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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">Compound Angle Formulae</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/compound-angle/main.png" style="float: left; width: 100px; height: 99px;">In this page, we will will learn about the Compound Angle Formulae used in Trigonometry. These are really important because they open up so many other formulae in trigonometry. In particular, we can derive the double angle formula. There are 6 formula which are written in a shortened form in the IB formula booklet. You need to be very careful with positive/negative signs.</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">\(\large\sin(A+B)\equiv\sin A\cos B+ \cos A \sin B\\ \large\sin(A-B)\equiv\sin A\cos B- \cos A \sin B\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large\cos(A+B)\equiv\cos A\cos B- \sin A \sin B\\ \large\cos(A-B)\equiv\cos A\cos B+ \sin A \sin B\\\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \tan(A+B)≡\frac{\tan A +\tan B}{1-\tan A\tan B}\\ \large \tan(A-B)≡\frac{\tan A -\tan B}{1+\tan A\tan B}\)</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/compound-angle/compound-angle-formulae-hl.pdf" width="100%"></iframe></p> <p>Print from <a href="http://https://file-manager.studyib.net/mathsanalysis/files/trigonometry/trig-identities/compound-angle/compound-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 Compound Angle formulae for sin(A+B) , cos(A+B) and tan(A+B)</div> <div class="panel-body"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#f01c9c5e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="f01c9c5e"> <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%;"> Compund Angle Quiz 1 <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="11-907-2795" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\large \sin (A+B)\)</span> if A and B are acute angles such that</p><p><span class="math-tex">\(\large \sin A=\frac{3}{5}\\ \large \cos A=\frac{4}{5}\)</span> and <span class="math-tex">\(\large \sin B=\frac{5}{13}\\ \large \cos B=\frac{12}{13}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{33}{65}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{56}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{16}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{63}{65}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\sin(A+B)\equiv\sin A\cos B+ \cos A \sin B\)</span></p><p><span class="math-tex">\(\large\sin(A+B)=\frac{3}{5}\times\frac{12}{13}+\frac{4}{5}\times\frac{5}{13}=\frac{36+20}{65}=\frac{56}{65}\)</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>A and B are acute angles such that</p><p><span class="math-tex">\(\large \tan A=\frac{1}{4}\\ \large \tan B=\frac{3}{4}\)</span></p><p>If <span class="math-tex">\(\large \tan(A+B)=\frac{a}{13}\)</span> , find <strong><em>a</em></strong></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \tan(A+B)≡\frac{\tan A +\tan B}{1-\tan A\tan B}\)</span></p><p><span class="math-tex">\(\large \tan(A+B)=\frac{\frac{1}{4} +\frac{3}{4}}{1-\frac{1}{4} \times\frac{3}{4}}=\frac{1}{1-\frac{3}{16}}=\frac{1}{\frac{13}{16}}={\frac{16}{13}}\)</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>Find <span class="math-tex">\(\large \cos (A-B)\)</span> if A and B are angles such that</p><p><span class="math-tex">\(\large \sin A=-\frac{4}{5}\\ \large \cos A=-\frac{3}{5}\)</span> and <span class="math-tex">\(\large \sin B=\frac{7}{25}\\ \large \cos B=\frac{24}{25}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{3}{5}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large-\frac{4}{5}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{44}{125}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{117}{125}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\cos(A-B)\equiv\cos A\cos B+ \sin A \sin B\\\)</span></p><p><span class="math-tex">\(\large\cos(A-B)=-\frac{3}{5}\times\frac{24}{25}+(-\frac{4}{5})\times\frac{7}{25}=\frac{-72-28}{125}=-\frac{100}{125}=-\frac{4}{5}\)</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>Find <span class="math-tex">\(\large \tan (A+B)\)</span> if A and B are angles such that</p><p><span class="math-tex">\(\large \tan A=-\frac{3}{4}\\ \large \tan B=\frac{2}{3}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{17}{6}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large-\frac{1}{18}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{1}{12}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{17}{12}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \tan(A+B)≡\frac{\tan A +\tan B}{1-\tan A\tan B}\)</span></p><p><span class="math-tex">\(\large \tan(A+B)=\frac{-\frac{3}{4} +\frac{2}{3}}{1-(-\frac{3}{4}) \times\frac{2}{3}}=\frac{-\frac{1}{12}}{\frac{3}{2}}=-\frac{1}{18}\)</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>Find <span class="math-tex">\(\large \sin (A+B)\)</span> if A and B are acute angles such that</p><p><span class="math-tex">\(\large \sin A=\frac{\sqrt{7}}{4}\\ \large \cos A=\frac{3}{4}\)</span> and <span class="math-tex">\(\large \sin B=\frac{2}{3}\\ \large \cos B=\frac{\sqrt{5}}{3}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{\sqrt{35}+1}{2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{\sqrt{35}+6}{12}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{3\sqrt{5}+2\sqrt{7}}{12}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\large\frac{3\sqrt{5}-2\sqrt{7}}{12}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\sin(A+B)\equiv\sin A\cos B+ \cos A \sin B\)</span></p><p><span class="math-tex">\(\large\sin(A+B)=\frac{\sqrt{7}}{4}\times\frac{\sqrt{5}}{3}+\frac{3}{4}\times\frac{2}{3}=\frac{\sqrt{35}+6}{12}\)</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>Find <span class="math-tex">\(\large \sin (A-B)\)</span> if A and B are angles represented as follows</p><p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q11a.png" style="width: 200px; height: 196px;"> <img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q11b.png" style="width: 200px; height: 199px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{33}{65}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{16}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{56}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{63}{65}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\sin(A-B)\equiv\sin A\cos B- \cos A \sin B\)</span></p><p><span class="math-tex">\(\large\sin(A-B)=\frac{4}{5}\times(-\frac{5}{13})-\frac{3}{5}\times(-\frac{12}{13})=\frac{-20+36}{65}=\frac{16}{65}\)</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>Find <span class="math-tex">\(\large \sin (A+B)\)</span> if A and B are angles represented as follows</p><p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q13a.png" style="width: 200px; height: 197px;"> <img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q13b.png" style="width: 200px; height: 204px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{4\sqrt{6}-5\sqrt{5}}{21}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large\frac{4\sqrt{6}+5\sqrt{5}}{21}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{10-2\sqrt{30}}{21}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{10+2\sqrt{30}}{21}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\sin(A+B)\equiv\sin A\cos B+ \cos A \sin B\)</span></p><p><span class="math-tex">\(\large\sin(A+B)=\frac{2}{3}\times(\frac{2\sqrt{6}}{7})+(-\frac{\sqrt{5}}{3})\times(-\frac{5}{7})=\frac{4\sqrt{6}+5\sqrt{5}}{21}\)</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>Find <span class="math-tex">\(\large \cos(A+B)\)</span> if A and B are angles represented as follows</p><p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q15a.png" style="width: 200px; height: 190px;"> <img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q15b.png" style="width: 200px; height: 188px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large-\frac{16}{65}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large-\frac{33}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\large\frac{16}{65}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{33}{65}\)</span></label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large\cos(A+B)\equiv\cos A\cos B- \sin A \sin B\\\)</span></p><p><span class="math-tex">\(\large\cos(A+B)=-\frac{4}{5}\times\frac{12}{13}-(-\frac{3}{5})\times\frac{5}{13}=\frac{-48+15}{65}=-\frac{33}{65}\)</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>If A and B are acute angles such that</p><p><span class="math-tex">\(\large \tan A=0.2\\ \large \tan B=0.5\)</span></p><p>Find <strong><em>a</em></strong> if <span class="math-tex">\(\large\tan(A+B)=\frac{a}{9}\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \tan(A+B)≡\frac{\tan A +\tan B}{1-\tan A\tan B}\)</span></p><p><span class="math-tex">\(\large \tan(A+B)=\frac{0.2 +0.5}{1-0.2\times 0.5}={\frac{0.7}{0.9}}={\frac{7}{9}}\)</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>Find <span class="math-tex">\(\large \cos (A-B)\)</span>, given that</p><p><span class="math-tex">\(\large \cos A=-\frac{4}{5}\)</span> , <span class="math-tex">\(\large \frac{\pi}{2}\le A\le \pi\)</span><span class="math-tex"><a<\pi\)< span=""></a<\pi\)<></span></p><p><span class="math-tex">\(\large \tan B=\frac{7}{24}\)</span> , <span class="math-tex">\(0\le B\le \frac{\pi}{2}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{4}{5}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{117}{125}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large-\frac{3}{5}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large\frac{44}{125}\)</span></label></p></div><div class="q-explanation"><p>Let's put the information into triangles and find the missing lengths using Pythagoras' Theorem</p><p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q19a.png" style="width: 200px; height: 203px;"> <img alt="" src="../../files/trigonometry/trig-identities/compound-angle/quiz1/q19b.png" style="width: 200px; height: 200px;"></p><p> </p><p><span class="math-tex">\(\large\cos(A-B)\equiv\cos A\cos B+ \sin A \sin B\\\)</span></p><p><span class="math-tex">\(\large\cos(A-B)=-\frac{4}{5}\times\frac{24}{25}+\frac{3}{5}\times\frac{7}{25}=\frac{-96+21}{125}=-\frac{75}{125}=-\frac{3}{5}\)</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-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> <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> <div class="smart-object center" data-id="915"> <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">\(\large \sin A=\frac{4}{5}\)</span> , where <span class="math-tex">\(\large 0\le A\le\frac{\pi}{2}\)</span></p> <p>and <span class="math-tex">\(\large \cos B=-\frac{12}{13}\)</span> , where <span class="math-tex">\(\large \pi \le B\le\frac{3\pi}{2}\)</span></p> <p>work out <span class="math-tex">\(\large \cos(B-A)\)</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>You will need to work out cosA and sinB</p> <p><content>You will need to use the compound formula <span class="math-tex">\(\large\cos(B-A)\equiv\cos B\cos A+ \sin B \sin A\)</span></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><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq1a.png" style="width: 600px; height: 312px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq1b.png" style="width: 600px; height: 188px;"></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 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="916"> <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>If <span class="math-tex">\(\large \sin(x+30°)=2\cos(x+60°)\)</span>, then show that <span class="math-tex">\(\large \tan x=\frac{\sqrt{3}}{9}\)</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">You will need to remember the exact values for sin30°, cos30°, sin60° and cos60° <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/compound-angle/esq2a.png" style="width: 600px; height: 274px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq2b.png" style="width: 600px; height: 296px;"></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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="917"> <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) By writing 15° as 45° - 30° , find the value of sin15°</p> <p>b) <strong>Hence</strong>, show that the area of this triangle <span class="math-tex">\(\large =4(\sqrt{3}-1)\)</span></p> <p style="margin-left: 40px;"><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq3q.png" style="width: 350px; height: 173px;"></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) We need to use the compound angle formula to evaluate sin(45°-30°)</p> <p><content>b) We need to use the area of the triangle formula. <span class="math-tex">\(\large A= \frac{1}{2}ab\sin C\)</span></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><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq3a.png" style="width: 600px; height: 180px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq3b.png" style="width: 600px; height: 311px;"></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> <div class="smart-object center" data-id="918"> <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>Prove that</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{\sin(A+B)+\sin(A-B)}{\cos(A+B)+\cos(A-B)}=\tan A\)</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"><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq4a.png" style="width: 600px; height: 157px;"><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/compound-angle/esq4a.png" style="width: 600px; height: 157px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq4b.png" style="width: 600px; height: 254px;"></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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="919"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL 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>Prove that <span class="math-tex">\(\large \tan 3x\equiv \frac{3\tan x-\tan^3x}{1-3\tan^2x}\)</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>Use the compound angle formula to work out <span class="math-tex">\(\large \tan(2x + x)\)</span><content></content></p> <p>Very careful manipulation is required in this question. Take your time to make sure that you do not make a mistake.</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><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5a.png" style="width: 600px; height: 261px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5b.png" style="width: 600px; height: 136px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5c.png" style="width: 600px; height: 266px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5d.png" style="width: 600px; height: 44px;"></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 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="920"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL 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>a) Write <span class="math-tex">\(\large \cos4x\)</span> in terms of <span class="math-tex">\(\large\cos x\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large 8\cos^4x-8\cos^2x+1=\sin4x\)</span> , for <span class="math-tex">\(\large 0\le x\le\pi\)</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"> <p>a) Write <span class="math-tex">\(\large\cos4x\equiv \cos(2x+2x)\)</span> and use the double angle formula for <span class="math-tex">\(\large \cos2x\)</span><content></content> and <span class="math-tex">\(\large \sin2x\)</span></p> <p>b) Remember that <span class="math-tex">\(\large \tan x\equiv\frac{\sin x}{\cos x}\)</span></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>a)</p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6a.png" style="width: 600px; height: 230px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6b.png" style="width: 600px; height: 108px;"></p> <p>b)</p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6c.png" style="width: 600px; height: 235px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6d.png" style="width: 600px; height: 234px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2795"> <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>Compound Angle Formulae</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 class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div 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