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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">Pythagorean Identity SL</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/pythag-identities-sl/main3-1.png" style="float: left; width: 100px; height: 97px;"></p> <p>In this page, we will will learn about the Pythagorean Identities. 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"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about </p> <ul> <li>the Pythagorean identity <span></span><span class="math-tex">\(\large \cos^{ 2 }\theta +\sin^{ 2 }\theta \equiv 1\)</span><span></span><span></span></li> </ul> </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/pythag-identities-sl/revision-notes_pythag_identitysl.pdf" width="100%"></iframe></p> <p> Print from <a href="../../files/trigonometry/trig-identities/pythag-identities-sl/revision-notes_pythag_identitysl.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>This quiz is about the Pythagorean Identity sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1a3b928a"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1a3b928a"> <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%;"> Pythagorean Identity 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="11-373-2790" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Thinking about the trigonometric 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">\(sin²x + cos²x \equiv 1\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(1 – sin²x \equiv cos²x\)</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">\(cos2x \equiv 1 +sin²x\)</span></label></p></div><div class="q-explanation"><p>The only 2 statements that are true come from the Pythagorean Trigonometric identity sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</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>Thinking about the trigonometric 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 type="checkbox"> <span class="math-tex">\(sinx \equiv 1 - cos²x\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(cosx \equiv \sqrt{1-sin²x}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(cosx \equiv 1 - sinx\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(sinx \equiv \sqrt{1-cos²x}\)</span></label></p></div><div class="q-explanation"><p>The only 2 statements that are true come from the Pythagorean Trigonometric identity sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</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 = 0.7 , work out sin²x</p></div><div class="q-answer"><p>sin²x = <input type="text" style="height: auto;" data-c="0.51"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><hr class="hidden-separator"><p>sin²x <span class="math-tex">\(\equiv\)</span> 1 - cos²x</p><p>sin²x = 1 - 0.7²</p><p>= 1 - 0.49</p><p>= 0.51</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"> Which of the following is the best representation of the functions <strong>f(x) = sin²x + cos²x</strong></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q4a" style="width: 300px; height: 246px;"></label></p><p><label class="radio"><input type="radio"> <img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q4c" style="width: 300px; height: 246px;"></label></p><p><label class="radio"><input type="radio"> <img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q4d" style="width: 300px; height: 246px;"></label></p><p><label class="radio"><input type="radio"> <img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q4b" style="width: 300px; height: 246px;"></label></p></div><div class="q-explanation"><p><strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><p><strong>Therefore, </strong></p><p><strong>f(x) = 1</strong></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>Write <strong>3sin²x + 4cos²x</strong> in terms of <strong>cosx</strong> only</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 7tan²x</label></p><p><label class="radio"><input type="radio"> cosx + 3</label></p><p><label class="radio"><input type="radio"> 7cos²x - 3</label></p><p><label class="radio"><input class="c" type="radio"> 3 + cos²x</label></p></div><div class="q-explanation"><p>Use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><hr class="hidden-separator"><p>3sin²x + 4cos²x</p><p><span class="math-tex">\(\equiv\)</span> 3(1 - cos²x) + 4 cos²x</p><p><span class="math-tex">\(\equiv\)</span> 3 - 3cos²x + 4 cos²x</p><p><span class="math-tex">\(\equiv\)</span> 3 + cos²x</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>Write <strong>2cos²x</strong> <strong>- 3sin²x</strong> in terms of <strong>sinx</strong> only</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2 + 5sin²x</label></p><p><label class="radio"><input class="c" type="radio"> 2 - 5sin²x</label></p><p><label class="radio"><input type="radio"> 2 - sin²x</label></p><p><label class="radio"><input type="radio"> 2 - 4sin²x</label></p></div><div class="q-explanation"><p>Use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1 </strong>?</p><hr class="hidden-separator"><p>2cos²x - 3sin²x</p><p><span class="math-tex">\(\equiv\)</span> 2(1 - sin²x) - 3sin²x</p><p><span class="math-tex">\(\equiv\)</span> 2 - 2sin²x - 3sin²x</p><p><span class="math-tex">\(\equiv\)</span> 2 - 5sin²x</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 equivalent to <strong>(sinx - cosx)²</strong> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> sin²x - cos²x</label></p><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input class="c" type="radio"> 1 - sin2x</label></p><p><label class="radio"><input type="radio"> 1</label></p></div><div class="q-explanation"><p>Use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><hr class="hidden-separator"><p>(sinx - cosx)²</p><p><span class="math-tex">\(\equiv\)</span> (sinx - cosx)(sinx - cosx)</p><p><span class="math-tex">\(\equiv\)</span> sin²x - 2sinxcosx + cos²x</p><p><span class="math-tex">\(\equiv\)</span> 1 - 2sinxcosx</p><p><span class="math-tex">\(\equiv\)</span> 1 - sin2x</p><p>For the last step, we use the double ange for sin2x</p><p>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>Which of the following is equivalent to cosx + tanxsinx?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2cosx</label></p><p><label class="radio"><input class="c" type="radio"> secx</label></p><p><label class="radio"><input type="radio"> 1 - tan²x</label></p><p><label class="radio"><input type="radio"> cosx</label></p></div><div class="q-explanation"><p>We use the following 2 identities</p><p>tanx <span class="math-tex">\(\equiv \frac{sinx}{cosx}\)</span></p><p><strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><hr class="hidden-separator"><p>cosx + tanxsinx</p><p><span class="math-tex">\(\equiv cosx + \frac{sinx}{cosx}sinx\)</span></p><p><span class="math-tex">\(\equiv \frac{cos²x}{cosx}+ \frac{sin²x}{cosx}\)</span></p><p><span class="math-tex">\(\equiv \frac{cos²x +sin²x}{cosx}\)</span></p><p><span class="math-tex">\(\equiv \frac{1}{cosx}\)</span></p><p><span class="math-tex">\(\equiv\)</span> secx</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 identically equal to 2sin²x - 5cos²x</p><p>Select <strong><u>all</u></strong> the correct answers</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> -3sin²x - 5</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2 - 7cos²x</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 7sin²x - 5</label></p><p><label class="checkbox"><input type="checkbox"> 2 + 3cos²x</label></p></div><div class="q-explanation"><p>We use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><p>There are two correct identities</p><hr class="hidden-separator"><strong>Answer 1</strong><p>2sin²x - 5cos²x <span class="math-tex">\(\equiv\)</span> 2(1 - cos²x) - 5 cos²x</p><p><span class="math-tex">\(\equiv\)</span> 2 - 2cos²x - 5cos²x</p><p><span class="math-tex">\(\equiv\)</span> 2 - 7cos²x</p><hr class="hidden-separator"><p><strong>Answer 2</strong></p><p>2sin²x - 5cos²x <span class="math-tex">\(\equiv\)</span> 2sin²x - 5(1 - sin²x)</p><p><span class="math-tex">\(\equiv\)</span> 2sin²x - 5 + 5sin²x</p><p><span class="math-tex">\(\equiv\)</span> 7sin²x - 5</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 identically equal to <span class="math-tex">\(\frac{cosx}{1+sinx}-\frac{1-sinx}{cosx}\)</span></p><p>There is only <strong><u>one</u></strong> correct answer</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{cosx-1+sinx}{1+sinx-cosx}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{cos²x-sin²x-1-2sinx}{cosx(1+sinx)}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{cos²x-sin²x-1}{cosx(1+sinx)}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> 0</label></p></div><div class="q-explanation"><p>We use the trigonometric identity <strong>sin²x + cos²x <span class="math-tex">\(\equiv\)</span> 1</strong></p><p>Add the fractions together by making the denominator the same</p><p><span class="math-tex">\(\frac{cosx}{1+sinx}-\frac{1-sinx}{cosx}\equiv\frac{cos²x}{(1+sinx)cosx}-\frac{(1-sinx)(1+sinx)}{(1+sinx)cosx}\)</span></p><p><span class="math-tex">\(\equiv\frac{cos²x}{(1+sinx)cosx}-\frac{1+sinx-sinx-sin²x}{(1+sinx)cosx}\)</span></p><p><span class="math-tex">\(\equiv\frac{cos²x}{(1+sinx)cosx}-\frac{1-sin²x}{(1+sinx)cosx}\)</span></p><p><span class="math-tex">\(\equiv\frac{cos²x-1+sin²x}{(1+sinx)cosx}\)</span></p><p><span class="math-tex">\(\equiv\frac{1-1}{(1+sinx)cosx}\)</span></p><p><span class="math-tex">\(\equiv0\)</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="panel-body"> <div class="smart-object center" data-id="910"><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> <p>a) Show that the equation <span class="math-tex">\(\large 2 \sin^2x=3 \cos x\)</span> may be written in the form</p> <p><span class="math-tex">\(\large 2 \cos^2x+3 \cos x-2=0\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large 2 \sin^2x=3 \cos x\)</span> , for <span class="math-tex">\(\large 0\le x\le2\pi\)</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) Use <span class="math-tex">\(\large \cos^2 x+\sin ^2x\equiv1\)</span><content></content></p> <p>b) Use the answer from part a). This is a quadratic equation.</p> <p>Don't forget that <span class="math-tex">\(\large \cos^2 x\)</span> means <span class="math-tex">\(\large (\cos x)^2\)</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><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq1a.png" style="width: 600px; height: 229px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq1b_1.png" style="width: 600px; height: 295px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq1b_2.png" style="width: 600px; height: 142px;"></p> </section> <h4> </h4> </div> </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="panel-body"> <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> <p>Given that <span class="math-tex">\(x=\frac{2}{cos\theta}\)</span> and <span class="math-tex">\(y=3tan\theta\)</span></p> <p>show that <span class="math-tex">\(\frac{x^2}{4}-\frac{y^2}{9}=1\)</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"><content> </content><span class="math-tex">\(\tan^2\theta=\frac{sin²\theta}{cos²\theta}\)</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_identies1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-identities/esq_trig_identies1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-expandable panel-has-border"> <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="909"><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>The following diagram shows triangle ABC with AB = 4 and AC = 5</p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q3-question.png" style="width: 350px; height: 209px;"><strong>DIAGRAM NOT TO SCALE</strong></p> <p>a) Given that <span class="math-tex">\(\large \sin \hat A=\frac{3}{4}\)</span>, find the value of <span class="math-tex">\(\large \cos \hat A\)</span></p> <p>b) <strong>Hence</strong>, show that the length of <span class="math-tex">\(\large BC=\sqrt{41-10\sqrt{7}}\)</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) You can use the Pythagorean identity <span class="math-tex">\(\large \cos^2\theta+\sin^2\theta\equiv1\)</span><content></content>, to find <span class="math-tex">\(\large \cos \hat A\)</span></p> <p>b) Use the Cosine Rule, <em>c</em>² = a² + <em>b</em>² - 2<em>ab</em> cos<em>C</em></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/pythag-identities-sl/esq3a.png" style="width: 600px; height: 293px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq3b.png" style="width: 600px; height: 360px;"></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="911"><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>Prove that <span class="math-tex">\(\large \frac{\sin ^3\theta}{\tan \theta}+\cos^3\theta\equiv\cos\theta\)</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>This is a very difficult proof for SL students!</p> <p>There is always more than one way to carry out this proof.</p> <p><content>The easiest, is to start with the left hand side and consider that <span class="math-tex">\(\large \tan\theta\equiv\frac{\sin\theta}{\cos\theta}\)</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/pythag-identities-sl/esq4_1.png" style="width: 600px; height: 319px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq4_2.png" style="width: 600px; height: 313px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq4_3.png" style="width: 600px; height: 246px;"></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="2790"> <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>Pythagorean Identity SL</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" 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