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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Complex Numbers - de Moivre's Theorem</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/algebra/complex-numbers/de-moirve-theorem/main.png" style="float: left; width: 100px; height: 85px;"></p> <p>De Moivre's Theorem gives a formula for calculating complex numbers. It enables us to connect complex numbers and trigonometry. Most importantly, it is incredibly useful for finding powers and roots of complex numbers. It can be stated in a number of ways:</p> <p><span class="math-tex">\([r(cos\theta+isin\theta)]^n=r^n(cos\ n\theta+isin\ n\theta)\)</span></p> <p>Whilst we can prove the formula using proof by induction, it becomes clear when we write it in Euler Form: <span class="math-tex">\([re^{i\theta}]^n=r^ne^{in\theta}\)</span></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 to</p> <ul> <li>Find powers and roots of complex numbers using de Moivre's theorem</li> <li>Carry out a proof by induction of de Moivre's theorem</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"> <p>The following videos will help you understand all the concepts from this page</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>Proof of de Moivre's Theorem</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="442"> <p>In the following video, we look at an example of a proof by induction question applied to complex numbers. In it, we prove De Moivre's Theorem:</p> <p>Let <span class="math-tex">\(z=r(cosθ+isinθ)\)</span></p> <p>Prove that <span class="math-tex">\(z^{ n }≡r^{ n }[cos(nθ)+isin(nθ)]\ ,\ n\in \mathbb{Z^+}\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/280215846"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</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/algebra/proof-by-induction/proof-by-induction---complex-numbers.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-induction/proof-by-induction---complex-numbers.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </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/algebra/complex-numbers/de-moirve-theorem/revision-notes_de-moivres_theorem.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/complex-numbers/de-moirve-theorem/revision-notes_de-moivres_theorem.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="#3985d890"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="3985d890"> <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%;"> Complex Numbers - de Moivre's Theorem <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="11-448-674" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>z is a complex number.</p><p>If |z| = 4 , work out</p></div><div class="q-answer"><p>|z²| = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p><span class="math-tex">\(|z^{\frac{1}{2}}|\)</span> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Use the following property of the modulus of a complex number:</p><p><span class="math-tex">\(|z^n|={|z|}^n\)</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>z can be expressed in the form a + bi</p><p>If arg(z) = 60° , work out</p></div><div class="q-answer"><p>arg(z²) = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span> °</p><p><span class="math-tex">\(arg(z^{\frac{1}{2}})\)</span> = <input type="text" style="height: auto;" data-c="30"> <span class="review"></span> °</p></div><div class="q-explanation"><p>Use the following property of the argument of a complex number:</p><p><span class="math-tex">\(arg(z^n)=n\cdot arg(z)\)</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><em>Let z = 3(cos30° + isin30°)</em></p><p>Find the principal argument of the following complex numbers</p></div><div class="q-answer"><p>a) -z = <input type="text" style="height: auto;" data-c="-150"> <span class="review"></span> °</p><p>b) iz = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span> °</p><p>c) (1 + i)z = <input type="text" style="height: auto;" data-c="75"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) Consider z in Cartesian form. Multiplying by -1 will make the real and imaginary parts become negative. Note that multiplying a complex number by -1 results in a 180° rotation.</p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q3.bmp" style="width: 250px; height: 238px;"></p><p>b) Multiplying a complex number by i results in a 90° anticlockwise rotation.</p><p>c) arg(1 + i) = 45°</p><p>arg((1 + i)z) = 30° + 45°</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>Let <span class="math-tex">\(z=2(cos(\frac{\pi}{4})+isin(\frac{\pi}{4}))\)</span></p><p>Find the modulus of the following complex numbers</p></div><div class="q-answer"><p>a) -z = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>b) iz = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>c) <span class="math-tex">\((\sqrt{3}+i)z\)</span> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>a) the modulus of -z = modulus of z</p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q3.bmp" style="width: 250px; height: 238px;"></p><p>b) modulus of i = 1</p><p><span class="math-tex">\(|z\cdot w|=|z|\cdot|w|\)</span></p><p>|iz|=1x2=2</p><p>c) <span class="math-tex">\(|\sqrt{3}+i|=\sqrt{3+1²}=2\)</span></p><p><span class="math-tex">\(|z\cdot w|=|z|\cdot|w|\)</span></p><p><span class="math-tex">\(|(\sqrt{3}+i)z|\)</span> = 2x2=4</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>Let <span class="math-tex">\(z=\sqrt{2}cis(\frac{\pi}{4})\)</span></p><p><span class="math-tex">\(z^2=acis(\frac{\pi}{b})\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(z^2=(\sqrt{2})^2cis(2\cdot\frac{\pi}{4})\)</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>Let <span class="math-tex">\(z=2cis(\frac{\pi}{3})\)</span></p><p><span class="math-tex">\(z^3=acis(\frac{\pi}{b})\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(z^3=2^3cis(3\cdot\frac{\pi}{3})\)</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><span class="math-tex">\((\sqrt{2}cis(\frac{\pi}{4}))^8=a+bi\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((\sqrt{2}cis(\frac{\pi}{4}))^8\)</span></p><p><span class="math-tex">\(=(\sqrt{2})^8cis(8\cdot\frac{\pi}{4})\)</span></p><p><span class="math-tex">\(=16cis(2\pi)\)</span></p><p>= 16</p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q7.bmp" style="width: 250px; height: 244px;"></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><span class="math-tex">\((\sqrt[3]{3}cis(\frac{\pi}{6}))^9=a+bi\)</span></p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-27"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((\sqrt[3]{3}cis(\frac{\pi}{6}))^9\)</span></p><p><span class="math-tex">\(=({3}^\frac{1}{{3}})^9cis(9\cdot\frac{\pi}{6})\)</span></p><p><span class="math-tex">\(=3^3cis(\frac{3\pi}{2})\)</span></p><p><span class="math-tex">\(=27cis(-\frac{\pi}{2})\)</span></p><p>= -27i</p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q8.bmp" style="width: 250px; height: 238px;"></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>Work out <span class="math-tex">\((-1+i)^4\)</span></p></div><div class="q-answer"><p><span class="math-tex">\((-1+i)^4\)</span> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p>Let z = -1 + i</p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q9.bmp" style="width: 250px; height: 244px;"></p><p><span class="math-tex">\(z=\sqrt{2}cis\frac{3\pi}{4}\)</span></p><p><span class="math-tex">\(z^4=(\sqrt{2})^4cis(4\cdot\frac{3\pi}{4})\)</span></p><p><span class="math-tex">\(z^4=4cis(3\pi)\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q9ans.bmp" style="width: 250px; height: 250px;"></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 Argand diagrams represents the roots to the equation <span class="math-tex">\(z^3=64i\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/10_1.bmp" style="width: 250px; height: 240px;"> <img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/10_2.bmp" style="width: 250px; height: 238px;"></p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/10_3.bmp" style="width: 250px; height: 238px;"> <img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/10_4.bmp" style="width: 250px; height: 238px;"></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>D</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(z^3=64i\)</span></p><p><img alt="" src="../../files/algebra/complex-numbers/de-moirve-theorem/q10ans.bmp" style="width: 250px; height: 250px;"></p><p><span class="math-tex">\(z^3=64cis(\frac{\pi}{2}+2k\pi),k=0,1,2\)</span></p><p><span class="math-tex">\(z=4cis(\frac{\pi}{6}+\frac{2k\pi}{3}),k=0,1,2\)</span></p><p><span class="math-tex">\(z=4cis(\frac{\pi}{6}),4cis(\frac{5\pi}{6}),4cis(\frac{9\pi}{6})\)</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-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> <div class="smart-object center" data-id="837"> <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>Find the roots of the equation <span class="math-tex">\(z^3=8i \)</span></p> <p>Express your answers in Cartesian Form</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>Write <span class="math-tex">\(z^3\)</span> in cis form</p> <p><content></content><span class="math-tex">\(z^3=8cis(\frac{\pi}{2}+2k\pi)\)</span> , k = 0, 1, 2</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/algebra/complex-numbers/de-moirve-theorem/esq1_complex_de_moivres_th.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/de-moirve-theorem/esq1_complex_de_moivres_th.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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="838"> <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><span class="math-tex">\(z=-2+2\sqrt{3}i\)</span></p> <p>a) Find |z| and arg(z)</p> <p>b) Find <span class="math-tex">\(z^6\)</span> and simplify your answer</p> <p>c) Given that <span class="math-tex">\(w^4=z^3\)</span> , find the values of <span class="math-tex">\(w\)</span> giving your answers in the form <span class="math-tex">\(a+bi\)</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 an Argand diagram</p> <p><content>b) Use de Moivre's Theorem <span class="math-tex">\(z^6={|z|}^6cis6\theta\)</span></content></p> <p>c) Find the 4th roots of <span class="math-tex">\(z^3\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/de-moirve-theorem/esq2_complex_de_moivres_th.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/de-moirve-theorem/esq2_complex_de_moivres_th.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="839"> <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>Find the values of <strong><em>n</em></strong> such that <span class="math-tex">\((\sqrt{3}-i)^n\)</span> is a <strong>real </strong>number</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>Find the argument of <span class="math-tex">\(z=\sqrt{3}-i\)</span><content></content></p> <p>What argument make <span class="math-tex">\(Im(z^n)=0\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/de-moirve-theorem/esq3_complex_de_moivres_th.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/de-moirve-theorem/esq3_complex_de_moivres_th.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> <div class="smart-object center" data-id="840"> <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) By writing <span class="math-tex">\(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\)</span>, show that <span class="math-tex">\(sin(\frac{\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)</span></p> <p>b) Work out <span class="math-tex">\(cos(\frac{\pi}{12})\)</span></p> <p>c) Hence, find the roots of the equation <span class="math-tex">\(z^4=2+2\sqrt{3}i\)</span>, giving answers in the form <span class="math-tex">\(z=a+ib\)</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 the compound angle identity for sin(A+B)</p> <p>c) You will need to use de Moivre's Theorem and the answers from part a and b</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/algebra/complex-numbers/de-moirve-theorem/esq4_complex_de_moivres_th.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/de-moirve-theorem/esq4_complex_de_moivres_th.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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="841"> <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) Find the roots of the equation <span class="math-tex">\(z^4-1=0\)</span></p> <p>b) Find the roots of the equation <span class="math-tex">\(z^4+1=0\)</span></p> <p>c) Show that roots of <span class="math-tex">\(z^4-1=0\)</span> and <span class="math-tex">\(z^4+1=0\)</span> together make the roots of <span class="math-tex">\(z^8-1=0\)</span></p> <p>d) Hence, find all the roots to <span class="math-tex">\(z^6+z^4+z^2+1=0\)</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>c) Factorise <span class="math-tex">\(z^8-1\)</span> into the product of 2 quartic expressions</p> <p><content>d) One of the quartic expressions will factorise further giving 3 factors. Two of these multiple to give <span class="math-tex">\(z^6+z^4+z^2+1\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/de-moirve-theorem/esq5_complex_de_moivres_th.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/de-moirve-theorem/esq5_complex_de_moivres_th.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="674"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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