{"id":1510,"date":"2025-07-25T02:20:17","date_gmt":"2025-07-25T02:20:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1510"},"modified":"2026-03-24T06:57:55","modified_gmt":"2026-03-24T06:57:55","slug":"sine-and-cosine-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sine-and-cosine-functions-fresh-take\/","title":{"raw":"Sine and Cosine Functions: Fresh Take","rendered":"Sine and Cosine Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find function values for the sine and cosine of the special angles.<\/li>\r\n \t<li>Use reference angles to evaluate trigonometric functions.<\/li>\r\n \t<li>Evaluate sine and cosine values using a calculator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding Function Values<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nSome angles come up so often in trigonometry that we call them the <strong data-start=\"256\" data-end=\"274\">special angles<\/strong>: [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{6}[\/latex], [latex]\\dfrac{\\pi}{4}[\/latex], and [latex]\\dfrac{\\pi}{3}[\/latex] in radians). The sine and cosine values of these angles can be found using right triangles or the unit circle. Instead of memorizing long lists, you can use patterns and symmetry to recall them quickly. These special values form the backbone of the unit circle and are used throughout trigonometry.\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Special Sine and Cosine Values<\/strong>\r\n<ol>\r\n \t<li data-start=\"784\" data-end=\"981\">\r\n<p data-start=\"787\" data-end=\"815\"><strong data-start=\"787\" data-end=\"812\">The 30\u201360\u201390 Triangle<\/strong>:<\/p>\r\n\r\n<ul data-start=\"819\" data-end=\"981\">\r\n \t<li data-start=\"819\" data-end=\"898\">\r\n<p data-start=\"821\" data-end=\"898\">[latex]\\sin(30^\\circ)=\\dfrac{1}{2}[\/latex], [latex]\\cos(30^\\circ)=\\dfrac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"902\" data-end=\"981\">\r\n<p data-start=\"904\" data-end=\"981\">[latex]\\sin(60^\\circ)=\\dfrac{\\sqrt{3}}{2}[\/latex], [latex]\\cos(60^\\circ)=\\dfrac{1}{2}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"982\" data-end=\"1076\">\r\n<p data-start=\"985\" data-end=\"1013\"><strong data-start=\"985\" data-end=\"1010\">The 45\u201345\u201390 Triangle<\/strong>:<\/p>\r\n\r\n<ul data-start=\"1017\" data-end=\"1076\">\r\n \t<li data-start=\"1017\" data-end=\"1076\">\r\n<p data-start=\"1019\" data-end=\"1076\">[latex]\\sin(45^\\circ)=\\cos(45^\\circ)=\\dfrac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1077\" data-end=\"1171\">\r\n<p data-start=\"1080\" data-end=\"1171\"><strong data-start=\"1080\" data-end=\"1104\">Unit Circle Reminder<\/strong>: On the unit circle, sine = y-coordinate, cosine = x-coordinate.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1172\" data-end=\"1331\">\r\n<p data-start=\"1175\" data-end=\"1331\"><strong data-start=\"1175\" data-end=\"1192\">Pattern Trick<\/strong>: For sine of 30\u00b0, 45\u00b0, 60\u00b0, think [latex]\\dfrac{1}{2}, \\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{3}}{2}[\/latex]. For cosine, the order reverses.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1332\" data-end=\"1438\">\r\n<p data-start=\"1335\" data-end=\"1438\"><strong data-start=\"1335\" data-end=\"1362\">Radians Are Your Friend<\/strong>: Know the same values in radian form: [latex]\\dfrac{\\pi}{6}, \\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A \"perfect pizza slice\" trend suggests cutting the pie into [latex]6[\/latex] equal slices. The center line to a slice edge makes an angle of [latex]60^\\circ[\/latex]. Find [latex]\\sin(60^\\circ)[\/latex] and [latex]\\cos(60^\\circ)[\/latex].\r\n[reveal-answer q=\"8382\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8382\"]Use the [latex]30^\\circ - 60^\\circ - 90^\\circ[\/latex] triangle ratios.[\/hidden-answer]\r\n[reveal-answer q=\"186592\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"186592\"][latex]\\sin(60^\\circ) = \\dfrac{\\sqrt{3}}{2}[\/latex]\r\n[latex]\\cos(60^\\circ) = \\dfrac{1}{2}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A viral \"speed-quilt\" block uses a diagonal cut at [latex]45^\\circ[\/latex]. Evaluate [latex]\\sin(45^\\circ)[\/latex] and [latex]\\cos(45^\\circ)[\/latex].\r\n[reveal-answer q=\"48531\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"48531\"]Use the [latex]45^\\circ - 45^\\circ - 90^\\circ[\/latex] triangle ratios.[\/hidden-answer]\r\n[reveal-answer q=\"688948\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"688948\"][latex]\\sin(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]\r\n[latex]\\cos(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">At a community festival drum circle, Section A sits at [latex]0^\\circ[\/latex] and Section C sits at [latex]120^\\circ[\/latex]. Find [latex]\\sin(120^\\circ)[\/latex] and [latex]\\cos(120^\\circ)[\/latex].\r\n[reveal-answer q=\"239192\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"239192\"][latex]120^\\circ[\/latex] is in QII.\r\nRemember the pattern trick and pay attention to the signs. [latex](\\sin&gt;0,\\ \\cos&lt;0)[\/latex][\/hidden-answer]\r\n[reveal-answer q=\"763870\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"763870\"][latex]\\sin(120^\\circ) = \\sin(60^\\circ) = \\dfrac{\\sqrt{3}}{2}[\/latex]\r\n[latex]\\cos(120^\\circ) = \\cos(60^\\circ) = -\\dfrac{1}{2}[\/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-dcbhegce-Ly1IRU9j1kQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ly1IRU9j1kQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dcbhegce-Ly1IRU9j1kQ\" 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=13922541&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dcbhegce-Ly1IRU9j1kQ&vembed=0&video_id=Ly1IRU9j1kQ&video_target=tpm-plugin-dcbhegce-Ly1IRU9j1kQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Easy+way+to+remember+trig+values+at+special+angles_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEasy way to remember trig values at special angles\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Using Reference Angles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"280\" data-end=\"787\">A <strong data-start=\"282\" data-end=\"301\">reference angle<\/strong> is the acute angle formed between the terminal side of a given angle and the x-axis. Reference angles are useful because they let us connect any angle on the unit circle back to one of the familiar \u201cspecial angles\u201d ([latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex]). By knowing the sine and cosine values for these special angles and then adjusting the <strong data-start=\"680\" data-end=\"688\">sign<\/strong> based on the quadrant, we can evaluate trig functions for almost any angle without a calculator.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips:\u00a0<\/strong>\r\n<ol>\r\n \t<li data-start=\"839\" data-end=\"1111\"><strong data-start=\"839\" data-end=\"867\">Find the Reference Angle<\/strong>: First, find which quadrant the angle is in and then use the reference angle rule that applies:\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">Quadrant<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">Degrees<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">Radians<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">\r\n<p data-start=\"1353\" data-end=\"1383\">I<\/p>\r\n<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">given angle<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">given angle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">II<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]180^\\circ - [\/latex] given angle<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\pi - [\/latex] given angle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">III<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">given angle [latex] - 180^\\circ[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">given angle [latex] - \\pi[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">IV<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]360^\\circ - [\/latex] given angle<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]2\\pi - [\/latex] given angle<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li data-start=\"839\" data-end=\"1111\"><strong data-start=\"1115\" data-end=\"1143\">Use Special Angle Values<\/strong>: Match the reference angle to [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], or [latex]60^\\circ[\/latex] (or their radian equivalents) to recall sine and cosine values.<\/li>\r\n \t<li data-start=\"839\" data-end=\"1111\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"1320\" data-end=\"1344\">Apply Quadrant Signs<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">Quadrant<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\cos[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\sin[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">\r\n<p data-start=\"1353\" data-end=\"1383\">I<\/p>\r\n<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] + [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] + [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">II<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] - [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] + [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">III<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] - [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] - [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center;\">IV<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] + [\/latex]<\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\">[latex] - [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li data-start=\"1462\" data-end=\"1495\">\r\n<p data-start=\"1464\" data-end=\"1495\"><strong data-start=\"1499\" data-end=\"1519\">General Strategy<\/strong>: \u201cReference angle gives the value, quadrant gives the sign.\u201d<\/p>\r\n<\/li>\r\n \t<li data-start=\"1462\" data-end=\"1495\">\r\n<p data-start=\"1464\" data-end=\"1495\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"1586\" data-end=\"1609\">Check with Examples<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span><\/p>\r\n\r\n<ul style=\"list-style-type: circle;\">\r\n \t<li data-start=\"1462\" data-end=\"1495\">\r\n<p data-start=\"1464\" data-end=\"1495\"><strong data-start=\"293\" data-end=\"307\">Example 1:<\/strong> [latex]\\sin(150^\\circ)[\/latex]<\/p>\r\n\r\n<ul data-start=\"343\" data-end=\"677\">\r\n \t<li data-start=\"343\" data-end=\"430\">\r\n<p data-start=\"345\" data-end=\"430\">Step 1: Find the reference angle \u2192 [latex]180^\\circ - 150^\\circ = 30^\\circ[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"433\" data-end=\"497\">\r\n<p data-start=\"435\" data-end=\"497\">Step 2: Recall [latex]\\sin(30^\\circ) = \\dfrac{1}{2}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"500\" data-end=\"612\">\r\n<p data-start=\"502\" data-end=\"612\">Step 3: Determine the quadrant \u2192 [latex]150^\\circ[\/latex] is in Quadrant II, where sine values are positive.<\/p>\r\n<\/li>\r\n \t<li data-start=\"500\" data-end=\"612\">\r\n<p data-start=\"502\" data-end=\"612\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"617\" data-end=\"628\">Answer:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]\\sin(150^\\circ) = \\dfrac{1}{2}[\/latex].<\/span><\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"615\" data-end=\"677\">\r\n<p data-start=\"617\" data-end=\"677\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"681\" data-end=\"695\">Example 2:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]\\cos(\\dfrac{7\\pi}{6})[\/latex]<\/span><\/p>\r\n\r\n<ul data-start=\"731\" data-end=\"1083\">\r\n \t<li data-start=\"731\" data-end=\"818\">\r\n<p data-start=\"733\" data-end=\"818\">Step 1: Find the reference angle \u2192 [latex]\\dfrac{7\\pi}{6} - \\pi = \\dfrac{\\pi}{6}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"821\" data-end=\"892\">\r\n<p data-start=\"823\" data-end=\"892\">Step 2: Recall [latex]\\cos (\\dfrac{\\pi}{6}) = \\dfrac{\\sqrt{3}}{2}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"895\" data-end=\"1010\">\r\n<p data-start=\"897\" data-end=\"1010\">Step 3: Determine the quadrant \u2192 [latex]\\dfrac{7\\pi}{6}[\/latex] is in Quadrant III, where cosine values are negative.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1013\" data-end=\"1083\">\r\n<p data-start=\"1015\" data-end=\"1083\"><strong data-start=\"1015\" data-end=\"1026\">Answer:<\/strong> [latex]\\cos(\\dfrac{7\\pi}{6}) = - \\dfrac{\\sqrt{3}}{2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A transit map shows a detour bearing of [latex]-210^\\circ[\/latex]. Use a reference angle to find [latex]\\sin(-210^\\circ)[\/latex] and [latex]\\cos(-210^\\circ).[\/latex]\r\n[reveal-answer q=\"542713\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"542713\"]Find the reference angle of [latex]-210^\\circ[\/latex]. Remember that the original angle tells you which quadrant the angle is in and whether [latex]\\sin[\/latex] or [latex]\\cos[\/latex] is [latex] + [\/latex] or [latex]\u00a0 -\u00a0 [\/latex][\/hidden-answer]\r\n[reveal-answer q=\"869770\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"869770\"][latex]210^\\circ - 180^\\circ = 30^\\circ[\/latex]\r\n[latex]\\sin(-210^\\circ) = \\sin(30^\\circ) = \\dfrac{1}{2}[\/latex]\r\n[latex]\\cos(-210^\\circ) = \\cos(30^\\circ) = -\\dfrac{\\sqrt{3}}{2}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A viral line dance calls for a [latex]765^\\circ[\/latex] spin. Use a reference angle to evaluate [latex]\\sin(765^\\circ)[\/latex] and [latex]\\cos(765^\\circ)[\/latex].\r\n[reveal-answer q=\"467255\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"467255\"]Subtract [latex]360^\\circ[\/latex] until you get to a reference angle.[\/hidden-answer]\r\n[reveal-answer q=\"816522\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"816522\"][latex]\\sin(765^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]\r\n[latex]\\cos(765^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Turning a tight lid, you rotate the handle by [latex]\\dfrac{23\\pi}{6}[\/latex]. Using a reference angle, find [latex]\\sin\\dfrac{23\\pi}{6}[\/latex] and [latex]\\cos\\dfrac{23\\pi}{6}[\/latex].\r\n[reveal-answer q=\"408027\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"408027\"]Subtract [latex]2\\pi[\/latex] until you get to a reference angle.[\/hidden-answer]\r\n[reveal-answer q=\"66395\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"66395\"][latex]\\sin(\\dfrac{23\\pi}{6}) = -\\dfrac{1}{2}[\/latex]\r\n[latex]\\cos(\\dfrac{23\\pi}{6}) = \\dfrac{\\sqrt{3}}{2}[\/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-ggcffafg-FlTOnUsjw_0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FlTOnUsjw_0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggcffafg-FlTOnUsjw_0\" 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=13923945&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggcffafg-FlTOnUsjw_0&vembed=0&video_id=FlTOnUsjw_0&video_target=tpm-plugin-ggcffafg-FlTOnUsjw_0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+find+a+reference+angle!_transcript.txt\">transcript for \"How to find a reference angle!\" here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Evaluating Sine and Cosine Values<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"280\" data-end=\"787\">While special angles can be evaluated exactly, most angles require a calculator. A calculator lets us find approximate values of sine and cosine for any angle, but it\u2019s important to use the <strong data-start=\"366\" data-end=\"382\">correct mode<\/strong> (degrees or radians) depending on how the angle is given. Understanding how to set up your calculator and interpret its output ensures accurate results and prevents common mistakes.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips:\u00a0 Using a Calculator for Sine and Cosine<\/strong>\r\n<ol>\r\n \t<li data-start=\"631\" data-end=\"801\">\r\n<p data-start=\"634\" data-end=\"661\"><strong data-start=\"634\" data-end=\"658\">Check the Mode First<\/strong>:<\/p>\r\n\r\n<ul data-start=\"665\" data-end=\"801\">\r\n \t<li data-start=\"665\" data-end=\"731\">\r\n<p data-start=\"667\" data-end=\"731\">If the angle is in degrees, set the calculator to degree mode.<\/p>\r\n<\/li>\r\n \t<li data-start=\"735\" data-end=\"801\">\r\n<p data-start=\"737\" data-end=\"801\">If the angle is in radians, set the calculator to radian mode.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"802\" data-end=\"916\">\r\n<p data-start=\"805\" data-end=\"916\"><strong data-start=\"805\" data-end=\"833\">Enter the Angle Directly<\/strong>: Use [latex]\\sin(\\theta)[\/latex] or [latex]\\cos(\\theta)[\/latex] and press enter.<\/p>\r\n<\/li>\r\n \t<li data-start=\"917\" data-end=\"1034\">\r\n<p data-start=\"920\" data-end=\"1034\"><strong data-start=\"920\" data-end=\"939\">Expect Decimals<\/strong>: Calculators return approximate values (e.g., [latex]\\sin(40^\\circ) \\approx 0.6428[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1035\" data-end=\"1127\">\r\n<p data-start=\"1038\" data-end=\"1127\"><strong data-start=\"1038\" data-end=\"1055\">Round Smartly<\/strong>: Round answers to 3\u20134 decimal places unless more precision is needed.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1128\" data-end=\"1333\">\r\n<p data-start=\"1131\" data-end=\"1333\"><strong data-start=\"1131\" data-end=\"1149\">Estimate First<\/strong>: Compare to a nearby special angle so the result makes sense (e.g., [latex]\\sin(40^\\circ)[\/latex] should be close to [latex]\\sin(45^\\circ)=\\dfrac{\\sqrt{2}}{2}\\approx0.7071[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1334\" data-end=\"1461\">\r\n<p data-start=\"1337\" data-end=\"1461\"><strong data-start=\"1337\" data-end=\"1355\">Common Pitfall<\/strong>: Wrong mode = wrong answer. If your sine or cosine seems \u201cway off,\u201d double-check the calculator\u2019s mode.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A stadium \"wave\" model uses [latex]t = 1.12[\/latex] radians as the phase. Compute [latex]\\sin(1.12)[\/latex] and [latex]\\cos(1.12)[\/latex] and round to [latex]4[\/latex] decimals.\r\n[reveal-answer q=\"894753\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"894753\"]Confirm your calculator is in RADIAN mode.[\/hidden-answer]\r\n[reveal-answer q=\"4024\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"4024\"][latex]\\sin(1.12) \\approx 0.9001[\/latex]\r\n[latex]\\cos(1.12) \\approx 0.4357[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Your trail app shows a bearing of [latex]247^\\circ[\/latex]. Find [latex]\\sin(247^\\circ)[\/latex] and [latex]\\cos(247^\\circ)[\/latex] and round to [latex]4[\/latex] decimals.\r\n[reveal-answer q=\"433128\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"433128\"]Confirm your calculator is in DEGREE mode.[\/hidden-answer]\r\n[reveal-answer q=\"917505\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"917505\"][latex]\\sin(247^\\circ) \\approx -0.9205[\/latex]\r\n[latex]\\cos(247^\\circ) \\approx -0.3907[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A video game controller's thumbstick reads [latex]2.87[\/latex] radians for a circle-dash move. Compute [latex]\\sin(2.87)[\/latex] and [latex]\\cos(2.87)[\/latex] and round to [latex]4[\/latex] decimals.\r\n[reveal-answer q=\"96663\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"96663\"]Confirm your calculator is in RADIAN mode.[\/hidden-answer]\r\n[reveal-answer q=\"653256\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"653256\"][latex]\\sin(2.87) \\approx 0.2683[\/latex]\r\n[latex]\\cos(2.87) \\approx -0.9633[\/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-edcdhcef--0UM5YmWhnI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-0UM5YmWhnI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-edcdhcef--0UM5YmWhnI\" 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=13923998&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-edcdhcef--0UM5YmWhnI&vembed=0&video_id=-0UM5YmWhnI&video_target=tpm-plugin-edcdhcef--0UM5YmWhnI'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluating+Sine+and+Cosine+Functions+on+Graphing+Calculator_transcript.txt\">transcript for \"Evaluating Sine and Cosine Functions on Graphing Calculator\" 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>Find function values for the sine and cosine of the special angles.<\/li>\n<li>Use reference angles to evaluate trigonometric functions.<\/li>\n<li>Evaluate sine and cosine values using a calculator.<\/li>\n<\/ul>\n<\/section>\n<h2>Finding Function Values<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Some angles come up so often in trigonometry that we call them the <strong data-start=\"256\" data-end=\"274\">special angles<\/strong>: [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{6}[\/latex], [latex]\\dfrac{\\pi}{4}[\/latex], and [latex]\\dfrac{\\pi}{3}[\/latex] in radians). The sine and cosine values of these angles can be found using right triangles or the unit circle. Instead of memorizing long lists, you can use patterns and symmetry to recall them quickly. These special values form the backbone of the unit circle and are used throughout trigonometry.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Special Sine and Cosine Values<\/strong><\/p>\n<ol>\n<li data-start=\"784\" data-end=\"981\">\n<p data-start=\"787\" data-end=\"815\"><strong data-start=\"787\" data-end=\"812\">The 30\u201360\u201390 Triangle<\/strong>:<\/p>\n<ul data-start=\"819\" data-end=\"981\">\n<li data-start=\"819\" data-end=\"898\">\n<p data-start=\"821\" data-end=\"898\">[latex]\\sin(30^\\circ)=\\dfrac{1}{2}[\/latex], [latex]\\cos(30^\\circ)=\\dfrac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"902\" data-end=\"981\">\n<p data-start=\"904\" data-end=\"981\">[latex]\\sin(60^\\circ)=\\dfrac{\\sqrt{3}}{2}[\/latex], [latex]\\cos(60^\\circ)=\\dfrac{1}{2}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"982\" data-end=\"1076\">\n<p data-start=\"985\" data-end=\"1013\"><strong data-start=\"985\" data-end=\"1010\">The 45\u201345\u201390 Triangle<\/strong>:<\/p>\n<ul data-start=\"1017\" data-end=\"1076\">\n<li data-start=\"1017\" data-end=\"1076\">\n<p data-start=\"1019\" data-end=\"1076\">[latex]\\sin(45^\\circ)=\\cos(45^\\circ)=\\dfrac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1077\" data-end=\"1171\">\n<p data-start=\"1080\" data-end=\"1171\"><strong data-start=\"1080\" data-end=\"1104\">Unit Circle Reminder<\/strong>: On the unit circle, sine = y-coordinate, cosine = x-coordinate.<\/p>\n<\/li>\n<li data-start=\"1172\" data-end=\"1331\">\n<p data-start=\"1175\" data-end=\"1331\"><strong data-start=\"1175\" data-end=\"1192\">Pattern Trick<\/strong>: For sine of 30\u00b0, 45\u00b0, 60\u00b0, think [latex]\\dfrac{1}{2}, \\dfrac{\\sqrt{2}}{2}, \\dfrac{\\sqrt{3}}{2}[\/latex]. For cosine, the order reverses.<\/p>\n<\/li>\n<li data-start=\"1332\" data-end=\"1438\">\n<p data-start=\"1335\" data-end=\"1438\"><strong data-start=\"1335\" data-end=\"1362\">Radians Are Your Friend<\/strong>: Know the same values in radian form: [latex]\\dfrac{\\pi}{6}, \\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A &#8220;perfect pizza slice&#8221; trend suggests cutting the pie into [latex]6[\/latex] equal slices. The center line to a slice edge makes an angle of [latex]60^\\circ[\/latex]. Find [latex]\\sin(60^\\circ)[\/latex] and [latex]\\cos(60^\\circ)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8382\">Hint<\/button><\/p>\n<div id=\"q8382\" class=\"hidden-answer\" style=\"display: none\">Use the [latex]30^\\circ - 60^\\circ - 90^\\circ[\/latex] triangle ratios.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q186592\">Show Answer<\/button><\/p>\n<div id=\"q186592\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(60^\\circ) = \\dfrac{\\sqrt{3}}{2}[\/latex]<br \/>\n[latex]\\cos(60^\\circ) = \\dfrac{1}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A viral &#8220;speed-quilt&#8221; block uses a diagonal cut at [latex]45^\\circ[\/latex]. Evaluate [latex]\\sin(45^\\circ)[\/latex] and [latex]\\cos(45^\\circ)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q48531\">Hint<\/button><\/p>\n<div id=\"q48531\" class=\"hidden-answer\" style=\"display: none\">Use the [latex]45^\\circ - 45^\\circ - 90^\\circ[\/latex] triangle ratios.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q688948\">Show Answer<\/button><\/p>\n<div id=\"q688948\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]<br \/>\n[latex]\\cos(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">At a community festival drum circle, Section A sits at [latex]0^\\circ[\/latex] and Section C sits at [latex]120^\\circ[\/latex]. Find [latex]\\sin(120^\\circ)[\/latex] and [latex]\\cos(120^\\circ)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q239192\">Hint<\/button><\/p>\n<div id=\"q239192\" class=\"hidden-answer\" style=\"display: none\">[latex]120^\\circ[\/latex] is in QII.<br \/>\nRemember the pattern trick and pay attention to the signs. [latex](\\sin>0,\\ \\cos<0)[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q763870\">Show Answer<\/button><\/p>\n<div id=\"q763870\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(120^\\circ) = \\sin(60^\\circ) = \\dfrac{\\sqrt{3}}{2}[\/latex]<br \/>\n[latex]\\cos(120^\\circ) = \\cos(60^\\circ) = -\\dfrac{1}{2}[\/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-dcbhegce-Ly1IRU9j1kQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ly1IRU9j1kQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dcbhegce-Ly1IRU9j1kQ\" 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=13922541&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dcbhegce-Ly1IRU9j1kQ&#38;vembed=0&#38;video_id=Ly1IRU9j1kQ&#38;video_target=tpm-plugin-dcbhegce-Ly1IRU9j1kQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Easy+way+to+remember+trig+values+at+special+angles_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEasy way to remember trig values at special angles\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Using Reference Angles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"280\" data-end=\"787\">A <strong data-start=\"282\" data-end=\"301\">reference angle<\/strong> is the acute angle formed between the terminal side of a given angle and the x-axis. Reference angles are useful because they let us connect any angle on the unit circle back to one of the familiar \u201cspecial angles\u201d ([latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex]). By knowing the sine and cosine values for these special angles and then adjusting the <strong data-start=\"680\" data-end=\"688\">sign<\/strong> based on the quadrant, we can evaluate trig functions for almost any angle without a calculator.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips:\u00a0<\/strong><\/p>\n<ol>\n<li data-start=\"839\" data-end=\"1111\"><strong data-start=\"839\" data-end=\"867\">Find the Reference Angle<\/strong>: First, find which quadrant the angle is in and then use the reference angle rule that applies:<br \/>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">Quadrant<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">Degrees<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">Radians<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">\n<p data-start=\"1353\" data-end=\"1383\">I<\/p>\n<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">given angle<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">given angle<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">II<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]180^\\circ -[\/latex] given angle<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\pi -[\/latex] given angle<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">III<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">given angle [latex]- 180^\\circ[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">given angle [latex]- \\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">IV<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]360^\\circ -[\/latex] given angle<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]2\\pi -[\/latex] given angle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li data-start=\"839\" data-end=\"1111\"><strong data-start=\"1115\" data-end=\"1143\">Use Special Angle Values<\/strong>: Match the reference angle to [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], or [latex]60^\\circ[\/latex] (or their radian equivalents) to recall sine and cosine values.<\/li>\n<li data-start=\"839\" data-end=\"1111\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"1320\" data-end=\"1344\">Apply Quadrant Signs<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span><br \/>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">Quadrant<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\cos[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]\\sin[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">\n<p data-start=\"1353\" data-end=\"1383\">I<\/p>\n<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]+[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]+[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">II<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]-[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]+[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">III<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]-[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]-[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; text-align: center;\">IV<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]+[\/latex]<\/td>\n<td style=\"width: 33.3333%; text-align: center;\">[latex]-[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li data-start=\"1462\" data-end=\"1495\">\n<p data-start=\"1464\" data-end=\"1495\"><strong data-start=\"1499\" data-end=\"1519\">General Strategy<\/strong>: \u201cReference angle gives the value, quadrant gives the sign.\u201d<\/p>\n<\/li>\n<li data-start=\"1462\" data-end=\"1495\">\n<p data-start=\"1464\" data-end=\"1495\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"1586\" data-end=\"1609\">Check with Examples<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span><\/p>\n<ul style=\"list-style-type: circle;\">\n<li data-start=\"1462\" data-end=\"1495\">\n<p data-start=\"1464\" data-end=\"1495\"><strong data-start=\"293\" data-end=\"307\">Example 1:<\/strong> [latex]\\sin(150^\\circ)[\/latex]<\/p>\n<ul data-start=\"343\" data-end=\"677\">\n<li data-start=\"343\" data-end=\"430\">\n<p data-start=\"345\" data-end=\"430\">Step 1: Find the reference angle \u2192 [latex]180^\\circ - 150^\\circ = 30^\\circ[\/latex].<\/p>\n<\/li>\n<li data-start=\"433\" data-end=\"497\">\n<p data-start=\"435\" data-end=\"497\">Step 2: Recall [latex]\\sin(30^\\circ) = \\dfrac{1}{2}[\/latex].<\/p>\n<\/li>\n<li data-start=\"500\" data-end=\"612\">\n<p data-start=\"502\" data-end=\"612\">Step 3: Determine the quadrant \u2192 [latex]150^\\circ[\/latex] is in Quadrant II, where sine values are positive.<\/p>\n<\/li>\n<li data-start=\"500\" data-end=\"612\">\n<p data-start=\"502\" data-end=\"612\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"617\" data-end=\"628\">Answer:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]\\sin(150^\\circ) = \\dfrac{1}{2}[\/latex].<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"615\" data-end=\"677\">\n<p data-start=\"617\" data-end=\"677\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\" data-start=\"681\" data-end=\"695\">Example 2:<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]\\cos(\\dfrac{7\\pi}{6})[\/latex]<\/span><\/p>\n<ul data-start=\"731\" data-end=\"1083\">\n<li data-start=\"731\" data-end=\"818\">\n<p data-start=\"733\" data-end=\"818\">Step 1: Find the reference angle \u2192 [latex]\\dfrac{7\\pi}{6} - \\pi = \\dfrac{\\pi}{6}[\/latex].<\/p>\n<\/li>\n<li data-start=\"821\" data-end=\"892\">\n<p data-start=\"823\" data-end=\"892\">Step 2: Recall [latex]\\cos (\\dfrac{\\pi}{6}) = \\dfrac{\\sqrt{3}}{2}[\/latex].<\/p>\n<\/li>\n<li data-start=\"895\" data-end=\"1010\">\n<p data-start=\"897\" data-end=\"1010\">Step 3: Determine the quadrant \u2192 [latex]\\dfrac{7\\pi}{6}[\/latex] is in Quadrant III, where cosine values are negative.<\/p>\n<\/li>\n<li data-start=\"1013\" data-end=\"1083\">\n<p data-start=\"1015\" data-end=\"1083\"><strong data-start=\"1015\" data-end=\"1026\">Answer:<\/strong> [latex]\\cos(\\dfrac{7\\pi}{6}) = - \\dfrac{\\sqrt{3}}{2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A transit map shows a detour bearing of [latex]-210^\\circ[\/latex]. Use a reference angle to find [latex]\\sin(-210^\\circ)[\/latex] and [latex]\\cos(-210^\\circ).[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q542713\">Hint<\/button><\/p>\n<div id=\"q542713\" class=\"hidden-answer\" style=\"display: none\">Find the reference angle of [latex]-210^\\circ[\/latex]. Remember that the original angle tells you which quadrant the angle is in and whether [latex]\\sin[\/latex] or [latex]\\cos[\/latex] is [latex]+[\/latex] or [latex]\u00a0 -\u00a0[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q869770\">Show Answer<\/button><\/p>\n<div id=\"q869770\" class=\"hidden-answer\" style=\"display: none\">[latex]210^\\circ - 180^\\circ = 30^\\circ[\/latex]<br \/>\n[latex]\\sin(-210^\\circ) = \\sin(30^\\circ) = \\dfrac{1}{2}[\/latex]<br \/>\n[latex]\\cos(-210^\\circ) = \\cos(30^\\circ) = -\\dfrac{\\sqrt{3}}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A viral line dance calls for a [latex]765^\\circ[\/latex] spin. Use a reference angle to evaluate [latex]\\sin(765^\\circ)[\/latex] and [latex]\\cos(765^\\circ)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q467255\">Hint<\/button><\/p>\n<div id=\"q467255\" class=\"hidden-answer\" style=\"display: none\">Subtract [latex]360^\\circ[\/latex] until you get to a reference angle.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q816522\">Show Answer<\/button><\/p>\n<div id=\"q816522\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(765^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]<br \/>\n[latex]\\cos(765^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Turning a tight lid, you rotate the handle by [latex]\\dfrac{23\\pi}{6}[\/latex]. Using a reference angle, find [latex]\\sin\\dfrac{23\\pi}{6}[\/latex] and [latex]\\cos\\dfrac{23\\pi}{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q408027\">Hint<\/button><\/p>\n<div id=\"q408027\" class=\"hidden-answer\" style=\"display: none\">Subtract [latex]2\\pi[\/latex] until you get to a reference angle.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q66395\">Show Answer<\/button><\/p>\n<div id=\"q66395\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(\\dfrac{23\\pi}{6}) = -\\dfrac{1}{2}[\/latex]<br \/>\n[latex]\\cos(\\dfrac{23\\pi}{6}) = \\dfrac{\\sqrt{3}}{2}[\/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-ggcffafg-FlTOnUsjw_0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/FlTOnUsjw_0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggcffafg-FlTOnUsjw_0\" 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=13923945&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ggcffafg-FlTOnUsjw_0&#38;vembed=0&#38;video_id=FlTOnUsjw_0&#38;video_target=tpm-plugin-ggcffafg-FlTOnUsjw_0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+find+a+reference+angle!_transcript.txt\">transcript for &#8220;How to find a reference angle!&#8221; here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Evaluating Sine and Cosine Values<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"280\" data-end=\"787\">While special angles can be evaluated exactly, most angles require a calculator. A calculator lets us find approximate values of sine and cosine for any angle, but it\u2019s important to use the <strong data-start=\"366\" data-end=\"382\">correct mode<\/strong> (degrees or radians) depending on how the angle is given. Understanding how to set up your calculator and interpret its output ensures accurate results and prevents common mistakes.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips:\u00a0 Using a Calculator for Sine and Cosine<\/strong><\/p>\n<ol>\n<li data-start=\"631\" data-end=\"801\">\n<p data-start=\"634\" data-end=\"661\"><strong data-start=\"634\" data-end=\"658\">Check the Mode First<\/strong>:<\/p>\n<ul data-start=\"665\" data-end=\"801\">\n<li data-start=\"665\" data-end=\"731\">\n<p data-start=\"667\" data-end=\"731\">If the angle is in degrees, set the calculator to degree mode.<\/p>\n<\/li>\n<li data-start=\"735\" data-end=\"801\">\n<p data-start=\"737\" data-end=\"801\">If the angle is in radians, set the calculator to radian mode.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"802\" data-end=\"916\">\n<p data-start=\"805\" data-end=\"916\"><strong data-start=\"805\" data-end=\"833\">Enter the Angle Directly<\/strong>: Use [latex]\\sin(\\theta)[\/latex] or [latex]\\cos(\\theta)[\/latex] and press enter.<\/p>\n<\/li>\n<li data-start=\"917\" data-end=\"1034\">\n<p data-start=\"920\" data-end=\"1034\"><strong data-start=\"920\" data-end=\"939\">Expect Decimals<\/strong>: Calculators return approximate values (e.g., [latex]\\sin(40^\\circ) \\approx 0.6428[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1035\" data-end=\"1127\">\n<p data-start=\"1038\" data-end=\"1127\"><strong data-start=\"1038\" data-end=\"1055\">Round Smartly<\/strong>: Round answers to 3\u20134 decimal places unless more precision is needed.<\/p>\n<\/li>\n<li data-start=\"1128\" data-end=\"1333\">\n<p data-start=\"1131\" data-end=\"1333\"><strong data-start=\"1131\" data-end=\"1149\">Estimate First<\/strong>: Compare to a nearby special angle so the result makes sense (e.g., [latex]\\sin(40^\\circ)[\/latex] should be close to [latex]\\sin(45^\\circ)=\\dfrac{\\sqrt{2}}{2}\\approx0.7071[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1334\" data-end=\"1461\">\n<p data-start=\"1337\" data-end=\"1461\"><strong data-start=\"1337\" data-end=\"1355\">Common Pitfall<\/strong>: Wrong mode = wrong answer. If your sine or cosine seems \u201cway off,\u201d double-check the calculator\u2019s mode.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A stadium &#8220;wave&#8221; model uses [latex]t = 1.12[\/latex] radians as the phase. Compute [latex]\\sin(1.12)[\/latex] and [latex]\\cos(1.12)[\/latex] and round to [latex]4[\/latex] decimals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q894753\">Hint<\/button><\/p>\n<div id=\"q894753\" class=\"hidden-answer\" style=\"display: none\">Confirm your calculator is in RADIAN mode.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4024\">Show Answer<\/button><\/p>\n<div id=\"q4024\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(1.12) \\approx 0.9001[\/latex]<br \/>\n[latex]\\cos(1.12) \\approx 0.4357[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Your trail app shows a bearing of [latex]247^\\circ[\/latex]. Find [latex]\\sin(247^\\circ)[\/latex] and [latex]\\cos(247^\\circ)[\/latex] and round to [latex]4[\/latex] decimals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q433128\">Hint<\/button><\/p>\n<div id=\"q433128\" class=\"hidden-answer\" style=\"display: none\">Confirm your calculator is in DEGREE mode.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q917505\">Show Answer<\/button><\/p>\n<div id=\"q917505\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(247^\\circ) \\approx -0.9205[\/latex]<br \/>\n[latex]\\cos(247^\\circ) \\approx -0.3907[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A video game controller&#8217;s thumbstick reads [latex]2.87[\/latex] radians for a circle-dash move. Compute [latex]\\sin(2.87)[\/latex] and [latex]\\cos(2.87)[\/latex] and round to [latex]4[\/latex] decimals.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96663\">Hint<\/button><\/p>\n<div id=\"q96663\" class=\"hidden-answer\" style=\"display: none\">Confirm your calculator is in RADIAN mode.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q653256\">Show Answer<\/button><\/p>\n<div id=\"q653256\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sin(2.87) \\approx 0.2683[\/latex]<br \/>\n[latex]\\cos(2.87) \\approx -0.9633[\/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-edcdhcef--0UM5YmWhnI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/-0UM5YmWhnI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-edcdhcef--0UM5YmWhnI\" 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=13923998&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-edcdhcef--0UM5YmWhnI&#38;vembed=0&#38;video_id=-0UM5YmWhnI&#38;video_target=tpm-plugin-edcdhcef--0UM5YmWhnI\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluating+Sine+and+Cosine+Functions+on+Graphing+Calculator_transcript.txt\">transcript for &#8220;Evaluating Sine and Cosine Functions on Graphing Calculator&#8221; here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Easy way to remember trig values at special angles\",\"author\":\"Daniel An\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Ly1IRU9j1kQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to find a reference angle!\",\"author\":\"\",\"organization\":\"Scalar Learning\",\"url\":\"https:\/\/youtu.be\/FlTOnUsjw_0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Evaluating Sine and Cosine Functions on Graphing 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