{"id":179,"date":"2025-02-13T22:44:41","date_gmt":"2025-02-13T22:44:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles\/"},"modified":"2025-10-08T16:45:02","modified_gmt":"2025-10-08T16:45:02","slug":"angles","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/angles\/","title":{"raw":"Angles: Learn It 1","rendered":"Angles: Learn It 1"},"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<h2>Drawing angles<\/h2>\r\nProperly defining an angle first requires that we define a ray. A\u00a0<strong>ray<\/strong>\u00a0consists of one point on a line and all points extending in one direction from that point. The first point is called the\u00a0<strong>endpoint<\/strong> of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray shown can be named as ray EF, or in symbolic form [latex]\\overrightarrow{EF}[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180159\/CNX_Precalc_Figure_05_01_0012.jpg\" alt=\"Illustration of Ray EF, with point F and endpoint E.\" width=\"487\" height=\"173\" \/>\r\n\r\nAn\u00a0<strong>angle<\/strong>\u00a0is the union of two rays having a common endpoint. The endpoint is called the\u00a0<strong>vertex<\/strong> of the angle, and the two rays are the sides of the angle. The angle in shown is formed from [latex]\\overrightarrow{ED}[\/latex] and [latex]\\overrightarrow{EF}[\/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[\/latex], or in symbol form [latex]\\angle{DEF}[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180201\/CNX_Precalc_Figure_05_01_0022.jpg\" alt=\"Illustration of Angle DEF, with vertex E and points D and F.\" width=\"487\" height=\"246\" \/>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>angle<\/h3>\r\nAn angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle.\r\n\r\n<span id=\"fs-id1165135192939\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180202\/CNX_Precalc_Figure_05_01_0032.jpg\" alt=\"Illustration of angle theta.\" \/><\/span>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Greek letters are often used as variables for the measure of an angle. The table below is a list of Greek letters commonly used to represent angles.\r\n<table id=\"Table_05_01_01\" style=\"height: 66px;\" summary=\"Two rows and five columns. First row shows symbols for theta, phi, alpha, beta, and gamma. Second row spells out name for each symbol.\">\r\n<tbody>\r\n<tr style=\"height: 53px;\">\r\n<td style=\"height: 53px; width: 130.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\theta[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 180.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\phi \\text{ or }\\varphi[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 106.676px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\alpha[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 102.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\beta[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<td style=\"height: 53px; width: 122.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\gamma[\/latex]<\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 13px;\">\r\n<td style=\"height: 13px; width: 130.313px;\">theta<\/td>\r\n<td style=\"height: 13px; width: 180.313px;\">phi<\/td>\r\n<td style=\"height: 13px; width: 106.676px;\">alpha<\/td>\r\n<td style=\"height: 13px; width: 102.131px;\">beta<\/td>\r\n<td style=\"height: 13px; width: 122.131px;\">gamma<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<figure id=\"Figure_05_01_003\" class=\"small\"><\/figure>\r\nAngle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the\u00a0<strong>initial side<\/strong>, and the rotated ray is the\u00a0<strong>terminal side<\/strong>. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex.<span id=\"fs-id1165137737991\">\r\n<\/span>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180205\/CNX_Precalc_Figure_05_01_0052.jpg\" alt=\"Graph of an angle in standard position with labels for the initial side and terminal side.\" width=\"487\" height=\"417\" \/>\r\n\r\nThe <strong>measure of an angle<\/strong>\u00a0is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One\u00a0<strong>degree<\/strong>\u00a0is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol \u00b0. For example, 90 degrees = 90\u00b0.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>angle measure<\/h3>\r\nThe angle measure is the amount of rotation from the initial side to the terminal side.\r\n\r\n<\/div>\r\n<\/section>To formalize our work, we will begin by drawing angles on an <em>x<\/em>-<em>y<\/em> coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in <strong>standard position<\/strong> if its vertex is located at the origin, and its initial side extends along the positive <em>x<\/em>-axis.\u00a0<span id=\"fs-id1165137804556\">\r\n<\/span>\r\n\r\nIf the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a <strong>positive angle<\/strong>. If the angle is measured in a clockwise direction, the angle is said to be a <strong>negative angl<\/strong><strong style=\"font-size: 1em;\">e<\/strong><span style=\"font-size: 1em;\">.<\/span>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180207\/CNX_Precalc_Figure_05_01_0062.jpg\" alt=\"Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs.\" width=\"731\" height=\"365\" \/><span id=\"fs-id1165134042853\"><\/span>\r\n\r\nSince we define an angle in <strong>standard position<\/strong> by its terminal side, we have a special type of angle whose terminal side lies on an axis, a <strong>quadrantal angle<\/strong>. This type of angle can have a measure of 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0 or 360\u00b0.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180209\/CNX_Precalc_Figure_05_01_0182.jpg\" alt=\"Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees.\" width=\"975\" height=\"237\" \/> Quadrantal angles have a terminal side that lies along an axis. Examples are shown.[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>quadrantal angles<\/h3>\r\nQuadrantal angles are angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, or 360\u00b0.\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>quadrants<\/h3>\r\nThe coordinate plane is divided into four quadrants. Quadrant numbering begins at the positive x-axis and rotates counterclockwise.\r\n\r\n<img src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250916.165151\/resources\/b60cd925724939299f52716cbf50b67c9a8ebf83\" alt=\"2.1 The Rectangular Coordinate Systems and Graphs - College Algebra 2e | OpenStax\" \/>\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: draw an angle in standard position<\/strong>\r\n<ol>\r\n \t<li>Determine which quadrant the angle belongs in:\r\n<ul>\r\n \t<li>Quadrant 1: Angles between 0\u00b0 and 90\u00b0<\/li>\r\n \t<li>Quadrant 2: Angles between 90\u00b0 and 180\u00b0<\/li>\r\n \t<li>Quadrant 3: Angles between 180\u00b0 and 270\u00b0<\/li>\r\n \t<li>Quadrant 4: Angles between 270\u00b0 and 360\u00b0<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Draw an angle that lands proportionately within the desired quadrant.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch an angle of 30\u00b0 in standard position.[reveal-answer q=\"899324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"899324\"]\r\n<ol>\r\n \t<li>30\u00b0 is between 0\u00b0 and 90\u00b0 so the angle will land in quadrant I. Since 30\u00b0 is one third of 90\u00b0 we can divide the quadrant into thirds:\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180211\/CNX_Precalc_Figure_05_01_0072.jpg\" alt=\"Graph of a 30 degree angle.\" width=\"487\" height=\"383\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Show an angle of 240\u00b0 on a circle in standard position.[reveal-answer q=\"862928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862928\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003504\/CNX_Precalc_Figure_05_01_0092.jpg\" alt=\"Graph of a 240 degree angle.\" \/>[\/hidden-answer]<\/section><section aria-label=\"Try It\"><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>negative angles<\/h3>\r\nA positive angle in standard position rotates\u00a0<strong>counterclockwise\u00a0<\/strong>from the positive x-axis. A negative angle in standard position rotates\u00a0<strong>clockwise\u00a0<\/strong>from the positive x-axis.\r\n\r\n<img src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250827.171612\/resources\/46c6a81b0687ad15613c48d312ff8b6fa2a93f20\" alt=\"7.1 Angles - Algebra and Trigonometry 2e | OpenStax\" \/>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch an angle of -135\u00b0 in standard position.\r\n\r\n[reveal-answer q=\"207462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"207462\"]135\u00b0 is between 90\u00b0 and 180\u00b0, but -135\u00b0 is between -90\u00b0 and -180\u00b0 going in the opposite direction, so it'll land in Quadrant 3. Since -135\u00b0 is exactly halfway between -90\u00b0 and -180\u00b0 we'll draw our angle halfway through the quadrant.\r\n\r\n<img src=\"https:\/\/math.libretexts.org\/@api\/deki\/files\/107261\/521b1374f77e48ebe74d6ea20b9ff700ac3e1bcf?revision=1\" alt=\"7.2: Angles - Mathematics LibreTexts\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>","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<h2>Drawing angles<\/h2>\n<p>Properly defining an angle first requires that we define a ray. A\u00a0<strong>ray<\/strong>\u00a0consists of one point on a line and all points extending in one direction from that point. The first point is called the\u00a0<strong>endpoint<\/strong> of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray shown can be named as ray EF, or in symbolic form [latex]\\overrightarrow{EF}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180159\/CNX_Precalc_Figure_05_01_0012.jpg\" alt=\"Illustration of Ray EF, with point F and endpoint E.\" width=\"487\" height=\"173\" \/><\/p>\n<p>An\u00a0<strong>angle<\/strong>\u00a0is the union of two rays having a common endpoint. The endpoint is called the\u00a0<strong>vertex<\/strong> of the angle, and the two rays are the sides of the angle. The angle in shown is formed from [latex]\\overrightarrow{ED}[\/latex] and [latex]\\overrightarrow{EF}[\/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[\/latex], or in symbol form [latex]\\angle{DEF}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180201\/CNX_Precalc_Figure_05_01_0022.jpg\" alt=\"Illustration of Angle DEF, with vertex E and points D and F.\" width=\"487\" height=\"246\" \/><\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>angle<\/h3>\n<p>An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle.<\/p>\n<p><span id=\"fs-id1165135192939\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180202\/CNX_Precalc_Figure_05_01_0032.jpg\" alt=\"Illustration of angle theta.\" \/><\/span><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Greek letters are often used as variables for the measure of an angle. The table below is a list of Greek letters commonly used to represent angles.<\/p>\n<table id=\"Table_05_01_01\" style=\"height: 66px;\" summary=\"Two rows and five columns. First row shows symbols for theta, phi, alpha, beta, and gamma. Second row spells out name for each symbol.\">\n<tbody>\n<tr style=\"height: 53px;\">\n<td style=\"height: 53px; width: 130.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\theta[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 180.313px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\phi \\text{ or }\\varphi[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 106.676px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\alpha[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 102.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\beta[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<td style=\"height: 53px; width: 122.131px;\"><span class=\"katex-html\"><span class=\"base textstyle uncramped\"><span class=\"reset-textstyle displaystyle textstyle uncramped\"><span class=\"mord mathit\">[latex]\\gamma[\/latex]<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr style=\"height: 13px;\">\n<td style=\"height: 13px; width: 130.313px;\">theta<\/td>\n<td style=\"height: 13px; width: 180.313px;\">phi<\/td>\n<td style=\"height: 13px; width: 106.676px;\">alpha<\/td>\n<td style=\"height: 13px; width: 102.131px;\">beta<\/td>\n<td style=\"height: 13px; width: 122.131px;\">gamma<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<figure id=\"Figure_05_01_003\" class=\"small\"><\/figure>\n<p>Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the\u00a0<strong>initial side<\/strong>, and the rotated ray is the\u00a0<strong>terminal side<\/strong>. In order to identify the different sides, we indicate the rotation with a small arc and arrow close to the vertex.<span id=\"fs-id1165137737991\"><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180205\/CNX_Precalc_Figure_05_01_0052.jpg\" alt=\"Graph of an angle in standard position with labels for the initial side and terminal side.\" width=\"487\" height=\"417\" \/><\/p>\n<p>The <strong>measure of an angle<\/strong>\u00a0is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One\u00a0<strong>degree<\/strong>\u00a0is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol \u00b0. For example, 90 degrees = 90\u00b0.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>angle measure<\/h3>\n<p>The angle measure is the amount of rotation from the initial side to the terminal side.<\/p>\n<\/div>\n<\/section>\n<p>To formalize our work, we will begin by drawing angles on an <em>x<\/em>&#8211;<em>y<\/em> coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in <strong>standard position<\/strong> if its vertex is located at the origin, and its initial side extends along the positive <em>x<\/em>-axis.\u00a0<span id=\"fs-id1165137804556\"><br \/>\n<\/span><\/p>\n<p>If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a <strong>positive angle<\/strong>. If the angle is measured in a clockwise direction, the angle is said to be a <strong>negative angl<\/strong><strong style=\"font-size: 1em;\">e<\/strong><span style=\"font-size: 1em;\">.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180207\/CNX_Precalc_Figure_05_01_0062.jpg\" alt=\"Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs.\" width=\"731\" height=\"365\" \/><span id=\"fs-id1165134042853\"><\/span><\/p>\n<p>Since we define an angle in <strong>standard position<\/strong> by its terminal side, we have a special type of angle whose terminal side lies on an axis, a <strong>quadrantal angle<\/strong>. This type of angle can have a measure of 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0 or 360\u00b0.<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180209\/CNX_Precalc_Figure_05_01_0182.jpg\" alt=\"Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees.\" width=\"975\" height=\"237\" \/><figcaption class=\"wp-caption-text\">Quadrantal angles have a terminal side that lies along an axis. Examples are shown.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>quadrantal angles<\/h3>\n<p>Quadrantal angles are angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, or 360\u00b0.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>quadrants<\/h3>\n<p>The coordinate plane is divided into four quadrants. Quadrant numbering begins at the positive x-axis and rotates counterclockwise.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250916.165151\/resources\/b60cd925724939299f52716cbf50b67c9a8ebf83\" alt=\"2.1 The Rectangular Coordinate Systems and Graphs - College Algebra 2e | OpenStax\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: draw an angle in standard position<\/strong><\/p>\n<ol>\n<li>Determine which quadrant the angle belongs in:\n<ul>\n<li>Quadrant 1: Angles between 0\u00b0 and 90\u00b0<\/li>\n<li>Quadrant 2: Angles between 90\u00b0 and 180\u00b0<\/li>\n<li>Quadrant 3: Angles between 180\u00b0 and 270\u00b0<\/li>\n<li>Quadrant 4: Angles between 270\u00b0 and 360\u00b0<\/li>\n<\/ul>\n<\/li>\n<li>Draw an angle that lands proportionately within the desired quadrant.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch an angle of 30\u00b0 in standard position.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q899324\">Show Solution<\/button><\/p>\n<div id=\"q899324\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>30\u00b0 is between 0\u00b0 and 90\u00b0 so the angle will land in quadrant I. Since 30\u00b0 is one third of 90\u00b0 we can divide the quadrant into thirds:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/923\/2015\/04\/25180211\/CNX_Precalc_Figure_05_01_0072.jpg\" alt=\"Graph of a 30 degree angle.\" width=\"487\" height=\"383\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Show an angle of 240\u00b0 on a circle in standard position.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q862928\">Show Solution<\/button><\/p>\n<div id=\"q862928\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003504\/CNX_Precalc_Figure_05_01_0092.jpg\" alt=\"Graph of a 240 degree angle.\" \/><\/div>\n<\/div>\n<\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>negative angles<\/h3>\n<p>A positive angle in standard position rotates\u00a0<strong>counterclockwise\u00a0<\/strong>from the positive x-axis. A negative angle in standard position rotates\u00a0<strong>clockwise\u00a0<\/strong>from the positive x-axis.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250827.171612\/resources\/46c6a81b0687ad15613c48d312ff8b6fa2a93f20\" alt=\"7.1 Angles - Algebra and Trigonometry 2e | OpenStax\" \/><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch an angle of -135\u00b0 in standard position.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q207462\">Show Solution<\/button><\/p>\n<div id=\"q207462\" class=\"hidden-answer\" style=\"display: none\">135\u00b0 is between 90\u00b0 and 180\u00b0, but -135\u00b0 is between -90\u00b0 and -180\u00b0 going in the opposite direction, so it&#8217;ll land in Quadrant 3. Since -135\u00b0 is exactly halfway between -90\u00b0 and -180\u00b0 we&#8217;ll draw our angle halfway through the quadrant.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/math.libretexts.org\/@api\/deki\/files\/107261\/521b1374f77e48ebe74d6ea20b9ff700ac3e1bcf?revision=1\" alt=\"7.2: Angles - Mathematics LibreTexts\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n","protected":false},"author":6,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Angles\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Animation: Angles in Standard Position\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/hpIjaKLOo6o\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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