{"id":2733,"date":"2025-08-13T18:33:08","date_gmt":"2025-08-13T18:33:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2733"},"modified":"2025-10-17T16:31:24","modified_gmt":"2025-10-17T16:31:24","slug":"trigonometric-identities-and-equations-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/trigonometric-identities-and-equations-background-youll-need-1\/","title":{"raw":"Trigonometric Identities and Equations: Background You'll Need 1","rendered":"Trigonometric Identities and Equations: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Evaluate sine and cosine functions at common points<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>sine and cosine<\/h3>\r\nThe sine and cosine functions describe coordinates on the unit circle, a circle of radius 1 centered at the origin.\r\nFor any angle [latex]\\theta[\/latex] measured in radians:\r\n[latex]\r\n\\sin(\\theta) = y \\quad \\text{and} \\quad \\cos(\\theta) = x\r\n[\/latex]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p data-start=\"733\" data-end=\"936\">Common angles on the unit circle: [latex]0, \\frac{\\pi}{6}, \\frac{\\pi}{4}, \\frac{\\pi}{3}, \\frac{\\pi}{2}[\/latex]<br data-start=\"843\" data-end=\"846\" \/>The corresponding coordinates [latex](\\cos\\theta, \\sin\\theta)[\/latex] are often memorized.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"958\" data-end=\"1059\">Find [latex]\\sin\\left(\\frac{\\pi}{3}\\right)[\/latex] and [latex]\\cos\\left(\\frac{\\pi}{3}\\right)[\/latex].<\/p>\r\n<p data-start=\"958\" data-end=\"1059\">\r\n[reveal-answer q=\"120729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"120729\"]<\/p>\r\n<p data-start=\"1061\" data-end=\"1171\">From the unit circle,<br data-start=\"1082\" data-end=\"1085\" \/>[latex](\\cos\\theta, \\sin\\theta) = \\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\r\n<p data-start=\"1173\" data-end=\"1299\">So:<br data-start=\"1176\" data-end=\"1179\" \/>[latex]\r\n\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}, \\quad \\cos\\left(\\frac{\\pi}{3}\\right) = \\frac{1}{2}\r\n[\/latex]<\/p>\r\n<p data-start=\"958\" data-end=\"1059\">[\/hidden-answer]<\/p>\r\n<p data-start=\"1173\" data-end=\"1299\"><\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"1321\" data-end=\"1380\">Find [latex]\\sin(\\pi)[\/latex] and [latex]\\cos(\\pi)[\/latex].<\/p>\r\n<p data-start=\"1382\" data-end=\"1507\">\r\n[reveal-answer q=\"427439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"427439\"]On the unit circle, [latex](\\cos\\pi, \\sin\\pi) = (-1, 0)[\/latex]. So: [latex] \\sin(\\pi) = 0, \\quad \\cos(\\pi) = -1 [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<ol data-start=\"1526\" data-end=\"1739\">\r\n \t<li data-start=\"1526\" data-end=\"1632\">\r\n<p data-start=\"1529\" data-end=\"1632\">Find [latex]\\sin\\left(\\frac{\\pi}{2}\\right)[\/latex] and [latex]\\cos\\left(\\frac{\\pi}{2}\\right)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1633\" data-end=\"1739\">\r\n<p data-start=\"1636\" data-end=\"1739\">Find [latex]\\sin\\left(\\frac{3\\pi}{2}\\right)[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">On the unit circle, sine corresponds to <\/span>y-values<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> and cosine to <\/span>x-values<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">.<\/span>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Evaluate sine and cosine functions at common points<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>sine and cosine<\/h3>\n<p>The sine and cosine functions describe coordinates on the unit circle, a circle of radius 1 centered at the origin.<br \/>\nFor any angle [latex]\\theta[\/latex] measured in radians:<br \/>\n[latex]\\sin(\\theta) = y \\quad \\text{and} \\quad \\cos(\\theta) = x[\/latex]<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p data-start=\"733\" data-end=\"936\">Common angles on the unit circle: [latex]0, \\frac{\\pi}{6}, \\frac{\\pi}{4}, \\frac{\\pi}{3}, \\frac{\\pi}{2}[\/latex]<br data-start=\"843\" data-end=\"846\" \/>The corresponding coordinates [latex](\\cos\\theta, \\sin\\theta)[\/latex] are often memorized.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"958\" data-end=\"1059\">Find [latex]\\sin\\left(\\frac{\\pi}{3}\\right)[\/latex] and [latex]\\cos\\left(\\frac{\\pi}{3}\\right)[\/latex].<\/p>\n<p data-start=\"958\" data-end=\"1059\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q120729\">Show Solution<\/button><\/p>\n<div id=\"q120729\" class=\"hidden-answer\" style=\"display: none\">\n<p data-start=\"1061\" data-end=\"1171\">From the unit circle,<br data-start=\"1082\" data-end=\"1085\" \/>[latex](\\cos\\theta, \\sin\\theta) = \\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n<p data-start=\"1173\" data-end=\"1299\">So:<br data-start=\"1176\" data-end=\"1179\" \/>[latex]\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}, \\quad \\cos\\left(\\frac{\\pi}{3}\\right) = \\frac{1}{2}[\/latex]<\/p>\n<p data-start=\"958\" data-end=\"1059\"><\/div>\n<\/div>\n<p data-start=\"1173\" data-end=\"1299\">\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"1321\" data-end=\"1380\">Find [latex]\\sin(\\pi)[\/latex] and [latex]\\cos(\\pi)[\/latex].<\/p>\n<p data-start=\"1382\" data-end=\"1507\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427439\">Show Solution<\/button><\/p>\n<div id=\"q427439\" class=\"hidden-answer\" style=\"display: none\">On the unit circle, [latex](\\cos\\pi, \\sin\\pi) = (-1, 0)[\/latex]. So: [latex]\\sin(\\pi) = 0, \\quad \\cos(\\pi) = -1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<ol data-start=\"1526\" data-end=\"1739\">\n<li data-start=\"1526\" data-end=\"1632\">\n<p data-start=\"1529\" data-end=\"1632\">Find [latex]\\sin\\left(\\frac{\\pi}{2}\\right)[\/latex] and [latex]\\cos\\left(\\frac{\\pi}{2}\\right)[\/latex].<\/p>\n<\/li>\n<li data-start=\"1633\" data-end=\"1739\">\n<p data-start=\"1636\" data-end=\"1739\">Find [latex]\\sin\\left(\\frac{3\\pi}{2}\\right)[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">On the unit circle, sine corresponds to <\/span>y-values<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> and cosine to <\/span>x-values<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">.<\/span><\/p>\n<\/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":201,"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\/2733"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2733\/revisions"}],"predecessor-version":[{"id":4711,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2733\/revisions\/4711"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2733\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2733"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2733"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2733"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2733"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}