li>Start on the positive x-axis: Place the initial side along the positive x-axis, vertex at the origin.<\/li>\r\n \t<li>Choose the direction of rotation:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Counterclockwise \u2192 positive angle<\/li>\r\n \t<li>Clockwise \u2192 negative angle<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Mark the rotation: Move the terminal side to the correct position according to the given measure.<\/li>\r\n \t<li>Identify the quadrant:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li style=\"text-align: left;\">Quandrant I: [latex]0^\\circ - 90^\\circ[\/latex]<\/li>\r\n \t<li>Quandrant II: [latex]90^\\circ - 180^\\circ[\/latex]<\/li>\r\n \t<li>Quandrant III: [latex]180^\\circ - 270^\\circ[\/latex]<\/li>\r\n \t<li>Quandrant IV: [latex]270^\\circ - 360^\\circ[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Extend beyond one rotation if needed: Angles greater than [latex]360^\\circ[\/latex] or less than [latex]0^\\circ[\/latex] wrap around, but still land on a terminal side in one of the four quadrants.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Draw an angle of -135\u00ba in standard position. Indicate the quadrant of its terminal side.\r\n[reveal-answer q=\"782027\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"782027\"]Remember the initial side is on the positive [latex]x[\/latex]-axis and that a negative angle rotates clockwise.[\/hidden-answer]\r\n[reveal-answer q=\"44984\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"44984\"]Start on the positive x-axis and rotate clockwise 135\u00ba degrees. The terminal side lands in Quadrant II (pointing halfway between the negative x-axis and the negative y-axis).\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3360\/2018\/06\/14234838\/CNX_Precalc_Figure_05_01_0082.jpg\" alt=\"Graph of a negative 135 degree angle.\" width=\"487\" height=\"383\" \/> Angle of -135\u00ba in standard position.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A drone's camera points from the origin through [latex](5,-2)[\/latex]. Sketch the angle in standard position and name its quadrant.\r\n[reveal-answer q=\"601541\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"601541\"]Use the signs of the given point to find the quadrant. In this example, we have [latex](+,-)[\/latex]. Which quadrant includes this point?[\/hidden-answer]\r\n[reveal-answer q=\"174014\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"174014\"][latex](5, -2)[\/latex] is right 5 units and down 2 units[latex]\\Rightarrow[\/latex] Quadrant IV. Draw the ray from the origin through [latex](5,-2).[\/latex]\r\n\r\n[caption id=\"attachment_4193\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-4193\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-300x300.png\" alt=\"Coordinate plane with a ray from the origin through (5, -2); small arc from positive x-axis to the ray; labeled Quadrant IV.\" width=\"300\" height=\"300\" \/> Angle through (5,-2) in Quadrant IV.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">On a DJ turntable, a sticker has rotated by [latex]\\frac{11\\pi}{6}[\/latex] from the positive [latex]x[\/latex]-axis. Sketch the angle and state which axis it is closest to.\r\n[reveal-answer q=\"28376\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"28376\"]Convert [latex]\\dfrac{11\\pi}{6}[\/latex] to degrees. You could also compare it to [latex]2\\pi[\/latex] (or [latex]\\dfrac{12\\pi}{6}[\/latex]) to judge which axis it is closest to.[\/hidden-answer]\r\n[reveal-answer q=\"176844\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"176844\"][latex]\\frac{11\\pi}{6} = 330^\\circ\\ \\Rightarrow[\/latex] Quadrant IV, closer to the positive [latex]x[\/latex]-axis [latex](360^\\circ)[\/latex] than the negative [latex]y[\/latex]-axis [latex](270^\\circ)[\/latex].[caption id=\"attachment_4198\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-4198\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-300x300.png\" alt=\"Coordinate plane with a ray from the origin at 11 pi over 6 (30 degrees below the positive x-axis).\" width=\"300\" height=\"300\" \/> Angle [latex]\\frac{11\\pi}{6}[\/latex] in standard position; terminal side in Quadrant IV, closest to the positive [latex]x[\/latex]-axis.[\/caption][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\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-aaeaadea-CNsRpez8fZE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/CNsRpez8fZE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aaeaadea-CNsRpez8fZE\" 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=13867583&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aaeaadea-CNsRpez8fZE&vembed=0&video_id=CNsRpez8fZE&video_target=tpm-plugin-aaeaadea-CNsRpez8fZE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Draw+an+Angle+in+Standard+Position_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Draw an Angle in Standard Position\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Converting Between Degrees and Radians<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nAngles can be measured in degrees or radians, and being able to switch between the two is essential in precalculus and beyond. A full circle is [latex]360^\\circ[\/latex], which equals [latex]2\\pi[\/latex] radians. A key conversion relationship to remember is: [latex]180^\\circ=\\pi[\/latex] radians. Degrees are more often seen in everyday use, while radians are the natural unit in higher mathematics because they link angle measure directly to arc length.\r\n\r\n<strong>Quick Tips: Converting Between Degrees and Radians<\/strong>\r\n<ol>\r\n \t<li>Memorize the core relationship: [latex]180^\\circ=\\pi[\/latex] radians.<\/li>\r\n \t<li>Degrees \u2192 Radians: Multiply by [latex]\\dfrac{\\pi}{180}[\/latex].\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Ex: [latex]60^\\circ \\cdot \\dfrac{\\pi}{180} = \\dfrac{\\pi}{3}[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Radians \u2192 Degrees: Multiply by [latex]\\dfrac{180}{\\pi}[\/latex].\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Ex: [latex]\\dfrac{3\\pi}{4} \\cdot \\dfrac{180}{\\pi} = 135^\\circ[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Keep It Exact (Unless Told Otherwise)<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>When converting to radians, leave your answer in terms of [latex]\\pi[\/latex] (for example, [latex]\\pi\/6[\/latex] instead of 0.52).<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Know the \"Common Angles\":<\/li>\r\n<\/ol>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 110px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; text-align: center; height: 22px;\"><strong>Degrees<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center; height: 22px;\"><strong>Radians<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]30^\\circ[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]45^\\circ[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]60^\\circ[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]90^\\circ[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese show up so often that it\u2019s worth memorizing them.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Convert [latex]37.5^\\circ[\/latex] to radians.\r\n[reveal-answer q=\"282667\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"282667\"]Multiply by [latex]\\dfrac{\\pi}{180^\\circ}[\/latex] and simplify the fraction carefully.[\/hidden-answer]\r\n[reveal-answer q=\"827432\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"827432\"][latex]\\begin{aligned} 37.5^\\circ \\Rightarrow \\text { radians} &amp;= 37.5^\\circ \\cdot \\dfrac{\\pi}{180^\\circ} \\\\ &amp;= \\dfrac{5\\pi}{24} \\end{aligned}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Convert [latex]-\\dfrac{5\\pi}{3}[\/latex] to degrees.\r\n[reveal-answer q=\"996357\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"996357\"]Multiply by [latex]\\dfrac{180^\\circ}{\\pi}[\/latex] and keep the negative sign all the way through.[\/hidden-answer]\r\n[reveal-answer q=\"571988\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"571988\"][latex]\\begin{aligned} -\\dfrac{5\\pi}{3} \\Rightarrow \\text { degrees} &amp;= -\\dfrac{5\\pi}{3} \\cdot \\dfrac{180^\\circ}{\\pi} \\\\ &amp;= -300^\\circ \\end{aligned}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A bike wheel turns [latex]2.4[\/latex] revolutions. Express this rotation in degrees and radians.\r\n[reveal-answer q=\"638993\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"638993\"]Remember that one revolution [latex] = 360^\\circ[\/latex] for degrees and also [latex] = 2\\pi[\/latex] for radians. Multiply [latex]2.4[\/latex] by each one.[\/hidden-answer]\r\n[reveal-answer q=\"713110\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"713110\"][latex]\\begin{aligned} 2.4 \\text { revolutions} \\Rightarrow \\text { degrees} &amp;= 2.4 \\cdot 360^\\circ \\\\ &amp;= 864^\\circ \\end{aligned}[\/latex]\r\n[latex]\\begin{aligned} 2.4 \\text { revolutions} \\Rightarrow \\text { radians} &amp;= 2.4 \\cdot 2\\pi \\\\ &amp;= 4.8\\pi \\\\ &amp;= \\dfrac{24\\pi}{5} \\text { radians} \\end{aligned}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\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-cgghebbc-1pjRy6M-KKg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1pjRy6M-KKg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cgghebbc-1pjRy6M-KKg\" 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=13867756&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cgghebbc-1pjRy6M-KKg&vembed=0&video_id=1pjRy6M-KKg&video_target=tpm-plugin-cgghebbc-1pjRy6M-KKg'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Degrees+to+Radians_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Degrees to Radians\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n&nbsp;\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-ddagfebd-aw6H3AntIaQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/aw6H3AntIaQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddagfebd-aw6H3AntIaQ\" 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=13867765&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddagfebd-aw6H3AntIaQ&vembed=0&video_id=aw6H3AntIaQ&video_target=tpm-plugin-ddagfebd-aw6H3AntIaQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Radians+to+Degrees_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Radians to Degrees\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Finding Coterminal Angles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nCoterminal angles are angles that <strong data-start=\"181\" data-end=\"206\">end in the same place<\/strong> \u2014 they share the same terminal side when drawn in standard position. Even though they may look different on paper (like [latex]-45^\\circ[\/latex] and [latex]315^\\circ[\/latex]), they represent the same rotation direction in the coordinate plane. Since a full circle is [latex]360^\\circ[\/latex] or [latex]2\\pi[\/latex] radians, we can always find new coterminal angles by adding or subtracting one or more full rotations. This makes it easier to work with very large or negative angles, because we can \u201cwrap\u201d them back into a friendlier version.\r\n\r\n<strong>Quick Tips: Finding Coterminal Angles<\/strong>\r\n<ol>\r\n \t<li>The Rule:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Degrees: [latex]\\theta_{\\text{cot}} = \\theta \\pm 360^\\circ k[\/latex]\r\n<ol style=\"list-style-type: lower-roman;\">\r\n \t<li>where [latex]k[\/latex] is any integer<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Radians: [latex]\\theta_{\\text{cot}} = \\theta \\pm 2\\pi k[\/latex]\r\n<ol style=\"list-style-type: lower-roman;\">\r\n \t<li>where [latex]k[\/latex] is any integer<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Positive and Negative Versions: Every angle has infinitely many coterminals. At least one positive and one negative.<\/li>\r\n \t<li>Shrink Big Angles: For large angles like [latex]1080^\\circ[\/latex], keep subtracting [latex]360^\\circ[\/latex] until you\u2019re within [latex]0^\\circ[\/latex] to [latex]360^\\circ[\/latex] (or [latex]0[\/latex] to [latex]2\\pi[\/latex] in radians).<\/li>\r\n \t<li>Use Coterminals to SImplify: When solving trig problems, replace a messy angle with a simple coterminal (like reducing [latex]-585^\\circ[\/latex] to [latex]135^\\circ[\/latex]).<\/li>\r\n \t<li>Think of a clock: On a clock, pointing at 3:00 is the same whether you went around once, twice, or even backwards. Coterminal angles work the same way; they all point to the same final direction.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Find the smallest positive angle coterminal with [latex]-785^\\circ.[\/latex]\r\n[reveal-answer q=\"318110\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"318110\"]Add [latex]360^\\circ[\/latex] until the result is between [latex]0^\\circ[\/latex] and [latex]360^\\circ.[\/latex][\/hidden-answer]\r\n[reveal-answer q=\"497034\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"497034\"][latex]-785^\\circ + 360^\\circ = -425^\\circ[\/latex]\r\n[latex]-425^\\circ + 360^\\circ = -65^\\circ[\/latex]\r\n[latex]-65^\\circ + 360^\\circ = 295^\\circ[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Give the angle in [latex][0,2\\pi)[\/latex] that is coterminal with [latex]\\dfrac{29\\pi}{8}[\/latex].\r\n[reveal-answer q=\"716740\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"716740\"]Convert [latex]2\\pi[\/latex] to a fraction with a common denominator and then subtract until the angle is in [latex][0,2\\pi).[\/latex][\/hidden-answer]\r\n[reveal-answer q=\"534572\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"534572\"][latex]\\begin{aligned} \\dfrac{29\\pi}{8} - 2\\pi &amp;= \\dfrac{29\\pi}{8} - \\dfrac{16\\pi}{8} \\\\ &amp;= \\dfrac{13\\pi}{8} \\end{aligned}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A skateboarder spind [latex]2.75[\/latex] revolutions counterclockwise. Give a negative coterminal angle in degrees.\r\n[reveal-answer q=\"102772\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"102772\"]First convert the revolutions to degrees, then subtract [latex]360^\\circ[\/latex] until you get a negative result.[\/hidden-answer]\r\n[reveal-answer q=\"818111\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"818111\"][latex]2.75 \\cdot 360^\\circ = 990^\\circ[\/latex]\r\n[latex]990^\\circ - 360^\\circ = 630^\\circ[\/latex]\r\n[latex]630^\\circ - 360^\\circ = 270^\\circ[\/latex]\r\n[latex]270^\\circ - 360^\\circ = -90^\\circ[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\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-hecfecce-TuyF8fFg3B0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TuyF8fFg3B0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hecfecce-TuyF8fFg3B0\" 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=13866804&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hecfecce-TuyF8fFg3B0&vembed=0&video_id=TuyF8fFg3B0&video_target=tpm-plugin-hecfecce-TuyF8fFg3B0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Determine+if+Two+Angles+Are+Coterminal_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine if Two Angles Are Coterminal\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>Draw angles in standard position.<\/li>\n<li>Convert between degrees and radians.<\/li>\n<li>Find coterminal angles.<\/li>\n<\/ul>\n<\/section>\n<h2>Angles in Standard Position<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>An angle in standard position has its vertex at the origin and its initial side lying along the positive x-axis. The direction of rotation determines the sign of the angle:<\/p>\n<ul>\n<li>counterclockwise rotations are positive<\/li>\n<li>clockwise rotations are negative<\/li>\n<\/ul>\n<p>Understanding an angle in standard position is important because it provides a consistent way to classify angles, no matter their size or sign. By using standard position, you can easily determine which quadrant an angle&#8217;s terminal side lands in and connect angles to trigonometric values.<\/p>\n<p><strong>Quick Tips: Drawing Angles in Standard Position<\/strong><\/p>\n<ol>\n<li>Start on the positive x-axis: Place the initial side along the positive x-axis, vertex at the origin.<\/li>\n<li>Choose the direction of rotation:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Counterclockwise \u2192 positive angle<\/li>\n<li>Clockwise \u2192 negative angle<\/li>\n<\/ol>\n<\/li>\n<li>Mark the rotation: Move the terminal side to the correct position according to the given measure.<\/li>\n<li>Identify the quadrant:\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"text-align: left;\">Quandrant I: [latex]0^\\circ - 90^\\circ[\/latex]<\/li>\n<li>Quandrant II: [latex]90^\\circ - 180^\\circ[\/latex]<\/li>\n<li>Quandrant III: [latex]180^\\circ - 270^\\circ[\/latex]<\/li>\n<li>Quandrant IV: [latex]270^\\circ - 360^\\circ[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Extend beyond one rotation if needed: Angles greater than [latex]360^\\circ[\/latex] or less than [latex]0^\\circ[\/latex] wrap around, but still land on a terminal side in one of the four quadrants.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Draw an angle of -135\u00ba in standard position. Indicate the quadrant of its terminal side.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q782027\">Hint<\/button><\/p>\n<div id=\"q782027\" class=\"hidden-answer\" style=\"display: none\">Remember the initial side is on the positive [latex]x[\/latex]-axis and that a negative angle rotates clockwise.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44984\">Show Answer<\/button><\/p>\n<div id=\"q44984\" class=\"hidden-answer\" style=\"display: none\">Start on the positive x-axis and rotate clockwise 135\u00ba degrees. The terminal side lands in Quadrant II (pointing halfway between the negative x-axis and the negative y-axis).<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3360\/2018\/06\/14234838\/CNX_Precalc_Figure_05_01_0082.jpg\" alt=\"Graph of a negative 135 degree angle.\" width=\"487\" height=\"383\" \/><figcaption class=\"wp-caption-text\">Angle of -135\u00ba in standard position.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A drone&#8217;s camera points from the origin through [latex](5,-2)[\/latex]. Sketch the angle in standard position and name its quadrant.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q601541\">Hint<\/button><\/p>\n<div id=\"q601541\" class=\"hidden-answer\" style=\"display: none\">Use the signs of the given point to find the quadrant. In this example, we have [latex](+,-)[\/latex]. Which quadrant includes this point?<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q174014\">Show Answer<\/button><\/p>\n<div id=\"q174014\" class=\"hidden-answer\" style=\"display: none\">[latex](5, -2)[\/latex] is right 5 units and down 2 units[latex]\\Rightarrow[\/latex] Quadrant IV. Draw the ray from the origin through [latex](5,-2).[\/latex]<\/p>\n<figure id=\"attachment_4193\" aria-describedby=\"caption-attachment-4193\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-4193\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-300x300.png\" alt=\"Coordinate plane with a ray from the origin through (5, -2); small arc from positive x-axis to the ray; labeled Quadrant IV.\" width=\"300\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-768x768.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2-350x350.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19163451\/angle_through_5_-2.png 1000w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-4193\" class=\"wp-caption-text\">Angle through (5,-2) in Quadrant IV.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">On a DJ turntable, a sticker has rotated by [latex]\\frac{11\\pi}{6}[\/latex] from the positive [latex]x[\/latex]-axis. Sketch the angle and state which axis it is closest to.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q28376\">Hint<\/button><\/p>\n<div id=\"q28376\" class=\"hidden-answer\" style=\"display: none\">Convert [latex]\\dfrac{11\\pi}{6}[\/latex] to degrees. You could also compare it to [latex]2\\pi[\/latex] (or [latex]\\dfrac{12\\pi}{6}[\/latex]) to judge which axis it is closest to.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q176844\">Show Answer<\/button><\/p>\n<div id=\"q176844\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{11\\pi}{6} = 330^\\circ\\ \\Rightarrow[\/latex] Quadrant IV, closer to the positive [latex]x[\/latex]-axis [latex](360^\\circ)[\/latex] than the negative [latex]y[\/latex]-axis [latex](270^\\circ)[\/latex].<\/p>\n<figure id=\"attachment_4198\" aria-describedby=\"caption-attachment-4198\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-4198\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-300x300.png\" alt=\"Coordinate plane with a ray from the origin at 11 pi over 6 (30 degrees below the positive x-axis).\" width=\"300\" height=\"300\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-1024x1024.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-768x768.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6-350x350.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/19165704\/angle_11pi_over_6.png 1040w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-4198\" class=\"wp-caption-text\">Angle [latex]\\frac{11\\pi}{6}[\/latex] in standard position; terminal side in Quadrant IV, closest to the positive [latex]x[\/latex]-axis.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\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-aaeaadea-CNsRpez8fZE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/CNsRpez8fZE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aaeaadea-CNsRpez8fZE\" 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=13867583&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aaeaadea-CNsRpez8fZE&#38;vembed=0&#38;video_id=CNsRpez8fZE&#38;video_target=tpm-plugin-aaeaadea-CNsRpez8fZE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Draw+an+Angle+in+Standard+Position_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Draw an Angle in Standard Position\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Converting Between Degrees and Radians<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Angles can be measured in degrees or radians, and being able to switch between the two is essential in precalculus and beyond. A full circle is [latex]360^\\circ[\/latex], which equals [latex]2\\pi[\/latex] radians. A key conversion relationship to remember is: [latex]180^\\circ=\\pi[\/latex] radians. Degrees are more often seen in everyday use, while radians are the natural unit in higher mathematics because they link angle measure directly to arc length.<\/p>\n<p><strong>Quick Tips: Converting Between Degrees and Radians<\/strong><\/p>\n<ol>\n<li>Memorize the core relationship: [latex]180^\\circ=\\pi[\/latex] radians.<\/li>\n<li>Degrees \u2192 Radians: Multiply by [latex]\\dfrac{\\pi}{180}[\/latex].\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Ex: [latex]60^\\circ \\cdot \\dfrac{\\pi}{180} = \\dfrac{\\pi}{3}[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Radians \u2192 Degrees: Multiply by [latex]\\dfrac{180}{\\pi}[\/latex].\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Ex: [latex]\\dfrac{3\\pi}{4} \\cdot \\dfrac{180}{\\pi} = 135^\\circ[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Keep It Exact (Unless Told Otherwise)<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>When converting to radians, leave your answer in terms of [latex]\\pi[\/latex] (for example, [latex]\\pi\/6[\/latex] instead of 0.52).<\/li>\n<\/ol>\n<\/li>\n<li>Know the &#8220;Common Angles&#8221;:<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%; height: 110px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; text-align: center; height: 22px;\"><strong>Degrees<\/strong><\/td>\n<td style=\"width: 50%; text-align: center; height: 22px;\"><strong>Radians<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]30^\\circ[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{6}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]45^\\circ[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]60^\\circ[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]90^\\circ[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px; text-align: center;\">[latex]\\dfrac{\\pi}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These show up so often that it\u2019s worth memorizing them.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Convert [latex]37.5^\\circ[\/latex] to radians.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q282667\">Hint<\/button><\/p>\n<div id=\"q282667\" class=\"hidden-answer\" style=\"display: none\">Multiply by [latex]\\dfrac{\\pi}{180^\\circ}[\/latex] and simplify the fraction carefully.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q827432\">Show Answer<\/button><\/p>\n<div id=\"q827432\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} 37.5^\\circ \\Rightarrow \\text { radians} &= 37.5^\\circ \\cdot \\dfrac{\\pi}{180^\\circ} \\\\ &= \\dfrac{5\\pi}{24} \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Convert [latex]-\\dfrac{5\\pi}{3}[\/latex] to degrees.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q996357\">Hint<\/button><\/p>\n<div id=\"q996357\" class=\"hidden-answer\" style=\"display: none\">Multiply by [latex]\\dfrac{180^\\circ}{\\pi}[\/latex] and keep the negative sign all the way through.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q571988\">Show Answer<\/button><\/p>\n<div id=\"q571988\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} -\\dfrac{5\\pi}{3} \\Rightarrow \\text { degrees} &= -\\dfrac{5\\pi}{3} \\cdot \\dfrac{180^\\circ}{\\pi} \\\\ &= -300^\\circ \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A bike wheel turns [latex]2.4[\/latex] revolutions. Express this rotation in degrees and radians.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q638993\">Hint<\/button><\/p>\n<div id=\"q638993\" class=\"hidden-answer\" style=\"display: none\">Remember that one revolution [latex]= 360^\\circ[\/latex] for degrees and also [latex]= 2\\pi[\/latex] for radians. Multiply [latex]2.4[\/latex] by each one.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q713110\">Show Answer<\/button><\/p>\n<div id=\"q713110\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} 2.4 \\text { revolutions} \\Rightarrow \\text { degrees} &= 2.4 \\cdot 360^\\circ \\\\ &= 864^\\circ \\end{aligned}[\/latex]<br \/>\n[latex]\\begin{aligned} 2.4 \\text { revolutions} \\Rightarrow \\text { radians} &= 2.4 \\cdot 2\\pi \\\\ &= 4.8\\pi \\\\ &= \\dfrac{24\\pi}{5} \\text { radians} \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\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-cgghebbc-1pjRy6M-KKg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1pjRy6M-KKg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cgghebbc-1pjRy6M-KKg\" 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=13867756&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cgghebbc-1pjRy6M-KKg&#38;vembed=0&#38;video_id=1pjRy6M-KKg&#38;video_target=tpm-plugin-cgghebbc-1pjRy6M-KKg\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Degrees+to+Radians_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Degrees to Radians\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\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-ddagfebd-aw6H3AntIaQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/aw6H3AntIaQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddagfebd-aw6H3AntIaQ\" 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=13867765&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ddagfebd-aw6H3AntIaQ&#38;vembed=0&#38;video_id=aw6H3AntIaQ&#38;video_target=tpm-plugin-ddagfebd-aw6H3AntIaQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Converting+Radians+to+Degrees_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConverting Radians to Degrees\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Finding Coterminal Angles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Coterminal angles are angles that <strong data-start=\"181\" data-end=\"206\">end in the same place<\/strong> \u2014 they share the same terminal side when drawn in standard position. Even though they may look different on paper (like [latex]-45^\\circ[\/latex] and [latex]315^\\circ[\/latex]), they represent the same rotation direction in the coordinate plane. Since a full circle is [latex]360^\\circ[\/latex] or [latex]2\\pi[\/latex] radians, we can always find new coterminal angles by adding or subtracting one or more full rotations. This makes it easier to work with very large or negative angles, because we can \u201cwrap\u201d them back into a friendlier version.<\/p>\n<p><strong>Quick Tips: Finding Coterminal Angles<\/strong><\/p>\n<ol>\n<li>The Rule:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Degrees: [latex]\\theta_{\\text{cot}} = \\theta \\pm 360^\\circ k[\/latex]\n<ol style=\"list-style-type: lower-roman;\">\n<li>where [latex]k[\/latex] is any integer<\/li>\n<\/ol>\n<\/li>\n<li>Radians: [latex]\\theta_{\\text{cot}} = \\theta \\pm 2\\pi k[\/latex]\n<ol style=\"list-style-type: lower-roman;\">\n<li>where [latex]k[\/latex] is any integer<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<li>Positive and Negative Versions: Every angle has infinitely many coterminals. At least one positive and one negative.<\/li>\n<li>Shrink Big Angles: For large angles like [latex]1080^\\circ[\/latex], keep subtracting [latex]360^\\circ[\/latex] until you\u2019re within [latex]0^\\circ[\/latex] to [latex]360^\\circ[\/latex] (or [latex]0[\/latex] to [latex]2\\pi[\/latex] in radians).<\/li>\n<li>Use Coterminals to SImplify: When solving trig problems, replace a messy angle with a simple coterminal (like reducing [latex]-585^\\circ[\/latex] to [latex]135^\\circ[\/latex]).<\/li>\n<li>Think of a clock: On a clock, pointing at 3:00 is the same whether you went around once, twice, or even backwards. Coterminal angles work the same way; they all point to the same final direction.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Find the smallest positive angle coterminal with [latex]-785^\\circ.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q318110\">Hint<\/button><\/p>\n<div id=\"q318110\" class=\"hidden-answer\" style=\"display: none\">Add [latex]360^\\circ[\/latex] until the result is between [latex]0^\\circ[\/latex] and [latex]360^\\circ.[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q497034\">Show Answer<\/button><\/p>\n<div id=\"q497034\" class=\"hidden-answer\" style=\"display: none\">[latex]-785^\\circ + 360^\\circ = -425^\\circ[\/latex]<br \/>\n[latex]-425^\\circ + 360^\\circ = -65^\\circ[\/latex]<br \/>\n[latex]-65^\\circ + 360^\\circ = 295^\\circ[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Give the angle in [latex][0,2\\pi)[\/latex] that is coterminal with [latex]\\dfrac{29\\pi}{8}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q716740\">Hint<\/button><\/p>\n<div id=\"q716740\" class=\"hidden-answer\" style=\"display: none\">Convert [latex]2\\pi[\/latex] to a fraction with a common denominator and then subtract until the angle is in [latex][0,2\\pi).[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q534572\">Show Answer<\/button><\/p>\n<div id=\"q534572\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} \\dfrac{29\\pi}{8} - 2\\pi &= \\dfrac{29\\pi}{8} - \\dfrac{16\\pi}{8} \\\\ &= \\dfrac{13\\pi}{8} \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A skateboarder spind [latex]2.75[\/latex] revolutions counterclockwise. Give a negative coterminal angle in degrees.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q102772\">Hint<\/button><\/p>\n<div id=\"q102772\" class=\"hidden-answer\" style=\"display: none\">First convert the revolutions to degrees, then subtract [latex]360^\\circ[\/latex] until you get a negative result.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q818111\">Show Answer<\/button><\/p>\n<div id=\"q818111\" class=\"hidden-answer\" style=\"display: none\">[latex]2.75 \\cdot 360^\\circ = 990^\\circ[\/latex]<br \/>\n[latex]990^\\circ - 360^\\circ = 630^\\circ[\/latex]<br \/>\n[latex]630^\\circ - 360^\\circ = 270^\\circ[\/latex]<br \/>\n[latex]270^\\circ - 360^\\circ = -90^\\circ[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\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-hecfecce-TuyF8fFg3B0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/TuyF8fFg3B0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hecfecce-TuyF8fFg3B0\" 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=13866804&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hecfecce-TuyF8fFg3B0&#38;vembed=0&#38;video_id=TuyF8fFg3B0&#38;video_target=tpm-plugin-hecfecce-TuyF8fFg3B0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Determine+if+Two+Angles+Are+Coterminal_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine if Two Angles Are Coterminal\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"How to Draw an Angle in Standard Position #angles #math #openstax\",\"author\":\"\",\"organization\":\"Scalar Learning\",\"url\":\"https:\/\/youtu.be\/CNsRpez8fZE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Converting Degrees to Radians\",\"author\":\"\",\"organization\":\"Jim Barnard Math Tutorials\",\"url\":\"https:\/\/youtu.be\/1pjRy6M-KKg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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