{"id":1513,"date":"2025-07-25T02:22:17","date_gmt":"2025-07-25T02:22:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1513"},"modified":"2026-03-12T05:38:24","modified_gmt":"2026-03-12T05:38:24","slug":"the-other-trigonometric-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions-fresh-take\/","title":{"raw":"The Other Trigonometric Functions: Fresh Take","rendered":"The Other Trigonometric Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\r\n \t<li>Use properties of even and odd trigonometric functions.<\/li>\r\n \t<li>Recognize and use fundamental identities.<\/li>\r\n \t<li>Evaluate trigonometric functions with a calculator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Other Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nBeyond sine and cosine, we often use four other trigonometric functions: <strong data-start=\"296\" data-end=\"340\">tangent, cotangent, secant, and cosecant<\/strong>. These functions are all defined in terms of sine and cosine, which means once you know [latex]\\sin(\\theta)[\/latex] and [latex]\\cos(\\theta)[\/latex], you can find the others. For special angles (30\u00b0, 45\u00b0, 60\u00b0, etc.), these values can be expressed exactly using square roots and fractions, not decimals.\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: Exact Values for the Other Trig Functions<\/strong>\r\n<ol>\r\n \t<li data-start=\"712\" data-end=\"1029\">\r\n<p data-start=\"715\" data-end=\"763\"><strong data-start=\"715\" data-end=\"761\">Definitions (in terms of sine and cosine):<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"767\" data-end=\"1029\">\r\n \t<li data-start=\"767\" data-end=\"835\">\r\n<p data-start=\"769\" data-end=\"835\">[latex]\\tan(\\theta) = \\dfrac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"839\" data-end=\"907\">\r\n<p data-start=\"841\" data-end=\"907\">[latex]\\cot(\\theta) = \\dfrac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"911\" data-end=\"968\">\r\n<p data-start=\"913\" data-end=\"968\">[latex]\\sec(\\theta) = \\dfrac{1}{\\cos(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"972\" data-end=\"1029\">\r\n<p data-start=\"974\" data-end=\"1029\">[latex]\\csc(\\theta) = \\dfrac{1}{\\sin(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1031\" data-end=\"1245\">\r\n<p data-start=\"1034\" data-end=\"1245\"><strong data-start=\"1034\" data-end=\"1064\">Start with Special Angles:<\/strong> Use exact sine and cosine values for [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{6}, \\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}[\/latex]).<\/p>\r\n[latex]\\begin{aligned}\\tan(45^\\circ) &amp;= \\dfrac{\\sin(45^\\circ)}{\\cos(45^\\circ)}\\\\ &amp;= \\dfrac{\\dfrac{\\sqrt{2}}{2}}{\\dfrac{\\sqrt{2}}{2}}\\\\ &amp;= 1 \\end{aligned}[\/latex]\r\n[latex]\\begin{aligned}\\sec(60^\\circ) &amp;= \\dfrac{1}{\\cos(60^\\circ)}\\\\ &amp;= \\dfrac{1}{\\dfrac{1}{2}}\\\\ &amp;= 2 \\end{aligned}[\/latex]\r\n[latex]\\begin{aligned}\\csc(30^\\circ) &amp;= \\dfrac{1}{\\sin(30^\\circ)}\\\\ &amp;= \\dfrac{1}{\\dfrac{1}{2}}\\\\ &amp;= 2 \\end{aligned}[\/latex]<\/li>\r\n \t<li data-start=\"1590\" data-end=\"1713\">\r\n<p data-start=\"1593\" data-end=\"1713\"><strong data-start=\"1593\" data-end=\"1614\">Undefined Values:<\/strong> Watch for division by zero \u2014 if sine or cosine = 0, then csc, sec, tan, or cot may be undefined.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1715\" data-end=\"1859\">\r\n<p data-start=\"1718\" data-end=\"1859\"><strong data-start=\"1718\" data-end=\"1736\">Keep It Exact:<\/strong> Always leave answers as simplified fractions or radicals (e.g., [latex]\\dfrac{\\sqrt{3}}{3}[\/latex] instead of decimals).<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">A theater's rear-right surround sound speaker is marked at [latex]300^\\circ[\/latex]. Find the exact values of [latex]\\sec(300^\\circ)[\/latex], [latex]\\csc(300^\\circ)[\/latex], [latex]\\tan(300^\\circ)[\/latex], and [latex]\\cot(300^\\circ)[\/latex].\r\n[reveal-answer q=\"840752\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"840752\"][latex]300^\\circ[\/latex] is in quadrant IV. Keep that in mind if you use a reference angle. Remember how each function is related to sine and cosine, and rationalize the denominator, if necessary.[\/hidden-answer]\r\n[reveal-answer q=\"535414\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"535414\"][latex]\\sec(300^\\circ) = 2[\/latex]\r\n[latex]\\begin{aligned} \\csc(300^\\circ) &amp;= -\\dfrac{2}{\\sqrt{3}}\\\\ &amp;= -\\dfrac{2\\sqrt{3}}{3} \\end{aligned}[\/latex]\r\n[latex]\\tan(300^\\circ) = -\\sqrt{3}[\/latex]\r\n[latex]\\begin{aligned} \\cot(300^\\circ) &amp;=-\\dfrac{1}{\\sqrt{3}}\\\\ &amp;= -\\dfrac{\\sqrt{3}}{3} \\end{aligned}[\/latex][\/hidden-answer]<\/section><section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddafbggf-u44h0LLqaRs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/u44h0LLqaRs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddafbggf-u44h0LLqaRs\" 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=14660548&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddafbggf-u44h0LLqaRs&vembed=0&video_id=u44h0LLqaRs&video_target=tpm-plugin-ddafbggf-u44h0LLqaRs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exact+values+of+sec%2C+cosec+and+cot_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExact values of sec, cosec and cot\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/section><\/div>\r\n<div><section class=\"textbox example\">An orientation leader's map encodes a direction by [latex]\\sec\\theta = - \\dfrac{5}{3}[\/latex] and specifies that the arrow points into QII. Find exact [latex]\\csc\\theta, \\tan\\theta, \\cot\\theta[\/latex].\r\n[reveal-answer q=\"53585\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"53585\"]Remember the definitions of the identities and apply them to the given values.[\/hidden-answer]\r\n[reveal-answer q=\"617802\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"617802\"][latex]\\sec\\theta = -\\dfrac{5}{3}[\/latex]\r\n[latex]\\csc\\theta = \\dfrac{5}{4}[\/latex]\r\n[latex]\\begin{aligned}\\tan\\theta &amp;= \\dfrac{4}{-3}\\\\ &amp;= -\\dfrac{4}{3} \\end{aligned}[\/latex]\r\n[latex]\\cot\\theta = -\\dfrac{3}{4}[\/latex][\/hidden-answer]<\/section><section><\/section><\/div>\r\n<div><section class=\"textbox example\">For a campus mural, a ray in standard position passes through the point [latex](-7,24)[\/latex]. Treat this as a terminal-side point for an angle [latex]\\theta[\/latex]. Find the exact values of [latex]\\sec\\theta, \\csc\\theta, \\tan\\theta, and \\cot\\theta[\/latex].\r\n[reveal-answer q=\"778358\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"778358\"]Use the given point and plug into [latex]x^2 + y^2 = r^2[\/latex] to find the hypotenuse first. Then, you can find the exact values using the definitions in terms of sine and cosine.[\/hidden-answer]\r\n[reveal-answer q=\"877887\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"877887\"][latex]\\sec\\theta=-\\dfrac{25}{7}[\/latex]\r\n[latex]\\csc\\theta = \\dfrac{25}{24}[\/latex]\r\n[latex]\\begin{aligned}\\tan\\theta &amp;= \\dfrac{24}{-7}\\\\ &amp;= -\\dfrac{24}{7} \\end{aligned}[\/latex]\r\n[latex]\\cot\\theta = -\\dfrac{7}{24}[\/latex][\/hidden-answer]<\/section><\/div>\r\n<div><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebfhhhca-uf4AyBBE1u0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uf4AyBBE1u0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebfhhhca-uf4AyBBE1u0\" 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=14660549&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ebfhhhca-uf4AyBBE1u0&vembed=0&video_id=uf4AyBBE1u0&video_target=tpm-plugin-ebfhhhca-uf4AyBBE1u0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Master+evaluating+the+six+trig+functions+when+given+a+point+on+the+unit+circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMaster evaluating the six trig functions when given a point on the unit circle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<\/div>\r\n<h2>Properties of Even and Odd Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"175\" data-end=\"483\">Trigonometric functions have symmetry that makes them easier to work with. Some are <strong data-start=\"259\" data-end=\"277\">even functions<\/strong>, which means their graphs are symmetric about the y-axis: [latex]f(-x) = f(x)[\/latex]. Others are <strong data-start=\"376\" data-end=\"393\">odd functions<\/strong>, which means their graphs are symmetric about the origin: [latex]f(-x) = -f(x)[\/latex].<\/p>\r\n\r\n<ul data-start=\"485\" data-end=\"612\">\r\n \t<li data-start=\"485\" data-end=\"538\">\r\n<p data-start=\"487\" data-end=\"538\"><strong data-start=\"487\" data-end=\"497\">Cosine<\/strong> and <strong data-start=\"502\" data-end=\"512\">secant<\/strong> are <strong data-start=\"517\" data-end=\"525\">even<\/strong> functions.<\/p>\r\n<\/li>\r\n \t<li data-start=\"539\" data-end=\"612\">\r\n<p data-start=\"541\" data-end=\"612\"><strong data-start=\"541\" data-end=\"570\">Sine, tangent, cotangent,<\/strong> and <strong data-start=\"575\" data-end=\"587\">cosecant<\/strong> are <strong data-start=\"592\" data-end=\"599\">odd<\/strong> functions.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"614\" data-end=\"763\">This property helps simplify expressions and quickly evaluate trig values for negative angles without needing to sketch the unit circle every time.<\/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: Even and Odd Trig Functions<\/strong>\r\n<ol>\r\n \t<li data-start=\"817\" data-end=\"1033\">\r\n<p data-start=\"820\" data-end=\"849\"><strong data-start=\"820\" data-end=\"847\">Cosine &amp; Secant (Even):<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"853\" data-end=\"1033\">\r\n \t<li data-start=\"853\" data-end=\"900\">\r\n<p data-start=\"855\" data-end=\"900\">[latex]\\cos(-\\theta) = \\cos(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"904\" data-end=\"951\">\r\n<p data-start=\"906\" data-end=\"951\">[latex]\\sec(-\\theta) = \\sec(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1035\" data-end=\"1384\">\r\n<p data-start=\"1038\" data-end=\"1085\"><strong data-start=\"1038\" data-end=\"1083\">Sine, Tangent, Cotangent, Cosecant (Odd):<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"1089\" data-end=\"1384\">\r\n \t<li data-start=\"1089\" data-end=\"1137\">\r\n<p data-start=\"1091\" data-end=\"1137\">[latex]\\sin(-\\theta) = -\\sin(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1141\" data-end=\"1189\">\r\n<p data-start=\"1143\" data-end=\"1189\">[latex]\\tan(-\\theta) = -\\tan(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1193\" data-end=\"1241\">\r\n<p data-start=\"1195\" data-end=\"1241\">[latex]\\cot(-\\theta) = -\\cot(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1245\" data-end=\"1293\">\r\n<p data-start=\"1247\" data-end=\"1293\">[latex]\\csc(-\\theta) = -\\csc(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1386\" data-end=\"1479\">\r\n<p data-start=\"1389\" data-end=\"1479\"><strong data-start=\"1389\" data-end=\"1402\">Shortcut:<\/strong> Negative angles don\u2019t need the unit circle \u2014 just apply the even\/odd rule.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1481\" data-end=\"1648\">\r\n<p data-start=\"1484\" data-end=\"1507\"><strong data-start=\"1484\" data-end=\"1505\">Graph Connection:<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"1511\" data-end=\"1648\">\r\n \t<li data-start=\"1511\" data-end=\"1566\">\r\n<p data-start=\"1513\" data-end=\"1566\">Even functions \u2192 mirror symmetry across the y-axis.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1570\" data-end=\"1648\">\r\n<p data-start=\"1572\" data-end=\"1648\">Odd functions \u2192 rotate 180\u00b0 about the origin and the graph looks the same.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1650\" data-end=\"1719\">\r\n<p data-start=\"1653\" data-end=\"1719\"><strong data-start=\"1653\" data-end=\"1670\">Memory Trick:<\/strong> \u201cCosine is the even one; most others are odd.\u201d<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/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-ahgaddae-YbU8Sq0quWE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YbU8Sq0quWE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahgaddae-YbU8Sq0quWE\" 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=13933451&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahgaddae-YbU8Sq0quWE&vembed=0&video_id=YbU8Sq0quWE&video_target=tpm-plugin-ahgaddae-YbU8Sq0quWE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Even+and+Odd+Trigonometric+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEven and Odd Trigonometric Identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Fundamental Identities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"175\" data-end=\"483\">Trigonometric identities are equations that are <strong data-start=\"207\" data-end=\"222\">always true<\/strong> for all values of the variable where both sides are defined. They let us rewrite trig expressions in simpler or more useful forms, making it easier to solve equations, prove relationships, and evaluate functions. The most important group is the <strong data-start=\"468\" data-end=\"494\">fundamental identities<\/strong>, which serve as the \u201cbuilding blocks\u201d for all other identities.<\/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: Fundamental Trig Identities<\/strong>\r\n<ol>\r\n \t<li data-start=\"614\" data-end=\"1013\">\r\n<p data-start=\"617\" data-end=\"644\"><strong data-start=\"617\" data-end=\"642\">Reciprocal Identities<\/strong><\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]\\sin(\\theta) = \\dfrac{1}{\\csc(\\theta)}[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]\\csc(\\theta) = \\dfrac{1}{\\sin(\\theta)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]\\cos(\\theta) = \\dfrac{1}{\\sec(\\theta)}[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]\\sec(\\theta) = \\dfrac{1}{\\cos(\\theta)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]\\tan(\\theta) = \\dfrac{1}{\\cot(\\theta)}[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]\\cot(\\theta) = \\dfrac{1}{\\tan(\\theta)}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li data-start=\"1015\" data-end=\"1187\">\r\n<p data-start=\"1018\" data-end=\"1043\"><strong data-start=\"1018\" data-end=\"1041\">Quotient Identities<\/strong><\/p>\r\n\r\n<ul data-start=\"1047\" data-end=\"1187\">\r\n \t<li data-start=\"1047\" data-end=\"1115\">\r\n<p data-start=\"1049\" data-end=\"1115\">[latex]\\tan(\\theta) = \\dfrac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1119\" data-end=\"1187\">\r\n<p data-start=\"1121\" data-end=\"1187\">[latex]\\cot(\\theta) = \\dfrac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1189\" data-end=\"1394\">\r\n<p data-start=\"1192\" data-end=\"1220\"><strong data-start=\"1192\" data-end=\"1218\">Pythagorean Identities<\/strong><\/p>\r\n\r\n<ul data-start=\"1224\" data-end=\"1394\">\r\n \t<li data-start=\"1224\" data-end=\"1278\">\r\n<p data-start=\"1226\" data-end=\"1278\">[latex]\\sin^2(\\theta) + \\cos^2(\\theta) = 1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1282\" data-end=\"1336\">\r\n<p data-start=\"1284\" data-end=\"1336\">[latex]1 + \\tan^2(\\theta) = \\sec^2(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1340\" data-end=\"1394\">\r\n<p data-start=\"1342\" data-end=\"1394\">[latex]1 + \\cot^2(\\theta) = \\csc^2(\\theta)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1396\" data-end=\"1584\">\r\n<p data-start=\"1399\" data-end=\"1420\"><strong data-start=\"1399\" data-end=\"1418\">Why They Matter<\/strong><\/p>\r\n\r\n<ul data-start=\"1424\" data-end=\"1584\">\r\n \t<li data-start=\"1424\" data-end=\"1496\">\r\n<p data-start=\"1426\" data-end=\"1496\">Simplify expressions (turn complicated fractions into simple forms).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1500\" data-end=\"1543\">\r\n<p data-start=\"1502\" data-end=\"1543\">Prove other identities by substitution.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1547\" data-end=\"1584\">\r\n<p data-start=\"1549\" data-end=\"1584\">Solve trig equations efficiently.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1586\" data-end=\"1802\">\r\n<p data-start=\"1589\" data-end=\"1607\"><strong data-start=\"1589\" data-end=\"1605\">Memory Trick<\/strong><\/p>\r\n\r\n<ul data-start=\"1611\" data-end=\"1802\">\r\n \t<li data-start=\"1611\" data-end=\"1671\">\r\n<p data-start=\"1613\" data-end=\"1671\">Think \u201cSine\u00b2 + Cosine\u00b2 = 1\u201d as the Pythagorean baseline.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1675\" data-end=\"1802\">\r\n<p data-start=\"1677\" data-end=\"1802\">Divide by [latex]\\cos^2[\/latex] to get the tangent identity; divide by [latex]\\sin^2[\/latex] to get the cotangent identity.<\/p>\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\">Simplify exactly: [latex]\\dfrac{(1 - \\cos x)(1 + cos x)}{\\sin x}[\/latex].\r\n[reveal-answer q=\"930931\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"930931\"]Remember the Pythagorean Identity that results when you multiply [latex](1 - \\cos x)(1 + \\cos x)[\/latex] and then reduce the fraction.[\/hidden-answer]\r\n[reveal-answer q=\"515600\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"515600\"][latex]\\begin{aligned} \\dfrac{(1 - \\cos x)(1 + \\cos x)}{\\sin x } &amp;= \\dfrac{1 -\u00a0 \\cos^2 x}{\\sin x}\\\\ &amp;= \\dfrac{\\sin^2 x}{\\sin x}\\\\ &amp;= \\sin x \\end{aligned}[\/latex][\/hidden-answer]<\/section><\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Rewrite as a single trig function: [latex]\\sec x \\cdot \\sin x[\/latex].\r\n[reveal-answer q=\"667465\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"667465\"]Remember [latex]\\sec x = \\dfrac{1}{\\cos x}[\/latex]. Multiply and simplify.[\/hidden-answer]\r\n[reveal-answer q=\"270744\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"270744\"][latex]\\begin{aligned}\\sec x \\cdot \\sin x &amp;= \\dfrac{1}{\\cos x} \\cdot \\sin x\\\\ &amp;= \\dfrac{\\sin x}{\\cos x}\\\\ &amp;= \\tan x \\end{aligned}[\/latex][\/hidden-answer]<\/section><section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfgdegdg-5MQsMvlD7OQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5MQsMvlD7OQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfgdegdg-5MQsMvlD7OQ\" 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=14660551&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfgdegdg-5MQsMvlD7OQ&vembed=0&video_id=5MQsMvlD7OQ&video_target=tpm-plugin-cfgdegdg-5MQsMvlD7OQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Simplifying+trig+expressions+by+using+the+reciprocal+identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimplifying trig expressions by using the reciprocal identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]\\tan \\theta = \\dfrac{5}{12}[\/latex] with [latex]\\theta[\/latex] in QIV, find exact values of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex].\r\n[reveal-answer q=\"304243\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"304243\"]Remember the quotient identity: [latex]\\tan = \\dfrac{\\sin}{\\cos}[\/latex] and make a triangle with given values (in QIV) to find the hypotenuse.[\/hidden-answer]\r\n[reveal-answer q=\"559643\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"559643\"][latex]\\begin{aligned} 5^2 + 12^2 &amp;= r^2\\\\\u00a0 25 + 144 &amp;= r^2\\\\ 169 &amp;= r^2\\\\ 13 &amp;= r \\end{aligned}[\/latex]\r\n[latex]\\sin\\theta = - \\dfrac{5}{13}[\/latex]\r\n[latex]\\cos\\theta = \\dfrac{12}{13}[\/latex][\/hidden-answer]<\/section><\/section><\/div>\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-gcffhaad-4BR_qUZ5jK0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4BR_qUZ5jK0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcffhaad-4BR_qUZ5jK0\" 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=13933484&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gcffhaad-4BR_qUZ5jK0&vembed=0&video_id=4BR_qUZ5jK0&video_target=tpm-plugin-gcffhaad-4BR_qUZ5jK0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/The+Reciprocal%2C+Quotient%2C+and+Pythagorean+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Reciprocal, Quotient, and Pythagorean Identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Evaluating Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"175\" data-end=\"483\">Not all angles have \u201cnice\u201d exact values like [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], or [latex]60^\\circ[\/latex]. For most angles, we rely on a calculator to approximate trig function values. The key is to make sure the calculator is in the <strong data-start=\"421\" data-end=\"437\">correct mode<\/strong> (degrees or radians) and to interpret the decimal output appropriately. Calculators give numerical approximations, but understanding what the output should look like helps catch errors and makes results more meaningful.<\/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: Using a Calculator for Trig Functions<\/strong>\r\n<ol>\r\n \t<li><strong data-start=\"726\" data-end=\"750\">Check the Mode First<\/strong>\r\n<ul data-start=\"756\" data-end=\"894\">\r\n \t<li data-start=\"756\" data-end=\"823\">\r\n<p data-start=\"758\" data-end=\"823\">If your angle is in degrees, set the calculator to degree mode.<\/p>\r\n<\/li>\r\n \t<li data-start=\"827\" data-end=\"894\">\r\n<p data-start=\"829\" data-end=\"894\">If your 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=\"896\" data-end=\"1073\">\r\n<p data-start=\"899\" data-end=\"932\"><strong data-start=\"899\" data-end=\"930\">Enter the Function Directly<\/strong><\/p>\r\n\r\n<ul data-start=\"936\" data-end=\"1073\">\r\n \t<li data-start=\"936\" data-end=\"993\">\r\n<p data-start=\"938\" data-end=\"993\">Example: [latex]\\sin(40^\\circ) \\approx 0.6428[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"997\" data-end=\"1073\">\r\n<p data-start=\"999\" data-end=\"1073\">Example: [latex]\\cos!\\left(\\dfrac{\\pi}{5}\\right) \\approx 0.8090[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1075\" data-end=\"1182\">\r\n<p data-start=\"1078\" data-end=\"1100\"><strong data-start=\"1078\" data-end=\"1098\">Expect a Decimal<\/strong><\/p>\r\n\r\n<ul data-start=\"1104\" data-end=\"1182\">\r\n \t<li data-start=\"1104\" data-end=\"1182\">\r\n<p data-start=\"1106\" data-end=\"1182\">Calculator results are approximations unless the angle is a special value.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1184\" data-end=\"1268\">\r\n<p data-start=\"1187\" data-end=\"1205\"><strong data-start=\"1187\" data-end=\"1203\">Round Wisely<\/strong><\/p>\r\n\r\n<ul data-start=\"1209\" data-end=\"1268\">\r\n \t<li data-start=\"1209\" data-end=\"1268\">\r\n<p data-start=\"1211\" data-end=\"1268\">Use 3\u20134 decimal places unless more precision is needed.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1270\" data-end=\"1507\">\r\n<p data-start=\"1273\" data-end=\"1293\"><strong data-start=\"1273\" data-end=\"1291\">Estimate First<\/strong><\/p>\r\n\r\n<ul data-start=\"1297\" data-end=\"1507\">\r\n \t<li data-start=\"1297\" data-end=\"1374\">\r\n<p data-start=\"1299\" data-end=\"1374\">Compare to a nearby special angle so you know if your answer makes sense.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1378\" data-end=\"1507\">\r\n<p data-start=\"1380\" data-end=\"1507\">Example: [latex]\\sin(40^\\circ)[\/latex] should be close to [latex]\\sin(45^\\circ) = \\dfrac{\\sqrt{2}}{2} \\approx 0.7071[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1509\" data-end=\"1655\">\r\n<p data-start=\"1512\" data-end=\"1532\"><strong data-start=\"1512\" data-end=\"1530\">Common Pitfall<\/strong><\/p>\r\n\r\n<ul data-start=\"1536\" data-end=\"1655\">\r\n \t<li data-start=\"1536\" data-end=\"1655\">\r\n<p data-start=\"1538\" data-end=\"1655\">Wrong mode = wrong answer. If a value looks way off, double-check whether your calculator is in degrees or radians.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\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-gdbddhge-rhRi_IuE_18\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/rhRi_IuE_18?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gdbddhge-rhRi_IuE_18\" 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=13933521&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gdbddhge-rhRi_IuE_18&vembed=0&video_id=rhRi_IuE_18&video_target=tpm-plugin-gdbddhge-rhRi_IuE_18'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+Trigonometric+Function+Values+on+the+Calculator_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Trigonometric Function Values on the Calculator\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\n<li>Use properties of even and odd trigonometric functions.<\/li>\n<li>Recognize and use fundamental identities.<\/li>\n<li>Evaluate trigonometric functions with a calculator.<\/li>\n<\/ul>\n<\/section>\n<h2>Other Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Beyond sine and cosine, we often use four other trigonometric functions: <strong data-start=\"296\" data-end=\"340\">tangent, cotangent, secant, and cosecant<\/strong>. These functions are all defined in terms of sine and cosine, which means once you know [latex]\\sin(\\theta)[\/latex] and [latex]\\cos(\\theta)[\/latex], you can find the others. For special angles (30\u00b0, 45\u00b0, 60\u00b0, etc.), these values can be expressed exactly using square roots and fractions, not decimals.<\/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: Exact Values for the Other Trig Functions<\/strong><\/p>\n<ol>\n<li data-start=\"712\" data-end=\"1029\">\n<p data-start=\"715\" data-end=\"763\"><strong data-start=\"715\" data-end=\"761\">Definitions (in terms of sine and cosine):<\/strong><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"767\" data-end=\"1029\">\n<li data-start=\"767\" data-end=\"835\">\n<p data-start=\"769\" data-end=\"835\">[latex]\\tan(\\theta) = \\dfrac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex]<\/p>\n<\/li>\n<li data-start=\"839\" data-end=\"907\">\n<p data-start=\"841\" data-end=\"907\">[latex]\\cot(\\theta) = \\dfrac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex]<\/p>\n<\/li>\n<li data-start=\"911\" data-end=\"968\">\n<p data-start=\"913\" data-end=\"968\">[latex]\\sec(\\theta) = \\dfrac{1}{\\cos(\\theta)}[\/latex]<\/p>\n<\/li>\n<li data-start=\"972\" data-end=\"1029\">\n<p data-start=\"974\" data-end=\"1029\">[latex]\\csc(\\theta) = \\dfrac{1}{\\sin(\\theta)}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1031\" data-end=\"1245\">\n<p data-start=\"1034\" data-end=\"1245\"><strong data-start=\"1034\" data-end=\"1064\">Start with Special Angles:<\/strong> Use exact sine and cosine values for [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], [latex]60^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{6}, \\dfrac{\\pi}{4}, \\dfrac{\\pi}{3}[\/latex]).<\/p>\n<p>[latex]\\begin{aligned}\\tan(45^\\circ) &= \\dfrac{\\sin(45^\\circ)}{\\cos(45^\\circ)}\\\\ &= \\dfrac{\\dfrac{\\sqrt{2}}{2}}{\\dfrac{\\sqrt{2}}{2}}\\\\ &= 1 \\end{aligned}[\/latex]<br \/>\n[latex]\\begin{aligned}\\sec(60^\\circ) &= \\dfrac{1}{\\cos(60^\\circ)}\\\\ &= \\dfrac{1}{\\dfrac{1}{2}}\\\\ &= 2 \\end{aligned}[\/latex]<br \/>\n[latex]\\begin{aligned}\\csc(30^\\circ) &= \\dfrac{1}{\\sin(30^\\circ)}\\\\ &= \\dfrac{1}{\\dfrac{1}{2}}\\\\ &= 2 \\end{aligned}[\/latex]<\/li>\n<li data-start=\"1590\" data-end=\"1713\">\n<p data-start=\"1593\" data-end=\"1713\"><strong data-start=\"1593\" data-end=\"1614\">Undefined Values:<\/strong> Watch for division by zero \u2014 if sine or cosine = 0, then csc, sec, tan, or cot may be undefined.<\/p>\n<\/li>\n<li data-start=\"1715\" data-end=\"1859\">\n<p data-start=\"1718\" data-end=\"1859\"><strong data-start=\"1718\" data-end=\"1736\">Keep It Exact:<\/strong> Always leave answers as simplified fractions or radicals (e.g., [latex]\\dfrac{\\sqrt{3}}{3}[\/latex] instead of decimals).<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A theater&#8217;s rear-right surround sound speaker is marked at [latex]300^\\circ[\/latex]. Find the exact values of [latex]\\sec(300^\\circ)[\/latex], [latex]\\csc(300^\\circ)[\/latex], [latex]\\tan(300^\\circ)[\/latex], and [latex]\\cot(300^\\circ)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q840752\">Hint<\/button><\/p>\n<div id=\"q840752\" class=\"hidden-answer\" style=\"display: none\">[latex]300^\\circ[\/latex] is in quadrant IV. Keep that in mind if you use a reference angle. Remember how each function is related to sine and cosine, and rationalize the denominator, if necessary.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q535414\">Show Answer<\/button><\/p>\n<div id=\"q535414\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sec(300^\\circ) = 2[\/latex]<br \/>\n[latex]\\begin{aligned} \\csc(300^\\circ) &= -\\dfrac{2}{\\sqrt{3}}\\\\ &= -\\dfrac{2\\sqrt{3}}{3} \\end{aligned}[\/latex]<br \/>\n[latex]\\tan(300^\\circ) = -\\sqrt{3}[\/latex]<br \/>\n[latex]\\begin{aligned} \\cot(300^\\circ) &=-\\dfrac{1}{\\sqrt{3}}\\\\ &= -\\dfrac{\\sqrt{3}}{3} \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddafbggf-u44h0LLqaRs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/u44h0LLqaRs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddafbggf-u44h0LLqaRs\" 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=14660548&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ddafbggf-u44h0LLqaRs&#38;vembed=0&#38;video_id=u44h0LLqaRs&#38;video_target=tpm-plugin-ddafbggf-u44h0LLqaRs\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exact+values+of+sec%2C+cosec+and+cot_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExact values of sec, cosec and cot\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox example\">An orientation leader&#8217;s map encodes a direction by [latex]\\sec\\theta = - \\dfrac{5}{3}[\/latex] and specifies that the arrow points into QII. Find exact [latex]\\csc\\theta, \\tan\\theta, \\cot\\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q53585\">Hint<\/button><\/p>\n<div id=\"q53585\" class=\"hidden-answer\" style=\"display: none\">Remember the definitions of the identities and apply them to the given values.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q617802\">Show Answer<\/button><\/p>\n<div id=\"q617802\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sec\\theta = -\\dfrac{5}{3}[\/latex]<br \/>\n[latex]\\csc\\theta = \\dfrac{5}{4}[\/latex]<br \/>\n[latex]\\begin{aligned}\\tan\\theta &= \\dfrac{4}{-3}\\\\ &= -\\dfrac{4}{3} \\end{aligned}[\/latex]<br \/>\n[latex]\\cot\\theta = -\\dfrac{3}{4}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section><\/section>\n<\/div>\n<div>\n<section class=\"textbox example\">For a campus mural, a ray in standard position passes through the point [latex](-7,24)[\/latex]. Treat this as a terminal-side point for an angle [latex]\\theta[\/latex]. Find the exact values of [latex]\\sec\\theta, \\csc\\theta, \\tan\\theta, and \\cot\\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q778358\">Hint<\/button><\/p>\n<div id=\"q778358\" class=\"hidden-answer\" style=\"display: none\">Use the given point and plug into [latex]x^2 + y^2 = r^2[\/latex] to find the hypotenuse first. Then, you can find the exact values using the definitions in terms of sine and cosine.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q877887\">Show Answer<\/button><\/p>\n<div id=\"q877887\" class=\"hidden-answer\" style=\"display: none\">[latex]\\sec\\theta=-\\dfrac{25}{7}[\/latex]<br \/>\n[latex]\\csc\\theta = \\dfrac{25}{24}[\/latex]<br \/>\n[latex]\\begin{aligned}\\tan\\theta &= \\dfrac{24}{-7}\\\\ &= -\\dfrac{24}{7} \\end{aligned}[\/latex]<br \/>\n[latex]\\cot\\theta = -\\dfrac{7}{24}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebfhhhca-uf4AyBBE1u0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uf4AyBBE1u0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebfhhhca-uf4AyBBE1u0\" 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=14660549&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ebfhhhca-uf4AyBBE1u0&#38;vembed=0&#38;video_id=uf4AyBBE1u0&#38;video_target=tpm-plugin-ebfhhhca-uf4AyBBE1u0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Master+evaluating+the+six+trig+functions+when+given+a+point+on+the+unit+circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMaster evaluating the six trig functions when given a point on the unit circle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<\/div>\n<h2>Properties of Even and Odd Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"175\" data-end=\"483\">Trigonometric functions have symmetry that makes them easier to work with. Some are <strong data-start=\"259\" data-end=\"277\">even functions<\/strong>, which means their graphs are symmetric about the y-axis: [latex]f(-x) = f(x)[\/latex]. Others are <strong data-start=\"376\" data-end=\"393\">odd functions<\/strong>, which means their graphs are symmetric about the origin: [latex]f(-x) = -f(x)[\/latex].<\/p>\n<ul data-start=\"485\" data-end=\"612\">\n<li data-start=\"485\" data-end=\"538\">\n<p data-start=\"487\" data-end=\"538\"><strong data-start=\"487\" data-end=\"497\">Cosine<\/strong> and <strong data-start=\"502\" data-end=\"512\">secant<\/strong> are <strong data-start=\"517\" data-end=\"525\">even<\/strong> functions.<\/p>\n<\/li>\n<li data-start=\"539\" data-end=\"612\">\n<p data-start=\"541\" data-end=\"612\"><strong data-start=\"541\" data-end=\"570\">Sine, tangent, cotangent,<\/strong> and <strong data-start=\"575\" data-end=\"587\">cosecant<\/strong> are <strong data-start=\"592\" data-end=\"599\">odd<\/strong> functions.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"614\" data-end=\"763\">This property helps simplify expressions and quickly evaluate trig values for negative angles without needing to sketch the unit circle every time.<\/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: Even and Odd Trig Functions<\/strong><\/p>\n<ol>\n<li data-start=\"817\" data-end=\"1033\">\n<p data-start=\"820\" data-end=\"849\"><strong data-start=\"820\" data-end=\"847\">Cosine &amp; Secant (Even):<\/strong><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"853\" data-end=\"1033\">\n<li data-start=\"853\" data-end=\"900\">\n<p data-start=\"855\" data-end=\"900\">[latex]\\cos(-\\theta) = \\cos(\\theta)[\/latex]<\/p>\n<\/li>\n<li data-start=\"904\" data-end=\"951\">\n<p data-start=\"906\" data-end=\"951\">[latex]\\sec(-\\theta) = \\sec(\\theta)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1035\" data-end=\"1384\">\n<p data-start=\"1038\" data-end=\"1085\"><strong data-start=\"1038\" data-end=\"1083\">Sine, Tangent, Cotangent, Cosecant (Odd):<\/strong><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"1089\" data-end=\"1384\">\n<li data-start=\"1089\" data-end=\"1137\">\n<p data-start=\"1091\" data-end=\"1137\">[latex]\\sin(-\\theta) = -\\sin(\\theta)[\/latex]<\/p>\n<\/li>\n<li data-start=\"1141\" data-end=\"1189\">\n<p data-start=\"1143\" data-end=\"1189\">[latex]\\tan(-\\theta) = -\\tan(\\theta)[\/latex]<\/p>\n<\/li>\n<li data-start=\"1193\" data-end=\"1241\">\n<p data-start=\"1195\" data-end=\"1241\">[latex]\\cot(-\\theta) = -\\cot(\\theta)[\/latex]<\/p>\n<\/li>\n<li data-start=\"1245\" data-end=\"1293\">\n<p data-start=\"1247\" data-end=\"1293\">[latex]\\csc(-\\theta) = -\\csc(\\theta)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1386\" data-end=\"1479\">\n<p data-start=\"1389\" data-end=\"1479\"><strong data-start=\"1389\" data-end=\"1402\">Shortcut:<\/strong> Negative angles don\u2019t need the unit circle \u2014 just apply the even\/odd rule.<\/p>\n<\/li>\n<li data-start=\"1481\" data-end=\"1648\">\n<p data-start=\"1484\" data-end=\"1507\"><strong data-start=\"1484\" data-end=\"1505\">Graph Connection:<\/strong><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"1511\" data-end=\"1648\">\n<li data-start=\"1511\" data-end=\"1566\">\n<p data-start=\"1513\" data-end=\"1566\">Even functions \u2192 mirror symmetry across the y-axis.<\/p>\n<\/li>\n<li data-start=\"1570\" data-end=\"1648\">\n<p data-start=\"1572\" data-end=\"1648\">Odd functions \u2192 rotate 180\u00b0 about the origin and the graph looks the same.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1650\" data-end=\"1719\">\n<p data-start=\"1653\" data-end=\"1719\"><strong data-start=\"1653\" data-end=\"1670\">Memory Trick:<\/strong> \u201cCosine is the even one; most others are odd.\u201d<\/p>\n<\/li>\n<\/ol>\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-ahgaddae-YbU8Sq0quWE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YbU8Sq0quWE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahgaddae-YbU8Sq0quWE\" 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=13933451&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ahgaddae-YbU8Sq0quWE&#38;vembed=0&#38;video_id=YbU8Sq0quWE&#38;video_target=tpm-plugin-ahgaddae-YbU8Sq0quWE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Even+and+Odd+Trigonometric+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEven and Odd Trigonometric Identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Fundamental Identities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"175\" data-end=\"483\">Trigonometric identities are equations that are <strong data-start=\"207\" data-end=\"222\">always true<\/strong> for all values of the variable where both sides are defined. They let us rewrite trig expressions in simpler or more useful forms, making it easier to solve equations, prove relationships, and evaluate functions. The most important group is the <strong data-start=\"468\" data-end=\"494\">fundamental identities<\/strong>, which serve as the \u201cbuilding blocks\u201d for all other identities.<\/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: Fundamental Trig Identities<\/strong><\/p>\n<ol>\n<li data-start=\"614\" data-end=\"1013\">\n<p data-start=\"617\" data-end=\"644\"><strong data-start=\"617\" data-end=\"642\">Reciprocal Identities<\/strong><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">[latex]\\sin(\\theta) = \\dfrac{1}{\\csc(\\theta)}[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]\\csc(\\theta) = \\dfrac{1}{\\sin(\\theta)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]\\cos(\\theta) = \\dfrac{1}{\\sec(\\theta)}[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]\\sec(\\theta) = \\dfrac{1}{\\cos(\\theta)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]\\tan(\\theta) = \\dfrac{1}{\\cot(\\theta)}[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]\\cot(\\theta) = \\dfrac{1}{\\tan(\\theta)}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li data-start=\"1015\" data-end=\"1187\">\n<p data-start=\"1018\" data-end=\"1043\"><strong data-start=\"1018\" data-end=\"1041\">Quotient Identities<\/strong><\/p>\n<ul data-start=\"1047\" data-end=\"1187\">\n<li data-start=\"1047\" data-end=\"1115\">\n<p data-start=\"1049\" data-end=\"1115\">[latex]\\tan(\\theta) = \\dfrac{\\sin(\\theta)}{\\cos(\\theta)}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1119\" data-end=\"1187\">\n<p data-start=\"1121\" data-end=\"1187\">[latex]\\cot(\\theta) = \\dfrac{\\cos(\\theta)}{\\sin(\\theta)}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1189\" data-end=\"1394\">\n<p data-start=\"1192\" data-end=\"1220\"><strong data-start=\"1192\" data-end=\"1218\">Pythagorean Identities<\/strong><\/p>\n<ul data-start=\"1224\" data-end=\"1394\">\n<li data-start=\"1224\" data-end=\"1278\">\n<p data-start=\"1226\" data-end=\"1278\">[latex]\\sin^2(\\theta) + \\cos^2(\\theta) = 1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1282\" data-end=\"1336\">\n<p data-start=\"1284\" data-end=\"1336\">[latex]1 + \\tan^2(\\theta) = \\sec^2(\\theta)[\/latex]<\/p>\n<\/li>\n<li data-start=\"1340\" data-end=\"1394\">\n<p data-start=\"1342\" data-end=\"1394\">[latex]1 + \\cot^2(\\theta) = \\csc^2(\\theta)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1396\" data-end=\"1584\">\n<p data-start=\"1399\" data-end=\"1420\"><strong data-start=\"1399\" data-end=\"1418\">Why They Matter<\/strong><\/p>\n<ul data-start=\"1424\" data-end=\"1584\">\n<li data-start=\"1424\" data-end=\"1496\">\n<p data-start=\"1426\" data-end=\"1496\">Simplify expressions (turn complicated fractions into simple forms).<\/p>\n<\/li>\n<li data-start=\"1500\" data-end=\"1543\">\n<p data-start=\"1502\" data-end=\"1543\">Prove other identities by substitution.<\/p>\n<\/li>\n<li data-start=\"1547\" data-end=\"1584\">\n<p data-start=\"1549\" data-end=\"1584\">Solve trig equations efficiently.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1586\" data-end=\"1802\">\n<p data-start=\"1589\" data-end=\"1607\"><strong data-start=\"1589\" data-end=\"1605\">Memory Trick<\/strong><\/p>\n<ul data-start=\"1611\" data-end=\"1802\">\n<li data-start=\"1611\" data-end=\"1671\">\n<p data-start=\"1613\" data-end=\"1671\">Think \u201cSine\u00b2 + Cosine\u00b2 = 1\u201d as the Pythagorean baseline.<\/p>\n<\/li>\n<li data-start=\"1675\" data-end=\"1802\">\n<p data-start=\"1677\" data-end=\"1802\">Divide by [latex]\\cos^2[\/latex] to get the tangent identity; divide by [latex]\\sin^2[\/latex] to get the cotangent identity.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Simplify exactly: [latex]\\dfrac{(1 - \\cos x)(1 + cos x)}{\\sin x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q930931\">Hint<\/button><\/p>\n<div id=\"q930931\" class=\"hidden-answer\" style=\"display: none\">Remember the Pythagorean Identity that results when you multiply [latex](1 - \\cos x)(1 + \\cos x)[\/latex] and then reduce the fraction.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q515600\">Show Answer<\/button><\/p>\n<div id=\"q515600\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} \\dfrac{(1 - \\cos x)(1 + \\cos x)}{\\sin x } &= \\dfrac{1 -\u00a0 \\cos^2 x}{\\sin x}\\\\ &= \\dfrac{\\sin^2 x}{\\sin x}\\\\ &= \\sin x \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Rewrite as a single trig function: [latex]\\sec x \\cdot \\sin x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q667465\">Hint<\/button><\/p>\n<div id=\"q667465\" class=\"hidden-answer\" style=\"display: none\">Remember [latex]\\sec x = \\dfrac{1}{\\cos x}[\/latex]. Multiply and simplify.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q270744\">Show Answer<\/button><\/p>\n<div id=\"q270744\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned}\\sec x \\cdot \\sin x &= \\dfrac{1}{\\cos x} \\cdot \\sin x\\\\ &= \\dfrac{\\sin x}{\\cos x}\\\\ &= \\tan x \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfgdegdg-5MQsMvlD7OQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5MQsMvlD7OQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfgdegdg-5MQsMvlD7OQ\" 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=14660551&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cfgdegdg-5MQsMvlD7OQ&#38;vembed=0&#38;video_id=5MQsMvlD7OQ&#38;video_target=tpm-plugin-cfgdegdg-5MQsMvlD7OQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Simplifying+trig+expressions+by+using+the+reciprocal+identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSimplifying trig expressions by using the reciprocal identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]\\tan \\theta = \\dfrac{5}{12}[\/latex] with [latex]\\theta[\/latex] in QIV, find exact values of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304243\">Hint<\/button><\/p>\n<div id=\"q304243\" class=\"hidden-answer\" style=\"display: none\">Remember the quotient identity: [latex]\\tan = \\dfrac{\\sin}{\\cos}[\/latex] and make a triangle with given values (in QIV) to find the hypotenuse.<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q559643\">Show Answer<\/button><\/p>\n<div id=\"q559643\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} 5^2 + 12^2 &= r^2\\\\\u00a0 25 + 144 &= r^2\\\\ 169 &= r^2\\\\ 13 &= r \\end{aligned}[\/latex]<br \/>\n[latex]\\sin\\theta = - \\dfrac{5}{13}[\/latex]<br \/>\n[latex]\\cos\\theta = \\dfrac{12}{13}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\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-gcffhaad-4BR_qUZ5jK0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4BR_qUZ5jK0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcffhaad-4BR_qUZ5jK0\" 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=13933484&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gcffhaad-4BR_qUZ5jK0&#38;vembed=0&#38;video_id=4BR_qUZ5jK0&#38;video_target=tpm-plugin-gcffhaad-4BR_qUZ5jK0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/The+Reciprocal%2C+Quotient%2C+and+Pythagorean+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Reciprocal, Quotient, and Pythagorean Identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Evaluating Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"175\" data-end=\"483\">Not all angles have \u201cnice\u201d exact values like [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], or [latex]60^\\circ[\/latex]. For most angles, we rely on a calculator to approximate trig function values. The key is to make sure the calculator is in the <strong data-start=\"421\" data-end=\"437\">correct mode<\/strong> (degrees or radians) and to interpret the decimal output appropriately. Calculators give numerical approximations, but understanding what the output should look like helps catch errors and makes results more meaningful.<\/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: Using a Calculator for Trig Functions<\/strong><\/p>\n<ol>\n<li><strong data-start=\"726\" data-end=\"750\">Check the Mode First<\/strong>\n<ul data-start=\"756\" data-end=\"894\">\n<li data-start=\"756\" data-end=\"823\">\n<p data-start=\"758\" data-end=\"823\">If your angle is in degrees, set the calculator to degree mode.<\/p>\n<\/li>\n<li data-start=\"827\" data-end=\"894\">\n<p data-start=\"829\" data-end=\"894\">If your angle is in radians, set the calculator to radian mode.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"896\" data-end=\"1073\">\n<p data-start=\"899\" data-end=\"932\"><strong data-start=\"899\" data-end=\"930\">Enter the Function Directly<\/strong><\/p>\n<ul data-start=\"936\" data-end=\"1073\">\n<li data-start=\"936\" data-end=\"993\">\n<p data-start=\"938\" data-end=\"993\">Example: [latex]\\sin(40^\\circ) \\approx 0.6428[\/latex]<\/p>\n<\/li>\n<li data-start=\"997\" data-end=\"1073\">\n<p data-start=\"999\" data-end=\"1073\">Example: [latex]\\cos!\\left(\\dfrac{\\pi}{5}\\right) \\approx 0.8090[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1075\" data-end=\"1182\">\n<p data-start=\"1078\" data-end=\"1100\"><strong data-start=\"1078\" data-end=\"1098\">Expect a Decimal<\/strong><\/p>\n<ul data-start=\"1104\" data-end=\"1182\">\n<li data-start=\"1104\" data-end=\"1182\">\n<p data-start=\"1106\" data-end=\"1182\">Calculator results are approximations unless the angle is a special value.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1184\" data-end=\"1268\">\n<p data-start=\"1187\" data-end=\"1205\"><strong data-start=\"1187\" data-end=\"1203\">Round Wisely<\/strong><\/p>\n<ul data-start=\"1209\" data-end=\"1268\">\n<li data-start=\"1209\" data-end=\"1268\">\n<p data-start=\"1211\" data-end=\"1268\">Use 3\u20134 decimal places unless more precision is needed.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1270\" data-end=\"1507\">\n<p data-start=\"1273\" data-end=\"1293\"><strong data-start=\"1273\" data-end=\"1291\">Estimate First<\/strong><\/p>\n<ul data-start=\"1297\" data-end=\"1507\">\n<li data-start=\"1297\" data-end=\"1374\">\n<p data-start=\"1299\" data-end=\"1374\">Compare to a nearby special angle so you know if your answer makes sense.<\/p>\n<\/li>\n<li data-start=\"1378\" data-end=\"1507\">\n<p data-start=\"1380\" data-end=\"1507\">Example: [latex]\\sin(40^\\circ)[\/latex] should be close to [latex]\\sin(45^\\circ) = \\dfrac{\\sqrt{2}}{2} \\approx 0.7071[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1509\" data-end=\"1655\">\n<p data-start=\"1512\" data-end=\"1532\"><strong data-start=\"1512\" data-end=\"1530\">Common Pitfall<\/strong><\/p>\n<ul data-start=\"1536\" data-end=\"1655\">\n<li data-start=\"1536\" data-end=\"1655\">\n<p data-start=\"1538\" data-end=\"1655\">Wrong mode = wrong answer. If a value looks way off, double-check whether your calculator is in degrees or radians.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\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-gdbddhge-rhRi_IuE_18\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/rhRi_IuE_18?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gdbddhge-rhRi_IuE_18\" 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=13933521&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gdbddhge-rhRi_IuE_18&#38;vembed=0&#38;video_id=rhRi_IuE_18&#38;video_target=tpm-plugin-gdbddhge-rhRi_IuE_18\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+Trigonometric+Function+Values+on+the+Calculator_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Trigonometric Function Values on the Calculator\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":32,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Exact values of sec, cosec and 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