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title="Mark as 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">Solving Trigonometric Equations</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-equations/main-1.jpg" style="float: left; width: 100px; height: 100px;">Solving trigonometric equations is a common topic on the examination. The key to solving them is a good knowledge of the trigonometric functions. Whether you prefer to use the <a href="../682/unit-circle.html" title="Unit Circle">Unit Circle</a> or the graphs of the functions, you need to a method that works for you. This topic often comes up on the non-calculator paper. Therefore, it is important that you do not solely rely on your graphical calculator to solve the equations and you need to remember the exact values for sin/cos/tan of 30°, 45°, 60°, ...</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</p> <ul> <li>solving trigonometric equations (including quadratic) both graphically and analytically</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 align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/revision-notes_trigequations.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/trigonometry/trig-equations/revision-notes_trigequations.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="#c75da45b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="c75da45b"> <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%;"> Trigonometric Equations <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="11-215-685" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Some of the following statements are TRUE and some are FALSE.</p><p>Type T (TRUE) or F (FALSE) next to each one</p></div><div class="q-answer"><p>a. sin x = sin (180 - x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>b. cos x = cos (180 - x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>c. sin x = sin (180 + x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>d. cos x = cos (360 + x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>e. tan x = tan (180 + x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>f. cos (180 - x) = cos (180 + x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>g. sin x = sin (-x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>h. sin x = cos (x - 180) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>i. cos x = cos ( -x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>j. tan (180 - x) = tan (180 + x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Some of the following statements are TRUE and some are FALSE.</p><p>Type T (TRUE) or F (FALSE) next to each one</p></div><div class="q-answer"><p>a. sin (-x) = sin (<span class="math-tex">\(2\pi\)</span> - x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>b. sin (<span class="math-tex">\(\pi\)</span> - x) = sin (<span class="math-tex">\(2\pi\)</span> + x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>c. tan x = tan (<span class="math-tex">\(\pi\)</span> - x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>d. tan x = tan (-x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>e. cos (-x) = -cosx <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>f. tan x = tan (<span class="math-tex">\(2\pi\)</span> + x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>g. cos x = cos (<span class="math-tex">\(2\pi\)</span> - x) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>h. cos x = cos (<span class="math-tex">\(\pi\)</span> + x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>i. tan (<span class="math-tex">\(\pi\)</span> - x) = tan (<span class="math-tex">\(\pi\)</span> + x) <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>j. cos (<span class="math-tex">\(2\pi\)</span> + x) = cos (<span class="math-tex">\(2\pi\)</span> - x)) <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Complete all the equations below using a different angle.</p><p>All angles are between 0° and 360°</p></div><div class="q-answer"><p>sin 40° = sin<input type="text" style="height: auto;" data-c="140"> <span class="review"></span>°</p><p>cos 80° = cos<input type="text" style="height: auto;" data-c="280"> <span class="review"></span>°</p><p>tan 25° = tan<input type="text" style="height: auto;" data-c="205"> <span class="review"></span>°</p><p>sin 100° = sin<input type="text" style="height: auto;" data-c="80"> <span class="review"></span>°</p><p>tan 269° = tan<input type="text" style="height: auto;" data-c="89"> <span class="review"></span>°</p></div><div class="q-explanation"><p>sinx = sin(180 - x)</p><p>cosx = cos(360 - x)</p><p>tanx = tan(180 + x)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Complete all the equations below using a different angle.</p><p>All angles are between 0° and 360°</p></div><div class="q-answer"><p>cos 359° = cos<input type="text" style="height: auto;" data-c="1"> <span class="review"></span>°</p><p>cos 135° = cos<input type="text" style="height: auto;" data-c="225"> <span class="review"></span>°</p><p>tan 102° = tan<input type="text" style="height: auto;" data-c="282"> <span class="review"></span>°</p><p>sin 208° = sin<input type="text" style="height: auto;" data-c="332"> <span class="review"></span>°</p><p>cos 251° = cos<input type="text" style="height: auto;" data-c="109"> <span class="review"></span>°</p></div><div class="q-explanation"><p>sinx = sin(180 - x)</p><p>cosx = cos(360 - x)</p><p>tanx = tan(180 + x)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the equations below using a different angle. All the angles are between 0 and <span class="math-tex">\(2\pi\)</span></p><p>Give your answers correct to 3 significant figures.</p></div><div class="q-answer"><p>cos 1.15 = cos<input type="text" style="height: auto;" data-c="5.13"> <span class="review"></span></p><p>sin 0.834 = sin<input type="text" style="height: auto;" data-c="2.31"> <span class="review"></span></p><p>tan 0.335 = tan<input type="text" style="height: auto;" data-c="3.48"> <span class="review"></span></p><p>tan 4.26 = tan<input type="text" style="height: auto;" data-c="1.12"> <span class="review"></span></p><p>cos 5.71 = cos<input type="text" style="height: auto;" data-c="0.574"> <span class="review"></span></p></div><div class="q-explanation"><p>sinx = sin(<span class="math-tex">\(\pi\)</span> - x)</p><p>cosx = cos(<span class="math-tex">\(2\pi\)</span> - x)</p><p>tanx = tan(<span class="math-tex">\(\pi\)</span> + x)</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the equations below using a different angle. All the angles are between 0 and <span class="math-tex">\(2\pi\)</span></p><p>Give your answers correct to 3 significant figures.</p></div><div class="q-answer"><p>sin 3.40 = sin<input type="text" style="height: auto;" data-c="6.03"> <span class="review"></span></p><p>tan 2.99 = tan<input type="text" style="height: auto;" data-c="6.13"> <span class="review"></span></p><p>cos 2.02 = cos<input type="text" style="height: auto;" data-c="4.26"> <span class="review"></span></p></div><div class="q-explanation"><p>sin (<span class="math-tex">\(\pi\)</span> + 2.58) = sin(2<span class="math-tex">\(\pi\)</span> - 2.58)</p><p>tan(<span class="math-tex">\(\pi\)</span> - 0.152) = tan(<span class="math-tex">\(2\pi\)</span> - 0.152)</p><p>cos(<span class="math-tex">\(\pi\)</span> - 1.12) = cos(<span class="math-tex">\(\pi\)</span> + 1.12)</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"> Solve <span class="math-tex">\(sin\theta = \frac{\sqrt{2}}{2}\)</span> for 0° < <span class="math-tex">\(\theta\)</span> < 360°</p><p><span class="math-tex">\(\theta_1<\theta_2\)</span></p></div><div class="q-answer"><p><span class="math-tex"><span class="math-tex">\(\theta_1\)</span> = <input type="text" style="height: auto;" data-c="45"> <span class="review"></span>°</span></p><p><span class="math-tex"><span class="math-tex">\(\theta_2\)</span> = <input type="text" style="height: auto;" data-c="225"> <span class="review"></span>°</span></p></div><div class="q-explanation"><span class="math-tex">\(arcsin( \frac{\sqrt{2}}{2})=45°\)</span><p><span class="math-tex">\(\theta_1\)</span> = 45°</p><p><span class="math-tex">\(\theta_2\)</span>= 180° - 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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The solutions to <span class="math-tex">\(cos\theta = -\frac{\sqrt{3}}{2}\)</span> for 0 < <span class="math-tex">\(\theta\)</span> < <span class="math-tex">\(2\pi\)</span></p><p>are <span class="math-tex">\(\frac { a\pi }{ 6 } \)</span> and <span class="math-tex">\(\frac { b\pi }{ 6 } \)</span> , where <strong>a </strong><<strong> b</strong></p><p>Find <strong>a</strong> and <strong>b</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(arccos(+ \frac{\sqrt{3}}{2})=\frac { \pi }{ 6} \)</span></p><p><span class="math-tex">\(\theta_1\)</span> = <span class="math-tex">\(\pi-\frac { \pi }{ 6 } \)</span> = <span class="math-tex">\(\frac { 5\pi }{ 6 } \)</span></p><p><span class="math-tex">\(\theta_2\)</span>= <span class="math-tex">\(\pi+\frac { \pi }{ 6 } \)</span> = <span class="math-tex">\(\frac { 7\pi }{ 6 } \)</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><p> <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"> The solutions to <span class="math-tex">\(sin2\theta = -\frac{1}{2}\)</span> for 0° < <span class="math-tex">\(\theta\)</span> < <span class="math-tex">\(2\pi\)</span></p><p>are <span class="math-tex">\(\frac { a\pi }{ 12 } ,\frac { b\pi }{ 12 } ,\frac { c\pi }{ 12 } ,\frac { d\pi }{ 12 } \)</span> , where a < b < c < d</p><p>Find <strong>a</strong> , <strong>b</strong> , <strong>c </strong>and <strong>d</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="11"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="19"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="23"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(sin2\theta = -\frac{1}{2}\)</span> for 0° < <span class="math-tex">\(2\theta\)</span> < <span class="math-tex">\(4\pi\)</span></p><p><span class="math-tex">\(arcsin(+ \frac{1}{2})=\frac{\pi}{6}\)</span></p><p><span class="math-tex">\(2\theta=\frac { \pi }{ 6 }\quad ,\quad \pi-\frac { \pi }{ 6 }\quad ,\quad 2\pi+\frac { \pi }{ 6 }\quad ,\quad 2\pi+(\pi-\frac { \pi }{ 6 })\)</span></p><p><span class="math-tex">\(2\theta =\frac { \pi }{ 6 } ,\frac { 5\pi }{ 6 } ,\frac { 13\pi }{ 6 } ,\frac { 17\pi }{ 6 } \)</span></p><p><span class="math-tex">\(\theta =\frac { \pi }{12 } ,\frac { 5\pi }{ 12 } ,\frac { 13\pi }{ 12 } ,\frac { 17\pi }{ 12 } \)</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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Solve <span class="math-tex">\(tan^2\theta = 1\)</span> for -180° < <span class="math-tex">\(\theta\)</span> < 180°</p><p><span></span><span class="math-tex">\(\theta_1<\theta_2<\theta_3<\theta_4\)</span><span></span></p></div><div class="q-answer"><p><span class="math-tex"><span class="math-tex">\(\theta_1\)</span></span>= <input type="text" style="height: auto;" data-c="-225"> <span class="review"></span>°</p><p><span class="math-tex">\(\theta_2\)</span> = <input type="text" style="height: auto;" data-c="-45"> <span class="review"></span>°</p><p><span class="math-tex">\(\theta_3\)</span> = <input type="text" style="height: auto;" data-c="45"> <span class="review"></span>°</p><p><span class="math-tex">\(\theta_4\)</span> = <input type="text" style="height: auto;" data-c="225"> <span class="review"></span>°</p></div><div class="q-explanation"><p><span></span><span class="math-tex">\(tan^2\theta = 1\)</span><span></span></p><p><span class="math-tex">\(tan\theta =\pm 1\)</span></p><p>arctan(+1) = 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> </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> <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="532"> <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>Let f(x)= cosx and g(x) = <span class="math-tex">\(\frac{2x^2}{1-x}\)</span></p> <p>a) Show that g∘f(x) = 1 can be written as 2cos²x + cosx - 1 = 0</p> <p>b) Hence solve g∘f(x)=1 for <span class="math-tex">\(-\pi\le x\le \pi\)</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) Start by finding the compostive function g∘f(x).</p> <p>b) This is a quadratic equation. Some students find it easier to factorise by substituting y = cosx and solving for y first.</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-equations/esq_esq_trig_equ1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_esq_trig_equ1.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="531"> <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>Solve <span class="math-tex">\(\log _{ 3 }{ sinx-\log _{ 3 }{ cosx=0.5 } } \)</span> for <span class="math-tex">\(0\le x\le 2\pi\)</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"> <p>Use log laws to simlify equation: <span class="math-tex">\(\log _{ a }{ \frac { x }{ y } =\log _{ a }{ x } -\log _{ b }{ y } } \)</span></p> <p><content>and convert log equation into index equation <span class="math-tex">\(a^x=b⇔x=log_ab\)</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/trigonometry/trig-equations/esq_trig_equ2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_trig_equ2.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="533"> <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>1 + cosx + cos²x + cos<sup>3</sup>x + ... = <strong><span class="math-tex">\(2 + \sqrt2\)</span></strong></p> <p>Find <strong><em>x</em></strong> given that <span class="math-tex">\(-\frac {\pi}{2}\le x\le \frac {\pi}{2}\)</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"> <p>1 + cosx + cos²x + cos<sup>3</sup>x + ... is an infinite geometric series. Can you find the sum?</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-equations/esq_esq_trig_equ3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_esq_trig_equ3.pdf" width="640"></iframe></p> </section> </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="685"> <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 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