Calculus I - Solving Trig Equations

<!DOCTYPE html> <html> <head><meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes" /><meta http-equiv="X-UA-Compatible" content="IE=edge" />  <meta name="google-site-verification" content="uLoA31CJfOhIVMJWBjCmQL8xNMmmLybZU3LRKavy9WQ" /><title> Calculus I - Solving Trig Equations </title>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-9SCXJM7BEJ"></script> <script> window.dataLayer = window.dataLayer || []; function gtag() { dataLayer.push(arguments); } gtag('js', new Date()); gtag('config', 'G-9SCXJM7BEJ'); </script> <link type="text/css" href="/css/jquery.mmenu.all.css" rel="stylesheet" /><link type="text/css" href="/css/jquery.dropdown.css" rel="stylesheet" /><link href="/FA/css/all.min.css" rel="stylesheet" /><link type="text/css" href="/css/notes-all.css" rel="stylesheet" /><link type="text/css" href="/css/notes-google.css" rel="stylesheet" /><link type="text/css" href="/css/notes-mmenu.css" rel="stylesheet" /><link type="text/css" href="/css/notes-dropdown.css" rel="stylesheet" /> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script> <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_CHTML-full"></script> <script type="text/javascript" src="/js/jquery_on.js"></script> <script type="text/javascript" src="/js/jquery.mmenu.all.js"></script> <script type="text/javascript" src="/js/jquery.dropdown.js"></script> <script type="text/javascript" src="/js/notes-all.js"></script> <script> (function () { var cx = '001004262401526223570:11yv6vpcqvy'; var gcse = document.createElement('script'); gcse.type = 'text/javascript'; gcse.async = true; gcse.src = 'https://cse.google.com/cse.js?cx=' + cx; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(gcse, s); })(); </script> <meta http-equiv="keywords" name="keywords" content="solve trig equations, solve trigonometry equations, trig equations, solve cosine, solve sine, solve cosine equations, solve sine equations, solve tangent equations" /><meta http-equiv="description" name="description" content="In this section we will discuss how to solve trig equations. The answers to the equations in this section will all be one of the “standard” angles that most students have memorized after a trig class. However, the process used here can be used for any answer regardless of it being one of the standard angles or not." /></head> <body onload="init({Notes: 'NoteMobile;8/21/2018;true'})"> <div id="page"> <div class="header"> <div class="header-row">  <div id="side-menu-icon" class="header-side-menu-icon"><a href="#menu"><span class="fas fa-bars fa-lg" aria-hidden="true" title="Main Menu - Change between topics, chapters and sections as well as a few extra pages."></span></a></div> <span class="header-title"><a href="/" class="header-title-link">Paul's Online Notes</a></span> <div class="header-spacer"></div> <div id="content-top-menu" class="top-menu"> <button id="content-type-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-type" title="View (Notes, Practice Problems or Assignment Problems, Show/Hide Solutions and/or Steps) Menu"> <span id="tab_top_menu_notes" class="top-menu-item-title">Notes</span> <span class="far fa-eye fa-lg" aria-hidden="true"></span> </button> <button id="quicknav-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-quicknav" title="Quick Navigation (Previous/Next Sections and Problems and Full Problem List) Menu"> <span class="top-menu-item-title">Quick Nav</span> <span class="fas fa-exchange fa-lg" aria-hidden="true"></span> </button> <button id="download-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-download" title="Download pdf Menu"> <span class="top-menu-item-title">Download</span> <span class="far fa-download fa-lg" aria-hidden="true"></span> </button> <button id="print-menu" class="top-menu-button top-menu-button-icon-only" data-jq-dropdown="#jq-print-download" title="Print Menu"> <span class="far fa-print fa-lg" aria-hidden="true"></span> </button> </div> <div id="header-google-search" class="header-search"> <gcse:search></gcse:search> </div> <div id="header-search-icon" title="Site Search" class="header-menu-icon"><span id="search-icon" class="fas fa-search" aria-hidden="true"></span></div> </div> </div> <div id="jq-dropdown-type" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_type_menu_goto" class="top-menu-nav-title">Go To</li> <li id="li_type_menu_notes"> <span id="type_menu_notes_span" title="Viewing the Notes for the current topic." class="top-menu-current">Notes</span> </li> <li id="li_type_menu_practice"> <a href="/Problems/CalcI/TrigEquations.aspx" id="type_menu_practice_link" title="Go to Practice Problems for current topic.">Practice Problems</a> </li> <li id="li_type_menu_asgn"> <a href="/ProblemsNS/CalcI/TrigEquations.aspx" id="type_menu_asgn_link" title="Go to Assignment Problems for current topic.">Assignment Problems</a> </li> <li id="li_type_menu_sh" class="top-menu-nav-title">Show/Hide</li> <li id="li_type_menu_show" title="Show any hidden solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(1,0)" id="view_menu_show">Show all Solutions/Steps/<em>etc.</em></a></li> <li id="li_type_menu_hide" title="Hide any visible solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(0,0)" id="view_menu_hide">Hide all Solutions/Steps/<em>etc.</em></a></li> </ul> </div> <div id="jq-dropdown-quicknav" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_nav_menu_sections" class="top-menu-nav-title">Sections</li> <li id="li_nav_menu_prev_section"><a href="/Classes/CalcI/TrigFcns.aspx" id="a_nav_menu_prev_section" class="top-menu-nav-link" title="Previous Section : Trig Functions"><span class="top-menu-prev fas fa-chevron-left"></span> Trig Functions</a></li> <li id="li_nav_menu_next_section"><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" id="a_nav_menu_next_section" class="top-menu-nav-link" title="Next Section : Trig Equations with Calculators, Part I"><span class="top-menu-prev-hidden fas fa-chevron-left"></span> Trig Equations with Calculators, Part I <span class="top-menu-next fas fa-chevron-right"></span></a></li> <li id="li_nav_menu_chapters" class="top-menu-nav-title">Chapters</li> <li id="li_nav_menu_next_chapter"><a href="/Classes/CalcI/limitsIntro.aspx" id="a_nav_menu_next_chapter" class="top-menu-nav-link" title="Next Chapter : Limits">Limits <span class="top-menu-next fas fa-chevron-right"></span><span class="top-menu-next fas fa-chevron-right"></span></a></li> <li id="li_nav_menu_classes" class="top-menu-nav-title">Classes</li> <li> <a href="/Classes/Alg/Alg.aspx" id="nav_menu_alg_link" title="Go To Algebra Notes">Algebra</a> </li> <li> <span id="nav_menu_calci_span" title="Currently Viewing Calculus I Material" class="top-menu-current">Calculus I</span> </li> <li> <a href="/Classes/CalcII/CalcII.aspx" id="nav_menu_calcii_link" title="Go To Calculus II Notes">Calculus II</a> </li> <li> <a href="/Classes/CalcIII/CalcIII.aspx" id="nav_menu_calciii_link" title="Go To Calculus III Notes">Calculus III</a> </li> <li> <a href="/Classes/DE/DE.aspx" id="nav_menu_de_link" title="Go To Differential Equations Notes">Differential Equations</a> </li> <li id="li_nav_menu_extras" class="top-menu-nav-title">Extras</li> <li> <a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" id="nav_menu_algtrig_link" title="Go To Algebra & Trig Review">Algebra & Trig Review</a> </li> <li> <a href="/Extras/CommonErrors/CommonMathErrors.aspx" id="nav_menu_commonerrors_link" title="Go To Common Math Errors">Common Math Errors</a> </li> <li> <a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" id="nav_menu_complexnumbers_link" title="Go To Complex Numbers Primer">Complex Number Primer</a> </li> <li> <a href="/Extras/StudyMath/HowToStudyMath.aspx" id="nav_menu_studymath_link" title="Go To How To Study Math">How To Study Math</a> </li> <li> <a href="/Extras/CheatSheets_Tables.aspx" id="nav_menu_cheattables_link" title="Go To List of Cheat Sheets and Tables">Cheat Sheets & Tables</a> </li> <li id="li_nav_menu_misc" class="top-menu-nav-title">Misc</li> <li><a href="/contact.aspx" id="nav_menu_contact" title="Contact Me!">Contact Me</a></li> <li><a href="/mathjax.aspx" id="nav_menu_mathjax" title="Info on MathJax and MathJax Configuration Menu">MathJax Help and Configuration</a></li> </ul> </div> <div id="jq-dropdown-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_download_menu_notes" class="top-menu-nav-title">Notes Downloads</li> <li id="li_download_menu_notes_book"><a href="/GetFile.aspx?file=B,20,N" id="download_menu_notes_book" data-PDF="Download - Menu$Notes - Book$Calculus$/Downloads/Calculus/Notes/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_practice" class="top-menu-nav-title">Practice Problems Downloads</li> <li id="li_download_menu_practice_book"><a href="/GetFile.aspx?file=B,20,P" id="download_menu_practice_book" data-PDF="Download - Menu$Practice Problems - Book$Calculus$/Downloads/Calculus/Practice Problems/Complete.pdf">Complete Book - Problems Only</a></li> <li id="li_download_menu_solutions_book"><a href="/GetFile.aspx?file=B,20,S" id="download_menu_solutions_book" data-PDF="Download - Menu$Practice Problems Solutions - Book$Calculus$/Downloads/Calculus/Practice Problems Solutions/Complete.pdf">Complete Book - Solutions</a></li> <li id="li_download_menu_asgn" class="top-menu-nav-title">Assignment Problems Downloads</li> <li id="li_download_menu_asgn_book"><a href="/GetFile.aspx?file=B,20,A" id="download_menu_asgn_book" data-PDF="Download - Menu$Assignment Problems - Book$Calculus$/Downloads/Calculus/Assignment Problems/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_other" class="top-menu-nav-title">Other Items</li> <li id="li_download_menu_urls"> <a href="/DownloadURLs.aspx?bi=20" id="download_menu_urls">Get URL's for Download Items</a> </li> </ul> </div> <div id="jq-print-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_print_menu_default"><a href="javascript:SHPrintPage()" id="print_menu_default">Print Page in Current Form (Default)</a></li> <li id="li_print_menu_show"><a href="javascript:SHPrintPage(1,1)" id="print_menu_show">Show all Solutions/Steps and Print Page</a></li> <li id="li_print_menu_hide"><a href="javascript:SHPrintPage(0,1)" id="print_menu_hide">Hide all Solutions/Steps and Print Page</a></li> </ul> </div> <nav id="menu" class="notes-nav"> <ul> <li><a href="/" class="mm-link">Home</a></li> <li><span>Classes</span></li> <li><a href="/Classes/Alg/Alg.aspx" class="mm-link">Algebra</a> <ul> <li><a href="/Classes/Alg/Preliminaries.aspx" class="mm-link">1. Preliminaries</a> <ul> <li><a href="/Classes/Alg/IntegerExponents.aspx" class="mm-link">1.1 Integer Exponents</a></li> <li><a href="/Classes/Alg/RationalExponents.aspx" class="mm-link">1.2 Rational Exponents</a></li> <li><a href="/Classes/Alg/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Classes/Alg/Polynomials.aspx" class="mm-link">1.4 Polynomials</a></li> <li><a href="/Classes/Alg/Factoring.aspx" class="mm-link">1.5 Factoring Polynomials</a></li> <li><a href="/Classes/Alg/RationalExpressions.aspx" class="mm-link">1.6 Rational Expressions</a></li> <li><a href="/Classes/Alg/ComplexNumbers.aspx" class="mm-link">1.7 Complex Numbers</a></li> </ul> </li> <li><a href="/Classes/Alg/Solving.aspx" class="mm-link">2. Solving Equations and Inequalities</a> <ul> <li><a href="/Classes/Alg/SolutionSets.aspx" class="mm-link">2.1 Solutions and Solution Sets</a></li> <li><a href="/Classes/Alg/SolveLinearEqns.aspx" class="mm-link">2.2 Linear Equations</a></li> <li><a href="/Classes/Alg/LinearApps.aspx" class="mm-link">2.3 Applications of Linear Equations</a></li> <li><a href="/Classes/Alg/SolveMultiVariable.aspx" class="mm-link">2.4 Equations With More Than One Variable</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsI.aspx" class="mm-link">2.5 Quadratic Equations - Part I</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsII.aspx" class="mm-link">2.6 Quadratic Equations - Part II</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnSummary.aspx" class="mm-link">2.7 Quadratic Equations : A Summary</a></li> <li><a href="/Classes/Alg/QuadraticApps.aspx" class="mm-link">2.8 Applications of Quadratic Equations</a></li> <li><a href="/Classes/Alg/ReducibleToQuadratic.aspx" class="mm-link">2.9 Equations Reducible to Quadratic in Form</a></li> <li><a href="/Classes/Alg/SolveRadicalEqns.aspx" class="mm-link">2.10 Equations with Radicals</a></li> <li><a href="/Classes/Alg/SolveLinearInequalities.aspx" class="mm-link">2.11 Linear Inequalities</a></li> <li><a href="/Classes/Alg/SolvePolyInequalities.aspx" class="mm-link">2.12 Polynomial Inequalities</a></li> <li><a href="/Classes/Alg/SolveRationalInequalities.aspx" class="mm-link">2.13 Rational Inequalities</a></li> <li><a href="/Classes/Alg/SolveAbsValueEqns.aspx" class="mm-link">2.14 Absolute Value Equations</a></li> <li><a href="/Classes/Alg/SolveAbsValueIneq.aspx" class="mm-link">2.15 Absolute Value Inequalities</a></li> </ul> </li> <li><a href="/Classes/Alg/Graphing_Functions.aspx" class="mm-link">3. Graphing and Functions</a> <ul> <li><a href="/Classes/Alg/Graphing.aspx" class="mm-link">3.1 Graphing</a></li> <li><a href="/Classes/Alg/Lines.aspx" class="mm-link">3.2 Lines</a></li> <li><a href="/Classes/Alg/Circles.aspx" class="mm-link">3.3 Circles</a></li> <li><a href="/Classes/Alg/FunctionDefn.aspx" class="mm-link">3.4 The Definition of a Function</a></li> <li><a href="/Classes/Alg/GraphFunctions.aspx" class="mm-link">3.5 Graphing Functions</a></li> <li><a href="/Classes/Alg/CombineFunctions.aspx" class="mm-link">3.6 Combining Functions</a></li> <li><a href="/Classes/Alg/InverseFunctions.aspx" class="mm-link">3.7 Inverse Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/CommonGraphs.aspx" class="mm-link">4. Common Graphs</a> <ul> <li><a href="/Classes/Alg/Lines_Circles_PWF.aspx" class="mm-link">4.1 Lines, Circles and Piecewise Functions</a></li> <li><a href="/Classes/Alg/Parabolas.aspx" class="mm-link">4.2 Parabolas</a></li> <li><a href="/Classes/Alg/Ellipses.aspx" class="mm-link">4.3 Ellipses</a></li> <li><a href="/Classes/Alg/Hyperbolas.aspx" class="mm-link">4.4 Hyperbolas</a></li> <li><a href="/Classes/Alg/MiscFunctions.aspx" class="mm-link">4.5 Miscellaneous Functions</a></li> <li><a href="/Classes/Alg/Transformations.aspx" class="mm-link">4.6 Transformations</a></li> <li><a href="/Classes/Alg/Symmetry.aspx" class="mm-link">4.7 Symmetry</a></li> <li><a href="/Classes/Alg/GraphRationalFcns.aspx" class="mm-link">4.8 Rational Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/PolynomialFunctions.aspx" class="mm-link">5. Polynomial Functions</a> <ul> <li><a href="/Classes/Alg/DividingPolynomials.aspx" class="mm-link">5.1 Dividing Polynomials</a></li> <li><a href="/Classes/Alg/ZeroesOfPolynomials.aspx" class="mm-link">5.2 Zeroes/Roots of Polynomials</a></li> <li><a href="/Classes/Alg/GraphingPolynomials.aspx" class="mm-link">5.3 Graphing Polynomials</a></li> <li><a href="/Classes/Alg/FindingZeroesOfPolynomials.aspx" class="mm-link">5.4 Finding Zeroes of Polynomials</a></li> <li><a href="/Classes/Alg/PartialFractions.aspx" class="mm-link">5.5 Partial Fractions</a></li> </ul> </li> <li><a href="/Classes/Alg/ExpAndLog.aspx" class="mm-link">6. Exponential and Logarithm Functions</a> <ul> <li><a href="/Classes/Alg/ExpFunctions.aspx" class="mm-link">6.1 Exponential Functions</a></li> <li><a href="/Classes/Alg/LogFunctions.aspx" class="mm-link">6.2 Logarithm Functions</a></li> <li><a href="/Classes/Alg/SolveExpEqns.aspx" class="mm-link">6.3 Solving Exponential Equations</a></li> <li><a href="/Classes/Alg/SolveLogEqns.aspx" class="mm-link">6.4 Solving Logarithm Equations</a></li> <li><a href="/Classes/Alg/ExpLogApplications.aspx" class="mm-link">6.5 Applications</a></li> </ul> </li> <li><a href="/Classes/Alg/Systems.aspx" class="mm-link">7. Systems of Equations</a> <ul> <li><a href="/Classes/Alg/SystemsTwoVrble.aspx" class="mm-link">7.1 Linear Systems with Two Variables</a></li> <li><a href="/Classes/Alg/SystemsThreeVrble.aspx" class="mm-link">7.2 Linear Systems with Three Variables</a></li> <li><a href="/Classes/Alg/AugmentedMatrix.aspx" class="mm-link">7.3 Augmented Matrices</a></li> <li><a href="/Classes/Alg/AugmentedMatrixII.aspx" class="mm-link">7.4 More on the Augmented Matrix</a></li> <li><a href="/Classes/Alg/NonlinearSystems.aspx" class="mm-link">7.5 Nonlinear Systems</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcI/CalcI.aspx" class="mm-link">Calculus I</a> <ul> <li><a href="/Classes/CalcI/ReviewIntro.aspx" class="mm-link">1. Review</a> <ul> <li><a href="/Classes/CalcI/Functions.aspx" class="mm-link">1.1 Functions</a></li> <li><a href="/Classes/CalcI/InverseFunctions.aspx" class="mm-link">1.2 Inverse Functions</a></li> <li><a href="/Classes/CalcI/TrigFcns.aspx" class="mm-link">1.3 Trig Functions</a></li> <li><a href="/Classes/CalcI/TrigEquations.aspx" class="mm-link">1.4 Solving Trig Equations</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" class="mm-link">1.5 Trig Equations with Calculators, Part I</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcII.aspx" class="mm-link">1.6 Trig Equations with Calculators, Part II</a></li> <li><a href="/Classes/CalcI/ExpFunctions.aspx" class="mm-link">1.7 Exponential Functions</a></li> <li><a href="/Classes/CalcI/LogFcns.aspx" class="mm-link">1.8 Logarithm Functions</a></li> <li><a href="/Classes/CalcI/ExpLogEqns.aspx" class="mm-link">1.9 Exponential and Logarithm Equations</a></li> <li><a href="/Classes/CalcI/CommonGraphs.aspx" class="mm-link">1.10 Common Graphs</a></li> </ul> </li> <li><a href="/Classes/CalcI/limitsIntro.aspx" class="mm-link">2. Limits</a> <ul> <li><a href="/Classes/CalcI/Tangents_Rates.aspx" class="mm-link">2.1 Tangent Lines and Rates of Change</a></li> <li><a href="/Classes/CalcI/TheLimit.aspx" class="mm-link">2.2 The Limit</a></li> <li><a href="/Classes/CalcI/OneSidedLimits.aspx" class="mm-link">2.3 One-Sided Limits</a></li> <li><a href="/Classes/CalcI/LimitsProperties.aspx" class="mm-link">2.4 Limit Properties</a></li> <li><a href="/Classes/CalcI/ComputingLimits.aspx" class="mm-link">2.5 Computing Limits</a></li> <li><a href="/Classes/CalcI/InfiniteLimits.aspx" class="mm-link">2.6 Infinite Limits</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityI.aspx" class="mm-link">2.7 Limits At Infinity, Part I</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityII.aspx" class="mm-link">2.8 Limits At Infinity, Part II</a></li> <li><a href="/Classes/CalcI/Continuity.aspx" class="mm-link">2.9 Continuity</a></li> <li><a href="/Classes/CalcI/DefnOfLimit.aspx" class="mm-link">2.10 The Definition of the Limit</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivativeIntro.aspx" class="mm-link">3. Derivatives</a> <ul> <li><a href="/Classes/CalcI/DefnOfDerivative.aspx" class="mm-link">3.1 The Definition of the Derivative</a></li> <li><a href="/Classes/CalcI/DerivativeInterp.aspx" class="mm-link">3.2 Interpretation of the Derivative</a></li> <li><a href="/Classes/CalcI/DiffFormulas.aspx" class="mm-link">3.3 Differentiation Formulas</a></li> <li><a href="/Classes/CalcI/ProductQuotientRule.aspx" class="mm-link">3.4 Product and Quotient Rule</a></li> <li><a href="/Classes/CalcI/DiffTrigFcns.aspx" class="mm-link">3.5 Derivatives of Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffExpLogFcns.aspx" class="mm-link">3.6 Derivatives of Exponential and Logarithm Functions</a></li> <li><a href="/Classes/CalcI/DiffInvTrigFcns.aspx" class="mm-link">3.7 Derivatives of Inverse Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffHyperFcns.aspx" class="mm-link">3.8 Derivatives of Hyperbolic Functions</a></li> <li><a href="/Classes/CalcI/ChainRule.aspx" class="mm-link">3.9 Chain Rule</a></li> <li><a href="/Classes/CalcI/ImplicitDIff.aspx" class="mm-link">3.10 Implicit Differentiation</a></li> <li><a href="/Classes/CalcI/RelatedRates.aspx" class="mm-link">3.11 Related Rates</a></li> <li><a href="/Classes/CalcI/HigherOrderDerivatives.aspx" class="mm-link">3.12 Higher Order Derivatives</a></li> <li><a href="/Classes/CalcI/LogDiff.aspx" class="mm-link">3.13 Logarithmic Differentiation</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivAppsIntro.aspx" class="mm-link">4. Applications of Derivatives</a> <ul> <li><a href="/Classes/CalcI/RateOfChange.aspx" class="mm-link">4.1 Rates of Change</a></li> <li><a href="/Classes/CalcI/CriticalPoints.aspx" class="mm-link">4.2 Critical Points</a></li> <li><a href="/Classes/CalcI/MinMaxValues.aspx" class="mm-link">4.3 Minimum and Maximum Values</a></li> <li><a href="/Classes/CalcI/AbsExtrema.aspx" class="mm-link">4.4 Finding Absolute Extrema</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtI.aspx" class="mm-link">4.5 The Shape of a Graph, Part I</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtII.aspx" class="mm-link">4.6 The Shape of a Graph, Part II</a></li> <li><a href="/Classes/CalcI/MeanValueTheorem.aspx" class="mm-link">4.7 The Mean Value Theorem</a></li> <li><a href="/Classes/CalcI/Optimization.aspx" class="mm-link">4.8 Optimization</a></li> <li><a href="/Classes/CalcI/MoreOptimization.aspx" class="mm-link">4.9 More Optimization Problems</a></li> <li><a href="/Classes/CalcI/LHospitalsRule.aspx" class="mm-link">4.10 L'Hospital's Rule and Indeterminate Forms</a></li> <li><a href="/Classes/CalcI/LinearApproximations.aspx" class="mm-link">4.11 Linear Approximations</a></li> <li><a href="/Classes/CalcI/Differentials.aspx" class="mm-link">4.12 Differentials</a></li> <li><a href="/Classes/CalcI/NewtonsMethod.aspx" class="mm-link">4.13 Newton's Method</a></li> <li><a href="/Classes/CalcI/BusinessApps.aspx" class="mm-link">4.14 Business Applications</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntegralsIntro.aspx" class="mm-link">5. Integrals</a> <ul> <li><a href="/Classes/CalcI/IndefiniteIntegrals.aspx" class="mm-link">5.1 Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/ComputingIndefiniteIntegrals.aspx" class="mm-link">5.2 Computing Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinite.aspx" class="mm-link">5.3 Substitution Rule for Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx" class="mm-link">5.4 More Substitution Rule</a></li> <li><a href="/Classes/CalcI/AreaProblem.aspx" class="mm-link">5.5 Area Problem</a></li> <li><a href="/Classes/CalcI/DefnOfDefiniteIntegral.aspx" class="mm-link">5.6 Definition of the Definite Integral</a></li> <li><a href="/Classes/CalcI/ComputingDefiniteIntegrals.aspx" class="mm-link">5.7 Computing Definite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleDefinite.aspx" class="mm-link">5.8 Substitution Rule for Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntAppsIntro.aspx" class="mm-link">6. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcI/AvgFcnValue.aspx" class="mm-link">6.1 Average Function Value</a></li> <li><a href="/Classes/CalcI/AreaBetweenCurves.aspx" class="mm-link">6.2 Area Between Curves</a></li> <li><a href="/Classes/CalcI/VolumeWithRings.aspx" class="mm-link">6.3 Volumes of Solids of Revolution / Method of Rings</a></li> <li><a href="/Classes/CalcI/VolumeWithCylinder.aspx" class="mm-link">6.4 Volumes of Solids of Revolution/Method of Cylinders</a></li> <li><a href="/Classes/CalcI/MoreVolume.aspx" class="mm-link">6.5 More Volume Problems</a></li> <li><a href="/Classes/CalcI/Work.aspx" class="mm-link">6.6 Work</a></li> </ul> </li> <li><a href="/Classes/CalcI/ExtrasIntro.aspx" class="mm-link">Appendix A. Extras</a> <ul> <li><a href="/Classes/CalcI/LimitProofs.aspx" class="mm-link">A.1 Proof of Various Limit Properties</a></li> <li><a href="/Classes/CalcI/DerivativeProofs.aspx" class="mm-link">A.2 Proof of Various Derivative Properties</a></li> <li><a href="/Classes/CalcI/ProofTrigDeriv.aspx" class="mm-link">A.3 Proof of Trig Limits</a></li> <li><a href="/Classes/CalcI/DerivativeAppsProofs.aspx" class="mm-link">A.4 Proofs of Derivative Applications Facts</a></li> <li><a href="/Classes/CalcI/ProofIntProp.aspx" class="mm-link">A.5 Proof of Various Integral Properties </a></li> <li><a href="/Classes/CalcI/Area_Volume_Formulas.aspx" class="mm-link">A.6 Area and Volume Formulas</a></li> <li><a href="/Classes/CalcI/TypesOfInfinity.aspx" class="mm-link">A.7 Types of Infinity</a></li> <li><a href="/Classes/CalcI/SummationNotation.aspx" class="mm-link">A.8 Summation Notation</a></li> <li><a href="/Classes/CalcI/ConstantofIntegration.aspx" class="mm-link">A.9 Constant of Integration</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcII/CalcII.aspx" class="mm-link">Calculus II</a> <ul> <li><a href="/Classes/CalcII/IntTechIntro.aspx" class="mm-link">7. Integration Techniques</a> <ul> <li><a href="/Classes/CalcII/IntegrationByParts.aspx" class="mm-link">7.1 Integration by Parts</a></li> <li><a href="/Classes/CalcII/IntegralsWithTrig.aspx" class="mm-link">7.2 Integrals Involving Trig Functions</a></li> <li><a href="/Classes/CalcII/TrigSubstitutions.aspx" class="mm-link">7.3 Trig Substitutions</a></li> <li><a href="/Classes/CalcII/PartialFractions.aspx" class="mm-link">7.4 Partial Fractions</a></li> <li><a href="/Classes/CalcII/IntegralsWithRoots.aspx" class="mm-link">7.5 Integrals Involving Roots</a></li> <li><a href="/Classes/CalcII/IntegralsWithQuadratics.aspx" class="mm-link">7.6 Integrals Involving Quadratics</a></li> <li><a href="/Classes/CalcII/IntegrationStrategy.aspx" class="mm-link">7.7 Integration Strategy</a></li> <li><a href="/Classes/CalcII/ImproperIntegrals.aspx" class="mm-link">7.8 Improper Integrals</a></li> <li><a href="/Classes/CalcII/ImproperIntegralsCompTest.aspx" class="mm-link">7.9 Comparison Test for Improper Integrals</a></li> <li><a href="/Classes/CalcII/ApproximatingDefIntegrals.aspx" class="mm-link">7.10 Approximating Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcII/IntAppsIntro.aspx" class="mm-link">8. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcII/ArcLength.aspx" class="mm-link">8.1 Arc Length</a></li> <li><a href="/Classes/CalcII/SurfaceArea.aspx" class="mm-link">8.2 Surface Area</a></li> <li><a href="/Classes/CalcII/CenterOfMass.aspx" class="mm-link">8.3 Center of Mass</a></li> <li><a href="/Classes/CalcII/HydrostaticPressure.aspx" class="mm-link">8.4 Hydrostatic Pressure</a></li> <li><a href="/Classes/CalcII/Probability.aspx" class="mm-link">8.5 Probability</a></li> </ul> </li> <li><a href="/Classes/CalcII/ParametricIntro.aspx" class="mm-link">9. Parametric Equations and Polar Coordinates</a> <ul> <li><a href="/Classes/CalcII/ParametricEqn.aspx" class="mm-link">9.1 Parametric Equations and Curves</a></li> <li><a href="/Classes/CalcII/ParaTangent.aspx" class="mm-link">9.2 Tangents with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArea.aspx" class="mm-link">9.3 Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArcLength.aspx" class="mm-link">9.4 Arc Length with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaSurfaceArea.aspx" class="mm-link">9.5 Surface Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/PolarCoordinates.aspx" class="mm-link">9.6 Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarTangents.aspx" class="mm-link">9.7 Tangents with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArea.aspx" class="mm-link">9.8 Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArcLength.aspx" class="mm-link">9.9 Arc Length with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarSurfaceArea.aspx" class="mm-link">9.10 Surface Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/ArcLength_SurfaceArea.aspx" class="mm-link">9.11 Arc Length and Surface Area Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcII/SeriesIntro.aspx" class="mm-link">10. Series & Sequences</a> <ul> <li><a href="/Classes/CalcII/Sequences.aspx" class="mm-link">10.1 Sequences</a></li> <li><a href="/Classes/CalcII/MoreSequences.aspx" class="mm-link">10.2 More on Sequences</a></li> <li><a href="/Classes/CalcII/Series_Basics.aspx" class="mm-link">10.3 Series - The Basics</a></li> <li><a href="/Classes/CalcII/ConvergenceOfSeries.aspx" class="mm-link">10.4 Convergence/Divergence of Series</a></li> <li><a href="/Classes/CalcII/Series_Special.aspx" class="mm-link">10.5 Special Series</a></li> <li><a href="/Classes/CalcII/IntegralTest.aspx" class="mm-link">10.6 Integral Test</a></li> <li><a href="/Classes/CalcII/SeriesCompTest.aspx" class="mm-link">10.7 Comparison Test/Limit Comparison Test</a></li> <li><a href="/Classes/CalcII/AlternatingSeries.aspx" class="mm-link">10.8 Alternating Series Test</a></li> <li><a href="/Classes/CalcII/AbsoluteConvergence.aspx" class="mm-link">10.9 Absolute Convergence</a></li> <li><a href="/Classes/CalcII/RatioTest.aspx" class="mm-link">10.10 Ratio Test</a></li> <li><a href="/Classes/CalcII/RootTest.aspx" class="mm-link">10.11 Root Test</a></li> <li><a href="/Classes/CalcII/SeriesStrategy.aspx" class="mm-link">10.12 Strategy for Series</a></li> <li><a href="/Classes/CalcII/EstimatingSeries.aspx" class="mm-link">10.13 Estimating the Value of a Series</a></li> <li><a href="/Classes/CalcII/PowerSeries.aspx" class="mm-link">10.14 Power Series</a></li> <li><a href="/Classes/CalcII/PowerSeriesandFunctions.aspx" class="mm-link">10.15 Power Series and Functions</a></li> <li><a href="/Classes/CalcII/TaylorSeries.aspx" class="mm-link">10.16 Taylor Series</a></li> <li><a href="/Classes/CalcII/TaylorSeriesApps.aspx" class="mm-link">10.17 Applications of Series</a></li> <li><a href="/Classes/CalcII/BinomialSeries.aspx" class="mm-link">10.18 Binomial Series</a></li> </ul> </li> <li><a href="/Classes/CalcII/VectorsIntro.aspx" class="mm-link">11. Vectors</a> <ul> <li><a href="/Classes/CalcII/Vectors_Basics.aspx" class="mm-link">11.1 Vectors - The Basics</a></li> <li><a href="/Classes/CalcII/VectorArithmetic.aspx" class="mm-link">11.2 Vector Arithmetic</a></li> <li><a href="/Classes/CalcII/DotProduct.aspx" class="mm-link">11.3 Dot Product</a></li> <li><a href="/Classes/CalcII/CrossProduct.aspx" class="mm-link">11.4 Cross Product</a></li> </ul> </li> <li><a href="/Classes/CalcII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcIII/CalcIII.aspx" class="mm-link">Calculus III</a> <ul> <li><a href="/Classes/CalcIII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcIII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcIII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcIII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcIII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcIII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcIII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcIII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcIII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcIII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcIII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcIII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcIII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivsIntro.aspx" class="mm-link">13. Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/Limits.aspx" class="mm-link">13.1 Limits</a></li> <li><a href="/Classes/CalcIII/PartialDerivatives.aspx" class="mm-link">13.2 Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/PartialDerivInterp.aspx" class="mm-link">13.3 Interpretations of Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/HighOrderPartialDerivs.aspx" class="mm-link">13.4 Higher Order Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/Differentials.aspx" class="mm-link">13.5 Differentials</a></li> <li><a href="/Classes/CalcIII/ChainRule.aspx" class="mm-link">13.6 Chain Rule</a></li> <li><a href="/Classes/CalcIII/DirectionalDeriv.aspx" class="mm-link">13.7 Directional Derivatives</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivAppsIntro.aspx" class="mm-link">14. Applications of Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/TangentPlanes.aspx" class="mm-link">14.1 Tangent Planes and Linear Approximations</a></li> <li><a href="/Classes/CalcIII/GradientVectorTangentPlane.aspx" class="mm-link">14.2 Gradient Vector, Tangent Planes and Normal Lines</a></li> <li><a href="/Classes/CalcIII/RelativeExtrema.aspx" class="mm-link">14.3 Relative Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/AbsoluteExtrema.aspx" class="mm-link">14.4 Absolute Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/LagrangeMultipliers.aspx" class="mm-link">14.5 Lagrange Multipliers</a></li> </ul> </li> <li><a href="/Classes/CalcIII/MultipleIntegralsIntro.aspx" class="mm-link">15. Multiple Integrals</a> <ul> <li><a href="/Classes/CalcIII/DoubleIntegrals.aspx" class="mm-link">15.1 Double Integrals</a></li> <li><a href="/Classes/CalcIII/IteratedIntegrals.aspx" class="mm-link">15.2 Iterated Integrals</a></li> <li><a href="/Classes/CalcIII/DIGeneralRegion.aspx" class="mm-link">15.3 Double Integrals over General Regions</a></li> <li><a href="/Classes/CalcIII/DIPolarCoords.aspx" class="mm-link">15.4 Double Integrals in Polar Coordinates</a></li> <li><a href="/Classes/CalcIII/TripleIntegrals.aspx" class="mm-link">15.5 Triple Integrals</a></li> <li><a href="/Classes/CalcIII/TICylindricalCoords.aspx" class="mm-link">15.6 Triple Integrals in Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/TISphericalCoords.aspx" class="mm-link">15.7 Triple Integrals in Spherical Coordinates</a></li> <li><a href="/Classes/CalcIII/ChangeOfVariables.aspx" class="mm-link">15.8 Change of Variables</a></li> <li><a href="/Classes/CalcIII/SurfaceArea.aspx" class="mm-link">15.9 Surface Area</a></li> <li><a href="/Classes/CalcIII/Area_Volume.aspx" class="mm-link">15.10 Area and Volume Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" class="mm-link">16. Line Integrals</a> <ul> <li><a href="/Classes/CalcIII/VectorFields.aspx" class="mm-link">16.1 Vector Fields</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtI.aspx" class="mm-link">16.2 Line Integrals - Part I</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtII.aspx" class="mm-link">16.3 Line Integrals - Part II</a></li> <li><a href="/Classes/CalcIII/LineIntegralsVectorFields.aspx" class="mm-link">16.4 Line Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/FundThmLineIntegrals.aspx" class="mm-link">16.5 Fundamental Theorem for Line Integrals</a></li> <li><a href="/Classes/CalcIII/ConservativeVectorField.aspx" class="mm-link">16.6 Conservative Vector Fields</a></li> <li><a href="/Classes/CalcIII/GreensTheorem.aspx" class="mm-link">16.7 Green's Theorem</a></li> </ul> </li> <li><a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" class="mm-link">17.Surface Integrals</a> <ul> <li><a href="/Classes/CalcIII/CurlDivergence.aspx" class="mm-link">17.1 Curl and Divergence</a></li> <li><a href="/Classes/CalcIII/ParametricSurfaces.aspx" class="mm-link">17.2 Parametric Surfaces</a></li> <li><a href="/Classes/CalcIII/SurfaceIntegrals.aspx" class="mm-link">17.3 Surface Integrals</a></li> <li><a href="/Classes/CalcIII/SurfIntVectorField.aspx" class="mm-link">17.4 Surface Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/StokesTheorem.aspx" class="mm-link">17.5 Stokes' Theorem</a></li> <li><a href="/Classes/CalcIII/DivergenceTheorem.aspx" class="mm-link">17.6 Divergence Theorem</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/DE/DE.aspx" class="mm-link">Differential Equations</a> <ul> <li><a href="/Classes/DE/IntroBasic.aspx" class="mm-link">1. Basic Concepts</a> <ul> <li><a href="/Classes/DE/Definitions.aspx" class="mm-link">1.1 Definitions</a></li> <li><a href="/Classes/DE/DirectionFields.aspx" class="mm-link">1.2 Direction Fields</a></li> <li><a href="/Classes/DE/FinalThoughts.aspx" class="mm-link">1.3 Final Thoughts</a></li> </ul> </li> <li><a href="/Classes/DE/IntroFirstOrder.aspx" class="mm-link">2. First Order DE's</a> <ul> <li><a href="/Classes/DE/Linear.aspx" class="mm-link">2.1 Linear Equations</a></li> <li><a href="/Classes/DE/Separable.aspx" class="mm-link">2.2 Separable Equations</a></li> <li><a href="/Classes/DE/Exact.aspx" class="mm-link">2.3 Exact Equations</a></li> <li><a href="/Classes/DE/Bernoulli.aspx" class="mm-link">2.4 Bernoulli Differential Equations</a></li> <li><a href="/Classes/DE/Substitutions.aspx" class="mm-link">2.5 Substitutions</a></li> <li><a href="/Classes/DE/IoV.aspx" class="mm-link">2.6 Intervals of Validity</a></li> <li><a href="/Classes/DE/Modeling.aspx" class="mm-link">2.7 Modeling with First Order DE's</a></li> <li><a href="/Classes/DE/EquilibriumSolutions.aspx" class="mm-link">2.8 Equilibrium Solutions</a></li> <li><a href="/Classes/DE/EulersMethod.aspx" class="mm-link">2.9 Euler's Method</a></li> </ul> </li> <li><a href="/Classes/DE/IntroSecondOrder.aspx" class="mm-link">3. Second Order DE's</a> <ul> <li><a href="/Classes/DE/SecondOrderConcepts.aspx" class="mm-link">3.1 Basic Concepts</a></li> <li><a href="/Classes/DE/RealRoots.aspx" class="mm-link">3.2 Real & Distinct Roots</a></li> <li><a href="/Classes/DE/ComplexRoots.aspx" class="mm-link">3.3 Complex Roots</a></li> <li><a href="/Classes/DE/RepeatedRoots.aspx" class="mm-link">3.4 Repeated Roots</a></li> <li><a href="/Classes/DE/ReductionofOrder.aspx" class="mm-link">3.5 Reduction of Order</a></li> <li><a href="/Classes/DE/FundamentalSetsofSolutions.aspx" class="mm-link">3.6 Fundamental Sets of Solutions</a></li> <li><a href="/Classes/DE/Wronskian.aspx" class="mm-link">3.7 More on the Wronskian</a></li> <li><a href="/Classes/DE/NonhomogeneousDE.aspx" class="mm-link">3.8 Nonhomogeneous Differential Equations</a></li> <li><a href="/Classes/DE/UndeterminedCoefficients.aspx" class="mm-link">3.9 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/VariationofParameters.aspx" class="mm-link">3.10 Variation of Parameters</a></li> <li><a href="/Classes/DE/Vibrations.aspx" class="mm-link">3.11 Mechanical Vibrations</a></li> </ul> </li> <li><a href="/Classes/DE/LaplaceIntro.aspx" class="mm-link">4. Laplace Transforms</a> <ul> <li><a href="/Classes/DE/LaplaceDefinition.aspx" class="mm-link">4.1 The Definition</a></li> <li><a href="/Classes/DE/LaplaceTransforms.aspx" class="mm-link">4.2 Laplace Transforms</a></li> <li><a href="/Classes/DE/InverseTransforms.aspx" class="mm-link">4.3 Inverse Laplace Transforms</a></li> <li><a href="/Classes/DE/StepFunctions.aspx" class="mm-link">4.4 Step Functions</a></li> <li><a href="/Classes/DE/IVPWithLaplace.aspx" class="mm-link">4.5 Solving IVP's with Laplace Transforms</a></li> <li><a href="/Classes/DE/IVPWithNonConstantCoefficient.aspx" class="mm-link">4.6 Nonconstant Coefficient IVP's</a></li> <li><a href="/Classes/DE/IVPWithStepFunction.aspx" class="mm-link">4.7 IVP's With Step Functions</a></li> <li><a href="/Classes/DE/DiracDeltaFunction.aspx" class="mm-link">4.8 Dirac Delta Function</a></li> <li><a href="/Classes/DE/ConvolutionIntegrals.aspx" class="mm-link">4.9 Convolution Integrals</a></li> <li><a href="/Classes/DE/Laplace_Table.aspx" class="mm-link">4.10 Table Of Laplace Transforms</a></li> </ul> </li> <li><a href="/Classes/DE/SystemsIntro.aspx" class="mm-link">5. Systems of DE's</a> <ul> <li><a href="/Classes/DE/LA_Systems.aspx" class="mm-link">5.1 Review : Systems of Equations</a></li> <li><a href="/Classes/DE/LA_Matrix.aspx" class="mm-link">5.2 Review : Matrices & Vectors</a></li> <li><a href="/Classes/DE/LA_Eigen.aspx" class="mm-link">5.3 Review : Eigenvalues & Eigenvectors</a></li> <li><a href="/Classes/DE/SystemsDE.aspx" class="mm-link">5.4 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/SolutionsToSystems.aspx" class="mm-link">5.5 Solutions to Systems</a></li> <li><a href="/Classes/DE/PhasePlane.aspx" class="mm-link">5.6 Phase Plane</a></li> <li><a href="/Classes/DE/RealEigenvalues.aspx" class="mm-link">5.7 Real Eigenvalues</a></li> <li><a href="/Classes/DE/ComplexEigenvalues.aspx" class="mm-link">5.8 Complex Eigenvalues</a></li> <li><a href="/Classes/DE/RepeatedEigenvalues.aspx" class="mm-link">5.9 Repeated Eigenvalues</a></li> <li><a href="/Classes/DE/NonhomogeneousSystems.aspx" class="mm-link">5.10 Nonhomogeneous Systems</a></li> <li><a href="/Classes/DE/SystemsLaplace.aspx" class="mm-link">5.11 Laplace Transforms</a></li> <li><a href="/Classes/DE/SystemsModeling.aspx" class="mm-link">5.12 Modeling</a></li> </ul> </li> <li><a href="/Classes/DE/SeriesIntro.aspx" class="mm-link">6. Series Solutions to DE's</a> <ul> <li><a href="/Classes/DE/PowerSeries.aspx" class="mm-link">6.1 Review : Power Series</a></li> <li><a href="/Classes/DE/TaylorSeries.aspx" class="mm-link">6.2 Review : Taylor Series</a></li> <li><a href="/Classes/DE/SeriesSolutions.aspx" class="mm-link">6.3 Series Solutions</a></li> <li><a href="/Classes/DE/EulerEquations.aspx" class="mm-link">6.4 Euler Equations</a></li> </ul> </li> <li><a href="/Classes/DE/IntroHigherOrder.aspx" class="mm-link">7. Higher Order Differential Equations</a> <ul> <li><a href="/Classes/DE/HOBasicConcepts.aspx" class="mm-link">7.1 Basic Concepts for <em>n</em><sup>th</sup> Order Linear Equations</a></li> <li><a href="/Classes/DE/HOHomogeneousDE.aspx" class="mm-link">7.2 Linear Homogeneous Differential Equations</a></li> <li><a href="/Classes/DE/HOUndeterminedCoeff.aspx" class="mm-link">7.3 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/HOVariationOfParam.aspx" class="mm-link">7.4 Variation of Parameters</a></li> <li><a href="/Classes/DE/HOLaplaceTransforms.aspx" class="mm-link">7.5 Laplace Transforms</a></li> <li><a href="/Classes/DE/HOSystems.aspx" class="mm-link">7.6 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/HOSeries.aspx" class="mm-link">7.7 Series Solutions</a></li> </ul> </li> <li><a href="/Classes/DE/IntroBVP.aspx" class="mm-link">8. Boundary Value Problems & Fourier Series</a> <ul> <li><a href="/Classes/DE/BoundaryValueProblem.aspx" class="mm-link">8.1 Boundary Value Problems</a></li> <li><a href="/Classes/DE/BVPEvals.aspx" class="mm-link">8.2 Eigenvalues and Eigenfunctions</a></li> <li><a href="/Classes/DE/PeriodicOrthogonal.aspx" class="mm-link">8.3 Periodic Functions & Orthogonal Functions</a></li> <li><a href="/Classes/DE/FourierSineSeries.aspx" class="mm-link">8.4 Fourier Sine Series</a></li> <li><a href="/Classes/DE/FourierCosineSeries.aspx" class="mm-link">8.5 Fourier Cosine Series</a></li> <li><a href="/Classes/DE/FourierSeries.aspx" class="mm-link">8.6 Fourier Series</a></li> <li><a href="/Classes/DE/ConvergenceFourierSeries.aspx" class="mm-link">8.7 Convergence of Fourier Series</a></li> </ul> </li> <li><a href="/Classes/DE/IntroPDE.aspx" class="mm-link">9. Partial Differential Equations </a> <ul> <li><a href="/Classes/DE/TheHeatEquation.aspx" class="mm-link">9.1 The Heat Equation</a></li> <li><a href="/Classes/DE/TheWaveEquation.aspx" class="mm-link">9.2 The Wave Equation</a></li> <li><a href="/Classes/DE/PDETerminology.aspx" class="mm-link">9.3 Terminology</a></li> <li><a href="/Classes/DE/SeparationofVariables.aspx" class="mm-link">9.4 Separation of Variables</a></li> <li><a href="/Classes/DE/SolvingHeatEquation.aspx" class="mm-link">9.5 Solving the Heat Equation</a></li> <li><a href="/Classes/DE/HeatEqnNonZero.aspx" class="mm-link">9.6 Heat Equation with Non-Zero Temperature Boundaries</a></li> <li><a href="/Classes/DE/LaplacesEqn.aspx" class="mm-link">9.7 Laplace's Equation</a></li> <li><a href="/Classes/DE/VibratingString.aspx" class="mm-link">9.8 Vibrating String</a></li> <li><a href="/Classes/DE/PDESummary.aspx" class="mm-link">9.9 Summary of Separation of Variables</a></li> </ul> </li> </ul> </li> <li><span>Extras</span></li> <li><a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" class="mm-link">Algebra & Trig Review</a> <ul> <li><a href="/Extras/AlgebraTrigReview/AlgebraIntro.aspx" class="mm-link">1. Algebra</a> <ul> <li><a href="/Extras/AlgebraTrigReview/Exponents.aspx" class="mm-link">1.1 Exponents </a></li> <li><a href="/Extras/AlgebraTrigReview/AbsoluteValue.aspx" class="mm-link">1.2 Absolute Value</a></li> <li><a href="/Extras/AlgebraTrigReview/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Extras/AlgebraTrigReview/Rationalizing.aspx" class="mm-link">1.4 Rationalizing </a></li> <li><a href="/Extras/AlgebraTrigReview/Functions.aspx" class="mm-link">1.5 Functions </a></li> <li><a href="/Extras/AlgebraTrigReview/MultPoly.aspx" class="mm-link">1.6 Multiplying Polynomials</a></li> <li><a href="/Extras/AlgebraTrigReview/Factoring.aspx" class="mm-link">1.7 Factoring</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpRatExp.aspx" class="mm-link">1.8 Simplifying Rational Expressions</a></li> <li><a href="/Extras/AlgebraTrigReview/Graphing.aspx" class="mm-link">1.9 Graphing and Common Graphs</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtI.aspx" class="mm-link">1.10 Solving Equations, Part I</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtII.aspx" class="mm-link">1.11 Solving Equations, Part II</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveSystems.aspx" class="mm-link">1.12 Solving Systems of Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveIneq.aspx" class="mm-link">1.13 Solving Inequalities</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveAbsValue.aspx" class="mm-link">1.14 Absolute Value Equations and Inequalities</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/TrigIntro.aspx" class="mm-link">2. Trigonometry</a> <ul> <li><a href="/Extras/AlgebraTrigReview/TrigFunctions.aspx" class="mm-link">2.1 Trig Function Evaluation</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigGraphs.aspx" class="mm-link">2.2 Graphs of Trig Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigFormulas.aspx" class="mm-link">2.3 Trig Formulas</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveTrigEqn.aspx" class="mm-link">2.4 Solving Trig Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/InverseTrig.aspx" class="mm-link">2.5 Inverse Trig Functions</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/ExpLogIntro.aspx" class="mm-link">3. Exponentials & Logarithms</a> <ul> <li><a href="/Extras/AlgebraTrigReview/ExponentialFcns.aspx" class="mm-link">3.1 Basic Exponential Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogarithmFcns.aspx" class="mm-link">3.2 Basic Logarithm Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogProperties.aspx" class="mm-link">3.3 Logarithm Properties</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpLogs.aspx" class="mm-link">3.4 Simplifying Logarithms</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveExpEqn.aspx" class="mm-link">3.5 Solving Exponential Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveLogEqn.aspx" class="mm-link">3.6 Solving Logarithm Equations</a></li> </ul> </li> </ul> </li> <li><a href="/Extras/CommonErrors/CommonMathErrors.aspx" class="mm-link">Common Math Errors</a> <ul> <li><a href="/Extras/CommonErrors/GeneralErrors.aspx" class="mm-link">1. General Errors</a> </li> <li><a href="/Extras/CommonErrors/AlgebraErrors.aspx" class="mm-link">2. Algebra Errors</a> </li> <li><a href="/Extras/CommonErrors/TrigErrors.aspx" class="mm-link">3. Trig Errors</a> </li> <li><a href="/Extras/CommonErrors/CommonErrors.aspx" class="mm-link">4. Common Errors</a> </li> <li><a href="/Extras/CommonErrors/CalculusErrors.aspx" class="mm-link">5. Calculus Errors</a> </li> </ul> </li> <li><a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" class="mm-link">Complex Number Primer</a> <ul> <li><a href="/Extras/ComplexPrimer/Definition.aspx" class="mm-link">1. The Definition</a> </li> <li><a href="/Extras/ComplexPrimer/Arithmetic.aspx" class="mm-link">2. Arithmetic</a> </li> <li><a href="/Extras/ComplexPrimer/ConjugateModulus.aspx" class="mm-link">3. Conjugate and Modulus</a> </li> <li><a href="/Extras/ComplexPrimer/Forms.aspx" class="mm-link">4. Polar and Exponential Forms</a> </li> <li><a href="/Extras/ComplexPrimer/Roots.aspx" class="mm-link">5. Powers and Roots</a> </li> </ul> </li> <li><a href="/Extras/StudyMath/HowToStudyMath.aspx" class="mm-link">How To Study Math</a> <ul> <li><a href="/Extras/StudyMath/GeneralTips.aspx" class="mm-link">1. General Tips</a> </li> <li><a href="/Extras/StudyMath/TakingNotes.aspx" class="mm-link">2. Taking Notes</a> </li> <li><a href="/Extras/StudyMath/GettingHelp.aspx" class="mm-link">3. Getting Help</a> </li> <li><a href="/Extras/StudyMath/Homework.aspx" class="mm-link">4. Doing Homework</a> </li> <li><a href="/Extras/StudyMath/ProblemSolving.aspx" class="mm-link">5. Problem Solving</a> </li> <li><a href="/Extras/StudyMath/StudyForExam.aspx" class="mm-link">6. Studying For an Exam</a> </li> <li><a href="/Extras/StudyMath/TakingExam.aspx" class="mm-link">7. Taking an Exam</a> </li> <li><a href="/Extras/StudyMath/Errors.aspx" class="mm-link">8. Learn From Your Errors</a> </li> </ul> </li> <li><span>Misc Links</span></li> <li><a href="/contact.aspx" class="mm-link">Contact Me</a></li> <li><a href="/links.aspx" class="mm-link">Links</a></li> <li><a href="/mathjax.aspx" class="mm-link">MathJax Help and Configuration</a></li> <li><a href="/privacy.aspx" class="mm-link">Privacy Statement</a></li> <li><a href="/help.aspx" class="mm-link">Site Help & FAQ</a></li> <li><a href="/terms.aspx" class="mm-link">Terms of Use</a></li> </ul> </nav> <div class="top-info-bar"> <span id="mobile-title" class="mobile-header-title header-title">Paul's Online Notes</span> <br /> <span class="top-menu-breadcrumb"> <a href="/" id="breadcrumb_home_link" title="Go to Main Page">Home</a> <span id="breadcrumb_h_b_sep_span">/ </span> <a href="/Classes/CalcI/CalcI.aspx" id="breadcrumb_book_link" title="Go to Book Introduction">Calculus I</a> <span id="breadcrumb_b_c_sep_span">/ </span> <a href="/Classes/CalcI/ReviewIntro.aspx" id="breadcrumb_chapter_link" title="Go to Chapter Introduction">Review</a> <span id="breadcrumb_section_span"> / Solving Trig Equations</span> </span> </div> <div id="nav_links" class="top-nav-bar"> <a href="/Classes/CalcI/TrigFcns.aspx" id="nav_links_prev_section" title="Goto Previous Section : Trig Functions"><span class="top-menu-prev fas fa-chevron-left"></span><span class="nav_desktop_extra_pn"> Prev. Section</span></a> <div class="top-nav-bar-link-spacer"></div> <span id="nav_current_notes">Notes</span> <a href="/Problems/CalcI/TrigEquations.aspx" id="nav_links_practice" title="Go to Practice Problems for current topic.">Practice<span class="nav_desktop_extra"> Problems</span></a> <a href="/ProblemsNS/CalcI/TrigEquations.aspx" id="nav_links_asgn" title="Go to Assignment Problems for current topic.">Assignment<span class="nav_desktop_extra"> Problems</span></a> <div class="top-nav-bar-link-spacer"></div> <a href="/Classes/CalcI/TrigEquations_CalcI.aspx" id="nav_links_next_section" title="Goto Next Section : Trig Equations with Calculators, Part I"><span class="nav_desktop_extra_pn"> Next Section </span><span class="top-menu-next fas fa-chevron-right"></span></a> </div> <div class="header-divider"></div> <div class="content"> <span id="SHLink_NoteMobile" class="SH-Link content-note-link-mobile">Show Mobile Notice</span> <span id="SHImg_NoteMobile" class="fas fa-caret-right SH-padding content-note-link-mobile" aria-hidden="true"></span> <span id="SHALink_S_Note" class="SH-Link SH-Hide SH-Bracket">Show All Notes</span> <span id="SHALink_H_Note" class="SH-Link SH-Hide SH-Bracket">Hide All Notes</span> <div id="SHObj_NoteMobile" class="content-note-container content-note-container-mobile"> <div class="content-note-header">Mobile Notice</div> <div class="content-note">You appear to be on a device with a "narrow" screen width (<em>i.e.</em> you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.</div> </div> <form method="post" action="./TrigEquations.aspx" id="ctl00"> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATE" id="__VIEWSTATE" value="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" /> </div> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATEGENERATOR" id="__VIEWSTATEGENERATOR" value="E937760B" /> </div> </form> <h3>Section 1.4 : Solving Trig Equations</h3> <p>In this section we will take a look at solving trig equations. This is something that you will be asked to do on a fairly regular basis in many classes.</p> <p>Let’s just jump into the examples and see how to solve trig equations.</p> <a class="anchor" name="Rev_TrigEqn_Ex1"></a> <div class="example"> <span class="example-title">Example 1</span> Solve $2\cos \left( t \right) = \sqrt 3 $. <div class="example-content"> <span id="SHLink_Soln1" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln1" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln1" class="soln-content"> <p>There’s really not a whole lot to do in solving this kind of trig equation. We first need to get the trig function on one side by itself. To do this all we need to do is divide both sides by 2.</p> \[\begin{align*}2\cos \left( t \right) & = \sqrt 3 \\ \cos \left( t \right) & = \frac{{\sqrt 3 }}{2}\end{align*}\] <p>We are looking for all the values of $t$ for which cosine will have the value of $\frac{{\sqrt 3 }}{2}$. So, let’s take a look at the following unit circle.</p> <div class="center-div"><img alt="A unit circle with lines representing the angles $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$ shown. The $\frac{\pi }{6}$ line intersects the unit circle at $\left( \frac{\sqrt{3}}{2},\frac{1}{2} \right)$. The $\frac{\pi }{2}$ line intersects the unit circle at $\left( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2} \right)$. The $\frac{\pi }{3}$ line intersects the unit circle at $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$. Also shown is a line which intersects the unit circle at $\left( \frac{\sqrt{3}}{2},-\frac{1}{2} \right)$. " height="401" src="TrigEquations_Files/image002.png" width="401" /></div> <p>From quick inspection we can see that $t = \frac{\pi }{6}$ is a solution. However, as we have shown on the unit circle there is another angle which will also be a solution. We need to determine what this angle is. When we look for these angles we typically want <em>positive</em> angles that lie between 0 and $2\pi $. This angle will not be the only possibility of course, but we typically look for angles that meet these conditions.</p> <p>To find this angle for this problem all we need to do is use a little geometry. The angle in the first quadrant makes an angle of $\frac{\pi }{6}$ with the positive $x$-axis, then so must the angle in the fourth quadrant. So, we have two options. We could use $ - \frac{\pi }{6}$, but again, it’s more common to use positive angles. To get a positive angle all we need to do is use the fact that the angle is $\frac{\pi }{6}$ with the positive $x$-axis (as noted above) and a positive angle will be $t = 2\pi - \frac{\pi }{6} = \frac{{11\pi }}{6}$.</p> <p>One way to remember how to get the positive form of the second angle is to think of making one full revolution from the positive $x$-axis (<em>i.e.</em> $2\pi $) and then backing off (<em>i.e.</em> subtracting) $\frac{\pi }{6}$.</p> <p>We aren’t done with this problem. As the discussion about finding the second angle has shown there are many ways to write any given angle on the unit circle. Sometimes it will be $ - \frac{\pi }{6}$ that we want for the solution and sometimes we will want both (or neither) of the listed angles. Therefore, since there isn’t anything in this problem (contrast this with the next problem) to tell us which is the correct solution we will need to list ALL possible solutions.</p> <p>This is very easy to do. Recall from the previous <a href="TrigFcns.aspx#All_Angles">section</a> and you’ll see there that we used</p> \[\frac{\pi }{6} + 2\pi \,n\,,\;\;n = 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots \] <p>to represent all the possible angles that can end at the same location on the unit circle, <em>i.e.</em> angles that end at $\frac{\pi }{6}$. Remember that all this says is that we start at $\frac{\pi }{6}$ then rotate around in the counter-clockwise direction ($n$ is positive) or clockwise direction ($n$ is negative) for $n$ complete rotations. The same thing can be done for the second solution.</p> <p>So, all together the complete solution to this problem is</p> \[\begin{gather*}\frac{\pi }{6} + 2\pi \,n\,,\;\;n = 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots \\ \frac{{11\pi }}{6} + 2\pi \,n\,,\;\;n = 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots \end{gather*}\] <p>As a final thought, notice that we can get $ - \frac{\pi }{6}$ by using $n = - 1$ in the second solution.</p> </div> </div> </div> <p>Now, in a calculus class this is not a typical trig equation that we’ll be asked to solve. A more typical example is the next one.</p> <a class="anchor" name="Rev_TrigEqn_Ex2"></a> <div class="example"> <span class="example-title">Example 2</span> Solve $2\cos \left( t \right) = \sqrt 3 $ on $[ - 2\pi ,2\pi ]$. <div class="example-content"> <span id="SHLink_Soln2" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2" class="soln-content"> <p>In a calculus class we are often more interested in only the solutions to a trig equation that fall in a certain interval. The first step in this kind of problem is to find all possible solutions. We did this in the previous example.</p> \[\begin{gather*}\frac{\pi }{6} + 2\pi \,n\,,\;\;n = 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots \\ \frac{{11\pi }}{6} + 2\pi \,n\,,\;\;n = 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \ldots \end{gather*}\] <p>Now, to find the solutions in the interval all we need to do is start picking values of $n$, plugging them in and getting the solutions that will fall into the interval that we’ve been given.</p> $n=0$. \[\begin{align*}& \frac{\pi }{6} + 2\pi \,\left( 0 \right)\, = \frac{\pi }{6} < 2\pi \\ &\frac{{11\pi }}{6} + 2\pi \,\left( 0 \right) = \frac{{11\pi }}{6} < 2\pi \end{align*}\] <p>Now, notice that if we take any positive value of $n$ we will be adding on positive multiples of $2\pi $ onto a positive quantity and this will take us past the upper bound of our interval so we don’t need to take any positive value of $n$.</p> <p>However, just because we aren’t going to take any positive value of $n$ doesn’t mean that we shouldn’t also look at negative values of $n$.</p> $n=-1$. \[\begin{align*}& \frac{\pi }{6} + 2\pi \,\left( { - 1} \right)\, = - \frac{{11\pi }}{6} > - 2\pi \\ & \frac{{11\pi }}{6} + 2\pi \,\left( { - 1} \right) = - \frac{\pi }{6} > - 2\pi \end{align*}\] <p>These are both greater than $ - 2\pi $and so are solutions, but if we subtract another $2\pi $ off (<em>i.e</em> use $n = - 2$) we will once again be outside of the interval so we’ve found all the possible solutions that lie inside the interval $[ - 2\pi ,2\pi ]$.</p> <p>So, the solutions are : $\frac{\pi }{6},\;\frac{{11\pi }}{6},\; - \frac{\pi }{6},\; - \frac{{11\pi }}{6}$.</p> </div> </div> </div> <p>So, let’s see if you’ve got all this down.</p> <a class="anchor" name="Rev_TrigEqn_Ex3"></a> <div class="example"> <span class="example-title">Example 3</span> Solve $2\sin \left( {5x} \right) = - \sqrt 3 $ on $[ - \pi ,2\pi ]$. <div class="example-content"> <span id="SHLink_Soln3" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln3" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln3" class="soln-content"> <p>This problem is very similar to the other problems in this section with a very important difference. We’ll start this problem in exactly the same way as we did in the first example. So, first get the sine on one side by itself.</p> \[\begin{align*}2\sin (5x) & = - \sqrt 3 \\ \sin (5x) & = \frac{{ - \sqrt 3 }}{2}\end{align*}\] <p>We are looking for angles that will give $ - \frac{{\sqrt 3 }}{2}$ out of the sine function. Let’s again go to our trusty unit circle.</p> <div class="center-div"><img alt="A unit circle with lines representing the angles $\frac{\pi }{6}$, $\frac{\pi }{4}$ and $\frac{\pi }{3}$ shown. The $\frac{\pi }{6}$ line intersects the unit circle at $\left( \frac{\sqrt{3}}{2},\frac{1}{2} \right)$. The $\frac{\pi }{2}$ line intersects the unit circle at $\left( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2} \right)$. The $\frac{\pi }{3}$ line intersects the unit circle at $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$. Also shown are lines representing which intersect the unit circle at $\left( -\frac{1}{2},-\frac{\sqrt{3}}{2} \right)$ and at $\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$." height="403" src="TrigEquations_Files/image004.png" width="403" /></div> <p>Now, there are no angles in the first quadrant for which sine has a value of $ - \frac{{\sqrt 3 }}{2}$. However, there are two angles in the lower half of the unit circle for which sine will have a value of $ - \frac{{\sqrt 3 }}{2}$. So, what are these angles?</p> <p>Notice that $\sin \left( {\frac{\pi }{3}} \right) = \frac{{\sqrt 3 }}{2}$. Given this we now know that the angle in the third quadrant will be $\frac{\pi }{3}$ below the <strong><em>negative</em></strong> $x$-axis or $\pi + \frac{\pi }{3} = \frac{{4\pi }}{3}$. An easy way to remember this is to notice that we’ll rotate half a revolution from the positive $x$-axis to get to the negative $x$-axis then add on $\frac{\pi }{3}$ to reach the angle we are looking for.</p> <p>Likewise, the angle in the fourth quadrant will $\frac{\pi }{3}$ below the <strong><em>positive</em></strong> $x$-axis. So, we could use $ - \frac{\pi }{3}$ or $2\pi - \frac{\pi }{3} = \frac{{5\pi }}{3}$. Remember that we’re typically looking for positive angles between 0 and $2\pi $ so we’ll use the positive angle. An easy way to remember how to the positive angle here is to rotate one full revolution from the positive $x$-axis (<em>i.e.</em> $2\pi $) and then backing off (<em>i.e.</em> subtracting) $\frac{\pi }{3}$.</p> <p>Now we come to the very important difference between this problem and the previous problems in this section. The solution is <strong>NOT</strong></p> \[\begin{align*}x & = \frac{{4\pi }}{3} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \\ x & = \frac{{5\pi }}{3} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \end{align*}\] <p>This is not the set of solutions because we are NOT looking for values of $x$ for which $\sin \left( x \right) = - \frac{{\sqrt 3 }}{2}$, but instead we are looking for values of $x$ for which $\sin \left( {5x} \right) = - \frac{{\sqrt 3 }}{2}$. Note the difference in the arguments of the sine function! One is $x$ and the other is $5x$. This makes all the difference in the world in finding the solution! Therefore, the set of solutions is</p> \[\begin{align*}5x & = \frac{{4\pi }}{3} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \\ 5x & = \frac{{5\pi }}{3} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \end{align*}\] <p>Well, actually, that’s not quite the solution. We are looking for values of $x$ so divide everything by 5 to get.</p> \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi n}}{5},\quad n = 0, \pm 1, \pm 2, \ldots \\ x & = \frac{\pi }{3} + \frac{{2\pi n}}{5},\quad n = 0, \pm 1, \pm 2, \ldots \end{align*}\] <p>Notice that we also divided the $2\pi n$by 5 as well! This is important! If we don’t do that you <strong>WILL</strong> miss solutions. For instance, take $n = 1$.</p> \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi }}{5} = \frac{{10\pi }}{{15}} = \frac{{2\pi }}{3} & \hspace{0.25in} & \Rightarrow \hspace{0.5in}\sin \left( {5\left( {\frac{{2\pi }}{3}} \right)} \right) = \sin \left( {\frac{{10\pi }}{3}} \right) = - \frac{{\sqrt 3 }}{2}\\ x & = \frac{\pi }{3} + \frac{{2\pi }}{5} = \frac{{11\pi }}{{15}} & \hspace{0.25in} & \Rightarrow \hspace{0.5in}\sin \left( {5\left( {\frac{{11\pi }}{{15}}} \right)} \right) = \sin \left( {\frac{{11\pi }}{3}} \right) = - \frac{{\sqrt 3 }}{2}\end{align*}\] <p>We’ll leave it to you to verify our work showing they are solutions. However, it makes the point. If you didn’t divide the $2\pi n$ by 5 you would have missed these solutions!</p> <p>Okay, now that we’ve gotten all possible solutions it’s time to find the solutions on the given interval. We’ll do this as we did in the previous problem. Pick values of $n$ and get the solutions.</p> $n = 0$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 0 \right)}}{5} = \frac{{4\pi }}{{15}} < 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 0 \right)}}{5} = \frac{\pi }{3} < 2\pi \end{align*}\] $n = 1$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 1 \right)}}{5} = \frac{{2\pi }}{3} < 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 1 \right)}}{5} = \frac{{11\pi }}{{15}} < 2\pi \end{align*}\] $n = 2$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 2 \right)}}{5} = \frac{{16\pi }}{{15}} < 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 2 \right)}}{5} = \frac{{17\pi }}{{15}} < 2\pi \end{align*}\] $n = 3$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 3 \right)}}{5} = \frac{{22\pi }}{{15}} < 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 3 \right)}}{5} = \frac{{23\pi }}{{15}} < 2\pi \end{align*}\] $n = 4$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 4 \right)}}{5} = \frac{{28\pi }}{{15}} < 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 4 \right)}}{5} = \frac{{29\pi }}{{15}} < 2\pi \end{align*}\] $n = 5$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( 5 \right)}}{5} = \frac{{34\pi }}{{15}} > 2\pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( 5 \right)}}{5} = \frac{{35\pi }}{{15}} > 2\pi \end{align*}\] <p>Okay, so we finally got past the right endpoint of our interval so we don’t need any more positive <i style="mso-bidi-font-style: normal">n</i>. Now let’s take a look at the negative $n$ and see what we’ve got.</p> $n = –1 $. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( { - 1} \right)}}{5} = - \frac{{2\pi }}{{15}} > - \pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( { - 1} \right)}}{5} = - \frac{\pi }{{15}} > - \pi \end{align*}\] $n = –2$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( { - 2} \right)}}{5} = - \frac{{8\pi }}{{15}} > - \pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( { - 2} \right)}}{5} = - \frac{{7\pi }}{{15}} > - \pi \end{align*}\] $n = –3$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( { - 3} \right)}}{5} = - \frac{{14\pi }}{{15}} > - \pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( { - 3} \right)}}{5} = - \frac{{13\pi }}{{15}} > - \pi \end{align*}\] $n = –4$. \[\begin{align*}x & = \frac{{4\pi }}{{15}} + \frac{{2\pi \left( { - 4} \right)}}{5} = - \frac{{4\pi }}{3} < - \pi \\ x & = \frac{\pi }{3} + \frac{{2\pi \left( { - 4} \right)}}{5} = - \frac{{19\pi }}{{15}} < - \pi \end{align*}\] <p>And we’re now past the left endpoint of the interval. Sometimes, there will be many solutions as there were in this example. Putting all of this together gives the following set of solutions that lie in the given interval.</p> \[\begin{align*} & \frac{{4\pi }}{{15}},\frac{\pi }{3},\frac{{2\pi }}{3},\frac{{11\pi }}{{15}},\frac{{16\pi }}{{15}},\frac{{17\pi }}{{15}},\frac{{22\pi }}{{15}},\frac{{23\pi }}{{15}},\frac{{28\pi }}{{15}},\frac{{29\pi }}{{15}}\\ & - \frac{\pi }{{15}}, - \frac{{2\pi }}{{15}}, - \frac{{7\pi }}{{15}}, - \frac{{8\pi }}{{15}}, - \frac{{13\pi }}{{15}}, - \frac{{14\pi }}{{15}}\end{align*}\] </div> </div> </div> <p>Let's work another example.</p> <a class="anchor" name="Rev_TrigEqn_Ex4"></a> <div class="example"> <span class="example-title">Example 4</span> Solve $\sin \left( {2x} \right) = - \cos \left( {2x} \right)$ on $\displaystyle \left[ { - \frac{{3\pi }}{2},\frac{{3\pi }}{2}} \right]$. <div class="example-content"> <span id="SHLink_Soln4" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln4" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln4" class="soln-content"> <p>This problem is a little different from the previous ones. First, we need to do some rearranging and simplification.</p> \[\begin{align*}\sin (2x) & = - \cos (2x)\\ \frac{{\sin (2x)}}{{\cos (2x)}} & = - 1\\ \tan \left( {2x} \right) & = - 1\end{align*}\] <p>So, solving $\sin (2x) = - \cos (2x)$ is the same as solving $\tan (2x) = - 1$. Hopefully, you’ll recall that the smallest positive angle where tangent it -1 is $\frac{{3\pi }}{4}$ and this angle is in the 2<sup>nd</sup> quadrant.</p> <p>There is also a second angle for which tangent will be -1 and we can use the unit circle to illustrate this second angle. Let’s take a look at the following unit circle.</p> <div class="center-div"><img border="0" height="414" src="TrigEquations_Files/image005.png" width="414" /></div> <p>As shown in this unit circle if we add $\pi $ to our first angle we get $\frac{{3\pi }}{4} + \pi = \frac{{7\pi }}{4}$ and we get an angle that is in the fourth quadrant and has the same coordinates except for opposite signs. This means that tangent will also have a value of -1 here and so is a second angle.</p> <p>This will always be true when solving tangent equations. Once we have one angle that will solve the equation a second angle will always be $\pi $ plus the first angle.</p> <p>All possible angles are then,</p> \[\begin{align*}2x & = \frac{{3\pi }}{4} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \\ 2x & = \frac{{7\pi }}{4} + 2\pi n,\quad n = 0, \pm 1, \pm 2, \ldots \end{align*}\] <p>Or, upon dividing by the 2 we get all possible solutions.</p> \[\begin{align*}x & = \frac{{3\pi }}{8} + \pi n,\quad n = 0, \pm 1, \pm 2, \ldots \\ x & = \frac{{7\pi }}{8} + \pi n,\quad n = 0, \pm 1, \pm 2, \ldots \end{align*}\] <p>Now, let’s determine the solutions that lie in the given interval.</p> $n = 0$. \[\begin{align*}x & = \frac{{3\pi }}{8} + \pi \left( 0 \right) = \frac{{3\pi }}{8} < \frac{{3\pi }}{2}\\ x & = \frac{{7\pi }}{8} + \pi \left( 0 \right) = \frac{{7\pi }}{8} < \frac{{3\pi }}{2}\end{align*}\] $n = 1$. \[\begin{align*}x & = \frac{{3\pi }}{8} + \pi \left( 1 \right) = \frac{{11\pi }}{8} < \frac{{3\pi }}{2}\\ x & = \frac{{7\pi }}{8} + \pi \left( 1 \right) = \frac{{15\pi }}{8} > \frac{{3\pi }}{2}\end{align*}\] <p>Unlike the previous example only one of these will be in the interval. This will happen occasionally so don’t always expect both answers from a particular $n$ to work. Also, we should now check $n=2$ for the first to see if it will be in or out of the interval. I’ll leave it to you to check that it’s out of the interval.</p> <p>Now, let’s check the negative $n$.</p> $n = –1$. \[\begin{align*}x & = \frac{{3\pi }}{8} + \pi \left( { - 1} \right) = - \frac{{5\pi }}{8} > - \frac{{3\pi }}{2}\\ x & = \frac{{7\pi }}{8} + \pi \left( { - 1} \right) = - \frac{\pi }{8} > - \frac{{3\pi }}{2}\end{align*}\] $n = –2$. \[\begin{align*}x & = \frac{{3\pi }}{8} + \pi \left( { - 2} \right) = - \frac{{13\pi }}{8} < - \frac{{3\pi }}{2}\\ x & = \frac{{7\pi }}{8} + \pi \left( { - 2} \right) = - \frac{{9\pi }}{8} > - \frac{{3\pi }}{2}\end{align*}\] <p>Again, only one will work here. I’ll leave it to you to verify that $n = –3$ will give two answers that are both out of the interval.</p> <p>The complete list of solutions is then,</p> \[ - \frac{{9\pi }}{8}, - \frac{{5\pi }}{8}, - \frac{\pi }{8},\frac{{3\pi }}{8},\frac{{7\pi }}{8},\frac{{11\pi }}{8}\] </div> </div> </div> <p>Before moving on we need to address one issue about the previous example. The solution method used there is not the “standard” solution method. Because the second angle is just $\pi $ plus the first and if we added $\pi $ onto the second angle we’d be back at the line representing the first angle the more standard solution method is to just add $\pi n$ onto the first angle to get,</p> \[2x = \frac{{3\pi }}{4} + \pi n,\quad n = 0, \pm 1, \pm 2, \ldots \] <p>Then dividing by 2 to get the full set of solutions,</p> \[x = \frac{{3\pi }}{8} + \frac{{\pi n}}{2},\quad n = 0, \pm 1, \pm 2, \ldots \] <p>This set of solutions is identical to the set of solutions we got in the example (we’ll leave it to you to plug in some $n$’s and verify that). So, why did we not use the method in the previous example? Simple. The method in the previous example more closely mirrors the solution method for cosine and sine (<em>i.e.</em> they both, generally, give two sets of angles) and so for students that aren’t comfortable with solving trig equations this gives a “consistent” solution method.</p> <p>Let’s work one more example so that we can make a point that needs to be understood when solving some trig equations.</p> <a class="anchor" name="Rev_TrigEqn_Ex5"></a> <div class="example"> <span class="example-title">Example 5</span> Solve $\cos \left( {3x} \right) = 2$. <div class="example-content"> <span id="SHLink_Soln5" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln5" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln5" class="soln-content"> <p>This example is designed to remind you of certain properties about sine and cosine. Recall that $ - 1 \le \cos \left( \theta \right) \le 1$ and $ - 1 \le \sin \left( \theta \right) \le 1$. Therefore, since cosine will never be greater that 1 it definitely can’t be 2. So <strong>THERE ARE NO SOLUTIONS</strong> to this equation!</p> <p>It is important to remember that not all trig equations will have solutions.</p> </div> </div> </div> <p>In this section we solved some simple trig equations. There are more complicated trig equations that we can solve so don’t leave this section with the feeling that there is nothing harder out there in the world to solve. In fact, we’ll see at least one of the more complicated problems in the next section. Also, every one of these problems came down to solutions involving one of the “common” or “standard” angles. Most trig equations won’t come down to one of those and will in fact need a calculator to solve. The next section is devoted to this kind of problem.</p> </div>  <div class="footer"> <div class="footer-links"> [<a href="/Contact.aspx">Contact Me</a>] [<a href="/Privacy.aspx">Privacy Statement</a>] [<a href="/Help.aspx">Site Help & FAQ</a>] [<a href="/Terms.aspx">Terms of Use</a>] </div> <div class="footer-dates"> <div class="footer-copyright"><span id="lblCopyRight">© 2003 - 2024 Paul Dawkins</span></div> <div class="footer-spacer"></div> <div class="footer-modified-date">Page Last Modified : <span id="lblModified">11/16/2022</span></div> </div> </div> </div>  </body> </html>

CINXE.COM

Calculus I - Solving Trig Equations