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class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Unit Circle HL</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/trigonometry/unit-circle/main-1.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>The Unit Circle is probably <strong>the </strong>most important topic to understand from the whole of trigonometry. Lots of the properties of the trigonometric functions can be found from the unit circle. All the work on this page will help us understand all of these properties. This is essential knowledge, for example, for us to be able to solve trigonometric equations.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>the definition of <span class="math-tex">\(\sin \theta\)</span> and <span class="math-tex">\(\cos \theta\)</span> in terms of the unit circle <ul> <li>If we consider a unit circle (a circle of radius 1), then</li> </ul> </li> </ul> <p style="text-align: center;"><span class="math-tex">\(sin\theta=\frac{opposite}{hypotenuse}=\frac{opposite}{1}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(cos\theta=\frac{adjacent}{hypotenuse}=\frac{adjacent}{1}\)</span></p> <ul> <li>the definition of <span class="math-tex">\(\tan \theta\)</span> as <span class="math-tex">\(\frac{opposite}{adjacent}=\frac{\sin\theta}{\cos\theta}\)</span></li> <li>exact values of trigonometric ratios of 0°, 30°, 45°, 60°, 90°</li> <li>the symmetry properties of graphs of trigonometric functions</li> <li>the reciprocal trigonometric ratios <span class="math-tex">\(cosec \theta\)</span>, <span class="math-tex">\(sec \theta\)</span>, <span class="math-tex">\(cot\theta\)</span></li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <p>The following applets will help you understand the definition of sine and cosine in terms of the unit circle</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sine</p> </div> </div> <div class="panel-body"> <div> <p><strong>sine</strong> of an angle is given by the <strong>opposite side</strong>, or the <strong>y coordinate</strong> on the unit circle</p> <p style="text-align: center"><iframe frameborder="0" height="394px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/zntdfysq/width/734/height/394/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Unit circle - SINE" width="734px"></iframe></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Cosine</p> </div> </div> <div class="panel-body"> <div> <p><strong>cosine</strong> of an angle is given by the <strong>adjacent side</strong>, or the <strong>x coordinate</strong> on the unit circle</p> <p style="text-align: center"><iframe height="396px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/wy4nzw3x/width/737/height/396/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Unit circle - COSINE" width="737px"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/revision-notes_unit-circlehl.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/trigonometry/unit-circle/revision-notes_unit-circlehl.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#8f7f1470"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="8f7f1470"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Unit Circle HL <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-243-869" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Drag the function next to the correct graph</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">sinx</span> <span class="q-text-draggable draggable" draggable="true">cos(-x)</span> <span class="q-text-draggable draggable" draggable="true">-cosx</span> <span class="q-text-draggable draggable" draggable="true">tanx</span> <span class="q-text-draggable draggable" draggable="true">-tanx</span> <span class="q-text-draggable draggable" draggable="true">-sinx</span> <span class="q-text-draggable draggable" draggable="true">sin(-x)</span></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><img alt="" src="../../files/trigonometry/unit-circle/quiz/g1.jpg" style="width: 250px; height: 220px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="sinx"> <span class="review"></span></td><td><img alt="" src="../../files/trigonometry/unit-circle/quiz/g2.jpg" style="width: 250px; height: 220px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="cos(-x)"> <span class="review"></span></td></tr><tr><td><img alt="" src="../../files/trigonometry/unit-circle/quiz/g3.jpg" style="width: 250px; height: 220px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="-cosx"> <span class="review"></span></td><td><img alt="" src="../../files/trigonometry/unit-circle/quiz/g5.jpg" style="width: 250px; height: 220px;"></td><td>f(x) = <input type="text" style="height: auto;" data-c="-tanx"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>Note that cosx = cos(-x)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>This question is about remembering the exact values for trigonometric ratios. Try to do it without your calculator.</p><p>Enter the correct angle. Each answer is an angle between 0 and 90°.</p><p>There is no need to enter the ° symbol</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\frac { 1 }{ 2 } \)</span> = cos <input type="text" style="height: auto;" data-c="60"> <span class="review"></span></td><td><span class="math-tex">\(\frac { \sqrt { 3 } }{ 2 } \)</span> = sin <input type="text" style="height: auto;" data-c="60"> <span class="review"></span></td><td>2 = cosec <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></td></tr><tr><td><span class="math-tex">\(\frac { \sqrt { 2 } }{ 2 } \)</span>= sin <input type="text" style="height: auto;" data-c="45"> <span class="review"></span></td><td><span class="math-tex">\(\sqrt{3}\)</span>= tan <input type="text" style="height: auto;" data-c="60"> <span class="review"></span></td><td>1 = cot <input type="text" style="height: auto;" data-c="45"> <span class="review"></span></td></tr><tr><td><span class="math-tex">\(\sqrt{3}\)</span>= cot <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></td><td><span class="math-tex">\(\sqrt{2}\)</span> =sec <input type="text" style="height: auto;" data-c="45"> <span class="review"></span></td><td><span class="math-tex">\(\frac { \sqrt { 3 } }{ 2 } \)</span> = cos <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Drag the answers into the correct places</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">sin150°</span> <span class="q-text-draggable draggable" draggable="true">cos120°</span> <span class="q-text-draggable draggable" draggable="true">tan225°</span> <span class="q-text-draggable draggable" draggable="true">sin270°</span> <span class="q-text-draggable draggable" draggable="true">cos330°</span> <span class="q-text-draggable draggable" draggable="true">cos210°</span> </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(\frac{1}{2}\)</span> =<input type="text" style="height: auto;" data-c="sin150°"> <span class="review"></span></td><td>1 =<input type="text" style="height: auto;" data-c="tan225°"> <span class="review"></span></td><td><span class="math-tex">\(\frac { \sqrt { 3 } }{ 2 } \)</span> =<input type="text" style="height: auto;" data-c="cos330°"> <span class="review"></span></td></tr><tr><td>-1 =<input type="text" style="height: auto;" data-c="sin270°"> <span class="review"></span></td><td><span class="math-tex">\(-\frac { \sqrt { 3 } }{ 2 } \)</span> =<input type="text" style="height: auto;" data-c="cos210°"> <span class="review"></span></td><td><span class="math-tex">\(-\frac{1}{2}\)</span> = <input type="text" style="height: auto;" data-c="cos120°"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Match up the following answers with their correct pair</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">cos150°</span> <span class="q-text-draggable draggable" draggable="true">sin330°</span> <span class="q-text-draggable draggable" draggable="true">sin120°</span></p><p>-sin60° = <input type="text" style="height: auto;" data-c="cos150°"> <span class="review"></span></p><p>cos120° = <input type="text" style="height: auto;" data-c="sin330°"> <span class="review"></span></p><p>cos330° = <input type="text" style="height: auto;" data-c="sin120°"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Match up the following answers with their correct pair</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">sin300°</span> <span class="q-text-draggable draggable" draggable="true">tan(-60°)</span> <span class="q-text-draggable draggable" draggable="true">-tan150°</span></p><p>sin240° = <input type="text" style="height: auto;" data-c="sin300°"> <span class="review"></span></p><p>tan120° = <input type="text" style="height: auto;" data-c="tan(-60°)"> <span class="review"></span></p><p>tan210° = <input type="text" style="height: auto;" data-c="-tan150°"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Match up the following answers with their correct pair</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">cosec135°</span> <span class="q-text-draggable draggable" draggable="true">cot315°</span> <span class="q-text-draggable draggable" draggable="true">sec225°</span></p><p>-sec 135° = <input type="text" style="height: auto;" data-c="cosec135°"> <span class="review"></span> </p><p>cot(-45°) = <input type="text" style="height: auto;" data-c="cot315°"> <span class="review"></span></p><p>cosec225° = <input type="text" style="height: auto;" data-c="sec225°"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(tan\theta = \frac{1}{3}\)</span> and <span class="math-tex">\(\pi<\theta<\frac{3\pi}{2}\)</span> , then</p><span class="math-tex">\(sin\theta\)</span> = <span class="math-tex">\(\frac{a}{\sqrt{10}}\)</span><p><span class="math-tex">\(cos\theta\)</span> = <span class="math-tex">\(\frac{b}{\sqrt{10}}\)</span></p><p>What are the values of a and b?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/unit-circle/quiz/q8.jpg" style="width: 268px; height: 275px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(sin\theta = \frac{3}{4}\)</span> and <span class="math-tex">\(90°< \theta <180°\)</span> ,</p><p>find <span class="math-tex">\(cos\theta\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(-\frac{\sqrt{7}} {4}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{\sqrt{7}} {4}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{4}{5}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-\frac{4}{5}\)</span></span></label> </p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/unit-circle/quiz/q7.jpg" style="width: 273px; height: 284px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><div class="q-question"><div class="q-question"><p>Given that <span class="math-tex">\(cos\theta = \frac{3}{5}\)</span> and <span class="math-tex">\(\frac{3\pi}{2}<\theta<2\pi\)</span> , then</p><p><span class="math-tex">\(sin\theta = \frac{a}{5}\)</span></p><p><span class="math-tex">\(tan\theta = \frac{b}{3}\)</span></p><p>What are the values of a and b?</p></div></div></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/unit-circle/quiz/q9.jpg" style="width: 272px; height: 276px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(tan\theta = \frac{5}{12}\)</span> and <span class="math-tex">\(\pi<\theta<\frac{3\pi}{2}\)</span> , then</p><p><span class="math-tex">\(cosec\theta = \frac{a}{5}\)</span></p><p><span class="math-tex">\(sec\theta = \frac{b}{5}\)</span></p><p>What are the values of a, and b?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-13"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="-12"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/unit-circle/quiz/q10.jpg" style="width: 270px; height: 273px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="530"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>Given that <span class="math-tex">\(cosx=-\frac{\sqrt{7}}{3}\)</span> and <span class="math-tex">\(\frac{\pi}{2}\le x\le \pi\)</span> , find the possible values of sinx and cotx</p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Draw a circle.</p> <p><content><strong><em>x</em></strong><em> </em>is in the second quadrant</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/unit-circle/esq_unit_circle1hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/esq_unit_circle1hl.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="529"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>If <span class="math-tex">\(tanx=\frac{12}{5}\)</span> and <span class="math-tex">\(\pi\le x\le \frac{3\pi}{2}\)</span> , find the value of cosecx</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Draw a circle.</p> <p><content><strong><em>x</em></strong><em> </em>is in the third quadrant</content></p> </section> <h4><span class="fa fa-pencil" 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