{"id":2772,"date":"2025-08-13T18:38:23","date_gmt":"2025-08-13T18:38:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2772"},"modified":"2026-03-25T03:35:00","modified_gmt":"2026-03-25T03:35:00","slug":"polar-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-functions-background-youll-need-1\/","title":{"raw":"Polar Functions: Background You'll Need 1","rendered":"Polar Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph angles in standard form<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p data-start=\"1232\" data-end=\"1339\">In trigonometry and now in polar coordinates, we describe an angle by how it rotates from the positive x-axis.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>standard position of an angle<\/h3>\r\n<p data-start=\"1363\" data-end=\"1405\">An angle is in standard position when:<\/p>\r\n\r\n<ul data-start=\"1406\" data-end=\"1572\">\r\n \t<li data-start=\"1406\" data-end=\"1450\">\r\n<p data-start=\"1408\" data-end=\"1450\">Its vertex is at the origin (0,0).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1451\" data-end=\"1506\">\r\n<p data-start=\"1453\" data-end=\"1506\">Its initial side lies on the positive x-axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1507\" data-end=\"1572\">\r\n<p data-start=\"1509\" data-end=\"1572\">Its terminal side is drawn after rotating the initial side.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<ul data-start=\"1595\" data-end=\"1689\">\r\n \t<li data-start=\"1595\" data-end=\"1645\">\r\n<p data-start=\"1597\" data-end=\"1645\">Positive angles rotate counterclockwise.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1646\" data-end=\"1689\">\r\n<p data-start=\"1648\" data-end=\"1689\">Negative angles rotate clockwise.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1703\" data-end=\"1753\">Draw a [latex]120^{\\circ}[\/latex] angle in standard position.<\/p>\r\n<p data-start=\"1703\" data-end=\"1753\">[reveal-answer q=\"20408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"20408\"]<\/p>\r\n\r\n<ol>\r\n \t<li data-start=\"1703\" data-end=\"1753\">Start along the positive x-axis.<\/li>\r\n \t<li data-start=\"1790\" data-end=\"1833\">\r\n<p data-start=\"1793\" data-end=\"1833\">Since [latex]120^{\\circ}[\/latex] is between [latex]90^{\\circ}[\/latex] and [latex]180^{\\circ}[\/latex] it will be in Quadrant II (closer to [latex]90^{\\circ}[\/latex]).\r\n<img src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcS0WolhnOj2j6tuKUU6iTIaheDD09vxOJF3WA&amp;s\" alt=\"Coterminal Angles \/ Ratios for Any Angle Angles On the Coordinate Plane Angles On the Coordinate Plane Coterminal Angles Cotermi\" \/><\/p>\r\n<p data-start=\"1703\" data-end=\"1753\">[\/hidden-answer]<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313874[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph angles in standard form<\/li>\n<\/ul>\n<\/section>\n<p data-start=\"1232\" data-end=\"1339\">In trigonometry and now in polar coordinates, we describe an angle by how it rotates from the positive x-axis.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>standard position of an angle<\/h3>\n<p data-start=\"1363\" data-end=\"1405\">An angle is in standard position when:<\/p>\n<ul data-start=\"1406\" data-end=\"1572\">\n<li data-start=\"1406\" data-end=\"1450\">\n<p data-start=\"1408\" data-end=\"1450\">Its vertex is at the origin (0,0).<\/p>\n<\/li>\n<li data-start=\"1451\" data-end=\"1506\">\n<p data-start=\"1453\" data-end=\"1506\">Its initial side lies on the positive x-axis.<\/p>\n<\/li>\n<li data-start=\"1507\" data-end=\"1572\">\n<p data-start=\"1509\" data-end=\"1572\">Its terminal side is drawn after rotating the initial side.<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<ul data-start=\"1595\" data-end=\"1689\">\n<li data-start=\"1595\" data-end=\"1645\">\n<p data-start=\"1597\" data-end=\"1645\">Positive angles rotate counterclockwise.<\/p>\n<\/li>\n<li data-start=\"1646\" data-end=\"1689\">\n<p data-start=\"1648\" data-end=\"1689\">Negative angles rotate clockwise.<\/p>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1703\" data-end=\"1753\">Draw a [latex]120^{\\circ}[\/latex] angle in standard position.<\/p>\n<p data-start=\"1703\" data-end=\"1753\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q20408\">Show Solution<\/button><\/p>\n<div id=\"q20408\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li data-start=\"1703\" data-end=\"1753\">Start along the positive x-axis.<\/li>\n<li data-start=\"1790\" data-end=\"1833\">\n<p data-start=\"1793\" data-end=\"1833\">Since [latex]120^{\\circ}[\/latex] is between [latex]90^{\\circ}[\/latex] and [latex]180^{\\circ}[\/latex] it will be in Quadrant II (closer to [latex]90^{\\circ}[\/latex]).<br \/>\n<img decoding=\"async\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcS0WolhnOj2j6tuKUU6iTIaheDD09vxOJF3WA&amp;s\" alt=\"Coterminal Angles \/ Ratios for Any Angle Angles On the Coordinate Plane Angles On the Coordinate Plane Coterminal Angles Cotermi\" \/><\/p>\n<p data-start=\"1703\" data-end=\"1753\"><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313874\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313874&theme=lumen&iframe_resize_id=ohm313874&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":247,"module-header":"background_you_need","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\/2772"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2772\/revisions"}],"predecessor-version":[{"id":6006,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2772\/revisions\/6006"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/247"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2772\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2772"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2772"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2772"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}