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favorite" onclick="return false;"><i 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">Double Angle Formulae 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/trig-identities/main-1.png" style="float: left; width: 100px; height: 100px;"></p> <p>In this page, we will will learn about the Double Angle Formulae used in Trigonometry. It is actually quite rare that exam questions are solely about these identities, but it is essential that you can use and manipulate them confidently because they are used in so many different parts of the course (so they do come up a lot!). You will learn what they are and how to use them.</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"> <div> <p>On this page, you should learn about the double angle identities for sine and cosine</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large\sin2\theta \equiv 2\sin \theta \cos \theta \)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large{\cos2\theta \equiv \cos^2\theta -\sin^2\theta\\ \cos2\theta \equiv 2\cos^2\theta -1\\ \cos2\theta \equiv 1 -2\sin^2\theta}\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \tan2θ≡\frac{2\tanθ}{1-\tan^2θ}\)</span></p> </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 frameborder="0" height="480" scrolling="no" src="../../files/trigonometry/trig-identities/double-angle-sl/double-angle-formulae-hl.pdf" width="100%"></iframe></p> <p>Print from <a href="http://https://file-manager.studyib.net/mathsanalysis/files/trigonometry/trig-identities/double-angle-sl/double-angle-formulae-hl.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">This quiz is about the Double Angle formulae for sin2x and cos2x <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d217521c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d217521c"> <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%;"> Double Angle SL <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="11-372-2793" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Consider the trig identities, which of the following is true?</p><p>Select <strong><u>all</u></strong> the correct answers.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(cos2x \equiv cos²x - sin²x\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(sin2x \equiv cos²x - sin²x\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(cos2x \equiv 1 + 2 sin²x\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(cos2x \equiv 2cos²x – 1\)</span></label></p></div><div class="q-explanation"><p>The only 2 statements that are true come from the double angle formula for cos2x</p><p><label class="checkbox"><span class="math-tex">\(cos2x \equiv cos²x - sin²x\)</span></label></p><p><label class="checkbox"><span class="math-tex">\(cos2x \equiv 2cos²x – 1\)</span></label></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>Consider the trig identities, which of the following is true?</p><p>Select <strong><u>all</u></strong> the correct answers.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(sin2x \equiv 2sinxcosx\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(sin4x \equiv 2sin2xcos2x\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(cos2x \equiv 2sinxcosx\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(sinx \equiv 2sin2xcos2x\)</span></label></p></div><div class="q-explanation"><p>The only 2 statements that are true come from the double angle formula for sin2x</p><p style="margin-left: 40px;">sin2x <span class="math-tex">\(\equiv \)</span> 2sinxcosx</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Given that cosx = 0.4, find the exact value of cos 2x</p></div><div class="q-answer"><p>cos2x = <input type="text" style="height: auto;" data-c="-0.68"> <span class="review"></span></p></div><div class="q-explanation"><p>cos2x <span class="math-tex">\(\equiv \)</span> 2cos²x - 1</p><p>cos2x = 2x0.4² - 1</p><p>= 2x0.16 - 1</p><p>= 0.32 - 1</p><p>= - 0.68</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Given that sinx = <span class="math-tex">\(\frac{\sqrt{5}}{4}\)</span>, find the exact value of cos 2x</p></div><div class="q-answer"><p>cos2x = <input type="text" style="height: auto;" data-c="-0.375"> <span class="review"></span></p></div><div class="q-explanation"><p>cos2x <span class="math-tex">\(\equiv \)</span> 1 - 2sin²x</p><p>cos2x = 1 - 2 <span class="math-tex">\((\frac{\sqrt{5}}{4})^2\)</span></p><p>=1 - 2 <span class="math-tex">\(\frac{5}{16}\)</span></p><p>= 1 - <span class="math-tex">\(\frac{5}{8}\)</span></p><p><span class="math-tex">\(=-\frac{3}{8}=-0.375\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Given that cos2x = <span class="math-tex">\(-\frac{7}{8}\)</span> , find the exact value of cosx, given that 0 < x < <span class="math-tex">\(\frac{\pi}{2}\)</span></p></div><div class="q-answer"><p>cosx = <input type="text" style="height: auto;" data-c="0.25"> <span class="review"></span></p></div><div class="q-explanation"><p>cos2x <span class="math-tex">\(\equiv \)</span> 2cos²x - 1</p><p><span class="math-tex">\(-\frac{7}{8}\)</span> = 2cos²x - 1</p><p><span class="math-tex">\(\frac{1}{8}\)</span>= 2cos²x</p><p><span class="math-tex">\(\frac{1}{16}\)</span>= cos²x</p><p>cosx = <span class="math-tex">\(\frac{1}{4}\)</span> = 0.25</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 cos2x = a and a < 0, find the exact value of sinx</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\sqrt{\frac{a+1}{2}}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\sqrt{\frac{1-a}{2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\sqrt{\frac{a-1}{2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\sqrt{\frac{1-a}{2}}\)</span></label></p></div><div class="q-explanation"><p>If a < 0 , then cos2x < 0</p><p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/q7" style="width: 300px; height: 205px;"></p><p>Therefore 90° < 2x < 270°</p><p>and so 45° < x < 135°</p><p>Hence 0 < sinx < 1</p><p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/q7a1.png" style="width: 300px; height: 205px;"></p><p>cos2x <span class="math-tex">\(\equiv \)</span> 1 - 2sin²x</p><p>a = 1 - 2sin²x</p><p>2sin²x = 1 - a</p><p>sin²x = <span class="math-tex">\(\frac{1-a}{2}\)</span></p><p>Since sinx is positve, we take the positive root</p><p>sinx = <span class="math-tex">\(\sqrt{\frac{1-a}{2}}\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> If sin x = <span class="math-tex">\(\frac{12}{13}\)</span> and 0 x < <span class="math-tex">\(\frac{\pi}{2}\)</span></p><p>then sin2x = <span class="math-tex">\(\frac{a}{169}\)</span>. Find <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/q8" style="width: 300px; height: 205px;"></p><p>cosx = <span class="math-tex">\(\frac{5}{13}\)</span></p><p>sin2x = 2sinxcosx</p><p>= <span class="math-tex">\(2 \times \frac{12}{13}\times\frac{5}{13}\)</span></p><p>= <span class="math-tex">\(\frac{120}{169}\)</span></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 cosx = <span class="math-tex">\(\frac{a}{b}\)</span> , work out cos2x</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(1-\frac { 2a^2 }{ b^ 2 } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { 2a^2-1 }{ b^ 2 } \)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac { 2a^2-b^2 }{ b^ 2 } \)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac { a^2-b^2 }{ b^ 2 } \)</span></label></p></div><div class="q-explanation"><p>cos2x <span class="math-tex">\(\equiv \)</span> 2cos²x - 1</p><p>cos2x = <span class="math-tex">\(2(\frac{a}{b})^2-1\)</span></p><p>= <span class="math-tex">\(\frac{2a^2}{b^2}-1\)</span></p><p>= <span class="math-tex">\(\frac{2a^2-b^2}{b^2}\)</span></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>Which of the following is true?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> sinx <span class="math-tex">\(\equiv \sqrt { \frac { 1-cos2x }{ 2 } } \)</span></label></p><p><label class="checkbox"><input type="checkbox"> sinx <span class="math-tex">\(\equiv \frac{sin2x}{2}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> cosx <span class="math-tex">\(\equiv \frac{cos2x}{2}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> cosx <span class="math-tex">\(\equiv \sqrt { \frac { cos2x+1 }{ 2 } } \)</span></label></p></div><div class="q-explanation"><p>The two correct answers are from rearranging the double angle formula for cos2x</p><p>cos2x <span class="math-tex">\(\equiv \)</span> 2cos²x - 1</p><p>cos2x + 1 <span class="math-tex">\(\equiv \)</span> 2 cos²x</p><p><span class="math-tex">\(\frac{cos2x+1}{2} \equiv\)</span> cos²x</p><p><span class="math-tex">\(\sqrt{\frac{cos2x+1}{2} }\equiv\)</span> cosx</p><hr class="hidden-separator"><p>cos2x <span class="math-tex">\(\equiv \)</span> 1 - 2sin²x</p><p>2sin²x <span class="math-tex">\(\equiv\)</span> 1 - cos2x</p><p>sin²x <span class="math-tex">\(\equiv \frac{1-cos2x}{2}\)</span></p><p>sinx <span class="math-tex">\(\equiv \sqrt{\frac{1-cos2x}{2}}\)</span></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 sinx = <span class="math-tex">\(\frac{a}{b}\)</span>, work out sin2x</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(1-\frac{2a^2}{b^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2a}{b}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{2a\sqrt{b^2-a^2}}{b^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{2a(b^2-a^2)}{b^2}\)</span></label></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/q10" style="width: 300px; height: 205px;"></p><p>If sinx = <span class="math-tex">\(\frac{a}{b}\)</span>, then cosx = <span class="math-tex">\(\frac{\sqrt{b^2-a^2}}{b}\)</span></p><p>sin2x <span class="math-tex">\(\equiv\)</span>2sinxcosx</p><p>sin2x = <span class="math-tex">\(2\frac{a}{b}\frac{\sqrt{b^2-a^2}}{b}\)</span></p><p>= <span class="math-tex">\(\frac{2a\sqrt{b^2-a^2}}{b^2}\)</span></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-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 1</p> </div> </div> <div class="panel-body"> <div> <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/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <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>Let f(x) = (cos2x - sin2x)²</p> <p>a) Show that f(x) can be expressed as 1 - sin4x</p> <p>b) Let f(x) = 1 - sin4x. Sketch the graph of <strong><em>f</em></strong> for <span class="math-tex">\(0\le x\le \pi \)</span></p> <p><img alt="" src="../../files/trigonometry/trig-identities/esq2.png" style="width: 600px; height: 388px;"></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>a) Expand the brackets</p> <p>Consider the following identities <span class="math-tex">\(cos^{ 2 }\theta +sin^{ 2 }\theta \equiv 1\\ sin2\theta \equiv 2sin\theta cos\theta \)</span></p> <p>b) In order to sketch this function, consider the transformations from the graph of y = sinx</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/trig-identities/esq_trig_identies2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-identities/esq_trig_identies2.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 panel-expandable"> <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> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Solve <span class="math-tex">\(cos2θ=sinθ\)</span> for <span class="math-tex">\(0\le \theta \le 2\pi \)</span></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">Use the identity <span class="math-tex">\(cos2\theta \equiv 1 -2sin^2\theta\)</span><content></content></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/trig-identities/esq_trig_identies4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-identities/esq_trig_identies4.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 panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>a) Show that <span class="math-tex">\(cos2\theta-3cos\theta+2\equiv 2{ cos }^{ 2 }\theta -3cos\theta +1\)</span></p> <p>b) <strong>Hence</strong><em>, solve <span class="math-tex">\(cos2\theta-3cos\theta+2=0\)</span> for <span class="math-tex">\(0\le \theta \le 2\pi \)</span></em></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"><content> </content>a) Use the following identity <span class="math-tex">\(cos2\theta \equiv 2cos^2\theta -1\)</span></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/trig-identities/esq_trig_identies5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-identities/esq_trig_identies5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Let <span class="math-tex">\(cos\theta=\frac{2}{3}\)</span>, where <span class="math-tex">\(0\le \theta \le \frac { \pi }{ 2 } \)</span></p> <p>Find the value of</p> <p>a) <span class="math-tex">\(sin\theta\)</span></p> <p>b) <span class="math-tex">\(sin2\theta\)</span></p> <p>c) <span class="math-tex">\(sin4\theta\)</span></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>a) Draw a triangle and work out <span class="math-tex">\(sin\theta\)</span></p> <p><content>b) Use the identity <span class="math-tex">\(sin2\theta \equiv 2sin\theta cos\theta \)</span></content></p> <p>c) Work out <span class="math-tex">\(cos2\theta\)</span> first. <span class="math-tex">\(2\theta\)</span> is obtuse</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/trig-identities/esq_trig_identies3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-identities/esq_trig_identies3.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 panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="914"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Show that <span class="math-tex">\(\large \tan 2x \cot 2x\equiv \frac{2}{1-\tan^2x}\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large \tan 2x \cot 2x=3\)</span> , for <span class="math-tex">\(\large -\frac{\pi}{2}<x<\frac{\pi}{2}\)< span=""></x<\frac{\pi}{2}\)<></span></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">a) Use the double angle formula for tan2x<content></content></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><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/esq5a.png" style="width: 600px; height: 188px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/esq5b_1.png" style="width: 600px; height: 313px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/esq5b_2.png" style="width: 600px; height: 151px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2793"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Double Angle Formulae HL</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div 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