{"id":813,"date":"2025-10-09T15:40:44","date_gmt":"2025-10-09T15:40:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=813"},"modified":"2025-11-20T18:50:45","modified_gmt":"2025-11-20T18:50:45","slug":"second-order-differential-equations-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/second-order-differential-equations-background-youll-need-3\/","title":{"raw":"Second-Order Differential Equations: Background You'll Need 3","rendered":"Second-Order Differential Equations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Understand and use the basic rules and relationships in trigonometry<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Trigonometric Identities<\/h2>\r\n<p id=\"fs-id1170572229415\">Trigonometric identities are the equalities that involve trigonometric functions and are true for any substitution of the variable where the sides of the equation are defined. Mastery of these identities is crucial for solving trigonometric equations, proving other mathematical statements, and is frequently necessary in calculus.<\/p>\r\nThe main trigonometric identities are listed below.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>trigonometric identities<\/h3>\r\n<p id=\"fs-id1170572208589\"><strong>Reciprocal identities<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572167703\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\tan \\theta =\\large \\frac{\\sin \\theta}{\\cos \\theta} &amp; &amp; &amp; \\cot \\theta =\\large \\frac{\\cos \\theta}{\\sin \\theta} \\\\ \\csc \\theta =\\large \\frac{1}{\\sin \\theta} &amp; &amp; &amp; \\sec \\theta =\\large \\frac{1}{\\cos \\theta} \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572140422\"><strong>Pythagorean identities<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572455704\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin^2 \\theta +\\cos^2 \\theta =1\\phantom{\\rule{2em}{0ex}}1+\\tan^2 \\theta =\\sec^2 \\theta \\phantom{\\rule{2em}{0ex}}1+\\cot^2 \\theta =\\csc^2 \\theta [\/latex]<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572480920\"><strong>Addition and subtraction formulas<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572216911\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin(\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta [\/latex]<\/div>\r\n<div id=\"fs-id1170572098934\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cos(\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta [\/latex]<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572240527\"><strong>Double-angle formulas<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572215732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin(2\\theta)=2\\sin \\theta \\cos \\theta [\/latex]<\/div>\r\n<div id=\"fs-id1170572130013\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cos(2\\theta)=2\\cos^2 \\theta -1=1-2\\sin^2 \\theta =\\cos^2 \\theta -\\sin^2 \\theta [\/latex]<\/div>\r\n<\/section>When facing a trigonometric identity, verification is key to ensuring the identity holds true for all permissible values of the variable. The process is a methodical one, where you manipulate one side of the equation until it matches the other.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How to: Given a Trigonometric Identity, Verify that it is True.\r\n<\/strong>\r\n<ol id=\"fs-id2191946\">\r\n \t<li>Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.<\/li>\r\n \t<li>Look for opportunities to factor expressions, square a binomial, or add fractions.<\/li>\r\n \t<li>Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.<\/li>\r\n \t<li>If these steps do not yield the desired result, try converting all terms to sines and cosines.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Verify [latex]\\tan{\\theta} \\cos{\\theta} = \\sin{\\theta}[\/latex].[reveal-answer q=\"663537\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"663537\"] We will start on the left side, as it is the more complicated side:\r\n\r\n<center>[latex] \\begin{align*} \\tan \\theta \\cos \\theta &amp;= (\\frac{\\sin \\theta}{\\cos \\theta}) \\cos \\theta &amp;&amp; \\text{Rewrite} \\tan \\theta \\text{ in terms of} \\sin \\theta \\text{ and} \\cos \\theta \\\\ &amp;= \\frac{\\sin \\theta}{\\cancel{\\cos \\theta}} \\cancel{\\cos \\theta} &amp;&amp; \\text{Multiply by} \\cos \\theta\\text{, then cancel} \\\\ &amp;= \\sin \\theta &amp;&amp; \\text{Simplify} \\end{align*} [\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170572550142\">Prove the trigonometric identity [latex]1+\\tan^2 \\theta =\\sec^2 \\theta[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572224130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572224130\"]\r\n<p id=\"fs-id1170572224130\">We start with the identity<\/p>\r\n\r\n<div id=\"fs-id1170572167191\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin^2 \\theta +\\cos^2 \\theta =1[\/latex]<\/div>\r\n<p id=\"fs-id1170572140245\">Dividing both sides of this equation by [latex]\\cos^2 \\theta[\/latex], we obtain<\/p>\r\n\r\n<div id=\"fs-id1170572481849\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\sin^2 \\theta}{\\cos^2 \\theta}+1=\\frac{1}{\\cos^2 \\theta}[\/latex]<\/div>\r\nSince [latex]\\frac{\\sin \\theta }{ \\cos \\theta} =\\tan \\theta[\/latex] and [latex]\\frac{1}{ \\cos \\theta} =\\sec \\theta[\/latex], we conclude that\r\n<div id=\"fs-id1170572481963\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tan^2 \\theta +1=\\sec^2 \\theta[\/latex].[\/hidden-answer]<\/div>\r\n<\/section>Algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.\r\n\r\n<section class=\"textbox example\">The equation [latex](\\sin x+1)(\\sin x\u22121)=0[\/latex] resembles the equation [latex](x+1)(x\u22121)=0[\/latex], which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve.\u00a0<\/section><section class=\"textbox questionHelp\"><strong>How to: Given a Trigonometric Equation, Solve using Algebra.<\/strong>\r\n<ol>\r\n \t<li><strong>Spot Patterns<\/strong>: Look for familiar algebraic cues in the equation, like notable identities or factors.<\/li>\r\n \t<li><strong>Substitute Variables<\/strong>: Temporarily replace trig expressions with a single variable to simplify the equation.<\/li>\r\n \t<li><strong>Solve Algebraically<\/strong>: Treat the simplified equation as you would a standard algebraic one.<\/li>\r\n \t<li><strong>Back-Substitute<\/strong>: Once you've solved for the temporary variable, revert to the original trigonometric terms.<\/li>\r\n \t<li><strong>Find Angles<\/strong>: Use inverse functions to solve for the angle, considering the function's period and domain.<\/li>\r\n \t<li><strong>Verify Solutions<\/strong>: Always check that your solutions satisfy the initial equation.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Write the following trigonometric expression as an algebraic expression:<center>[latex]2 \\cos^2{\\theta} + \\cos{\\theta} -1[\/latex]<\/center>[reveal-answer q=\"469876\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"469876\"]Notice that the pattern displayed has the same form as a standard quadratic expression, [latex]ax^2+bx+c[\/latex]. Letting [latex]\\cos{\\theta}=x[\/latex], we can rewrite the expression as follows:\r\n\r\n<center>[latex]2x^2+x-1[\/latex]<\/center>This expression can be factored as [latex](2x-1)(x+1)[\/latex]. If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for [latex]x[\/latex]. At this point, we would replace [latex]x[\/latex] with [latex]\\cos{\\theta}[\/latex] and solve for [latex]\\theta[\/latex]. [\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Understand and use the basic rules and relationships in trigonometry<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Trigonometric Identities<\/h2>\n<p id=\"fs-id1170572229415\">Trigonometric identities are the equalities that involve trigonometric functions and are true for any substitution of the variable where the sides of the equation are defined. Mastery of these identities is crucial for solving trigonometric equations, proving other mathematical statements, and is frequently necessary in calculus.<\/p>\n<p>The main trigonometric identities are listed below.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>trigonometric identities<\/h3>\n<p id=\"fs-id1170572208589\"><strong>Reciprocal identities<\/strong><\/p>\n<div id=\"fs-id1170572167703\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\tan \\theta =\\large \\frac{\\sin \\theta}{\\cos \\theta} & & & \\cot \\theta =\\large \\frac{\\cos \\theta}{\\sin \\theta} \\\\ \\csc \\theta =\\large \\frac{1}{\\sin \\theta} & & & \\sec \\theta =\\large \\frac{1}{\\cos \\theta} \\end{array}[\/latex]<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572140422\"><strong>Pythagorean identities<\/strong><\/p>\n<div id=\"fs-id1170572455704\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin^2 \\theta +\\cos^2 \\theta =1\\phantom{\\rule{2em}{0ex}}1+\\tan^2 \\theta =\\sec^2 \\theta \\phantom{\\rule{2em}{0ex}}1+\\cot^2 \\theta =\\csc^2 \\theta[\/latex]<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572480920\"><strong>Addition and subtraction formulas<\/strong><\/p>\n<div id=\"fs-id1170572216911\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin(\\alpha \\pm \\beta)=\\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta[\/latex]<\/div>\n<div id=\"fs-id1170572098934\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cos(\\alpha \\pm \\beta)=\\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta[\/latex]<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572240527\"><strong>Double-angle formulas<\/strong><\/p>\n<div id=\"fs-id1170572215732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin(2\\theta)=2\\sin \\theta \\cos \\theta[\/latex]<\/div>\n<div id=\"fs-id1170572130013\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cos(2\\theta)=2\\cos^2 \\theta -1=1-2\\sin^2 \\theta =\\cos^2 \\theta -\\sin^2 \\theta[\/latex]<\/div>\n<\/section>\n<p>When facing a trigonometric identity, verification is key to ensuring the identity holds true for all permissible values of the variable. The process is a methodical one, where you manipulate one side of the equation until it matches the other.<\/p>\n<section class=\"textbox questionHelp\"><strong>How to: Given a Trigonometric Identity, Verify that it is True.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id2191946\">\n<li>Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.<\/li>\n<li>Look for opportunities to factor expressions, square a binomial, or add fractions.<\/li>\n<li>Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.<\/li>\n<li>If these steps do not yield the desired result, try converting all terms to sines and cosines.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Verify [latex]\\tan{\\theta} \\cos{\\theta} = \\sin{\\theta}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q663537\">Show Answer<\/button><\/p>\n<div id=\"q663537\" class=\"hidden-answer\" style=\"display: none\"> We will start on the left side, as it is the more complicated side:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*} \\tan \\theta \\cos \\theta &= (\\frac{\\sin \\theta}{\\cos \\theta}) \\cos \\theta && \\text{Rewrite} \\tan \\theta \\text{ in terms of} \\sin \\theta \\text{ and} \\cos \\theta \\\\ &= \\frac{\\sin \\theta}{\\cancel{\\cos \\theta}} \\cancel{\\cos \\theta} && \\text{Multiply by} \\cos \\theta\\text{, then cancel} \\\\ &= \\sin \\theta && \\text{Simplify} \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572550142\">Prove the trigonometric identity [latex]1+\\tan^2 \\theta =\\sec^2 \\theta[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572224130\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572224130\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572224130\">We start with the identity<\/p>\n<div id=\"fs-id1170572167191\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sin^2 \\theta +\\cos^2 \\theta =1[\/latex]<\/div>\n<p id=\"fs-id1170572140245\">Dividing both sides of this equation by [latex]\\cos^2 \\theta[\/latex], we obtain<\/p>\n<div id=\"fs-id1170572481849\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\sin^2 \\theta}{\\cos^2 \\theta}+1=\\frac{1}{\\cos^2 \\theta}[\/latex]<\/div>\n<p>Since [latex]\\frac{\\sin \\theta }{ \\cos \\theta} =\\tan \\theta[\/latex] and [latex]\\frac{1}{ \\cos \\theta} =\\sec \\theta[\/latex], we conclude that<\/p>\n<div id=\"fs-id1170572481963\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tan^2 \\theta +1=\\sec^2 \\theta[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>Algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.<\/p>\n<section class=\"textbox example\">The equation [latex](\\sin x+1)(\\sin x\u22121)=0[\/latex] resembles the equation [latex](x+1)(x\u22121)=0[\/latex], which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve.\u00a0<\/section>\n<section class=\"textbox questionHelp\"><strong>How to: Given a Trigonometric Equation, Solve using Algebra.<\/strong><\/p>\n<ol>\n<li><strong>Spot Patterns<\/strong>: Look for familiar algebraic cues in the equation, like notable identities or factors.<\/li>\n<li><strong>Substitute Variables<\/strong>: Temporarily replace trig expressions with a single variable to simplify the equation.<\/li>\n<li><strong>Solve Algebraically<\/strong>: Treat the simplified equation as you would a standard algebraic one.<\/li>\n<li><strong>Back-Substitute<\/strong>: Once you&#8217;ve solved for the temporary variable, revert to the original trigonometric terms.<\/li>\n<li><strong>Find Angles<\/strong>: Use inverse functions to solve for the angle, considering the function&#8217;s period and domain.<\/li>\n<li><strong>Verify Solutions<\/strong>: Always check that your solutions satisfy the initial equation.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Write the following trigonometric expression as an algebraic expression:<\/p>\n<div style=\"text-align: center;\">[latex]2 \\cos^2{\\theta} + \\cos{\\theta} -1[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q469876\">Show Answer<\/button><\/p>\n<div id=\"q469876\" class=\"hidden-answer\" style=\"display: none\">Notice that the pattern displayed has the same form as a standard quadratic expression, [latex]ax^2+bx+c[\/latex]. Letting [latex]\\cos{\\theta}=x[\/latex], we can rewrite the expression as follows:<\/p>\n<div style=\"text-align: center;\">[latex]2x^2+x-1[\/latex]<\/div>\n<p>This expression can be factored as [latex](2x-1)(x+1)[\/latex]. If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for [latex]x[\/latex]. At this point, we would replace [latex]x[\/latex] with [latex]\\cos{\\theta}[\/latex] and solve for [latex]\\theta[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-813","chapter","type-chapter","status-publish","hentry"],"part":440,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/813","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions"}],"predecessor-version":[{"id":2691,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions\/2691"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/440"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/813\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=813"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=813"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=813"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=813"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}