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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">Radians, Arcs and Sectors</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/radians%2C-arcs-and-sectors/main1-1.png" style="float: left; width: 100px; height: 117px;"> In this page, we will look at another measure of angles, that is, radians. This is a far more useful measure than degrees. Formula for calculating arcs and sectors become simpler and most importantly, we need to use radian measure whenever we are doing Calculus with trigonometric functions. Calculating arc lengths and sector areas is straightforward, but take care, some of the questions on this topic can be challenging. Make sure that you try some of the more difficult exam-style questions.</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>radian measure for angles</li> <li>lengths of arcs</li> <li>areas of sectors</li> <li>areas of segments</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/radians%2C-arcs-and-sectors/radians_arc_and_sector-revision.pdf" width="100%"></iframe></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 converting degrees and radians</p> <p> </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#0d3f6e7c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="0d3f6e7c"> <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%;"> Radians 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="11-910-2798" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Convert <span class="math-tex">\(\large\frac{\pi}{4}\)</span>radians into degrees</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="45"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \pi\)</span> radians = 180°</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>Convert <span class="math-tex">\(\large\frac{\pi}{6}\)</span>radians into degrees</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \pi\)</span> radians = 180°</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>Convert <span class="math-tex">\(\large\frac{\pi}{5}\)</span>radians into degrees</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="36"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \pi\)</span> radians = 180°</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>Convert <span class="math-tex">\(\large\frac{2\pi}{3}\)</span>radians into degrees</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \pi\)</span> radians = 180°</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>Convert <span class="math-tex">\(\large\frac{7\pi}{4}\)</span>radians into degrees</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="315"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \pi\)</span> radians = 180°</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>90° = <span class="math-tex">\(\large \frac{\pi}{a}\)</span>radians</p><p>What is the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>180° = <span class="math-tex">\(\large \pi\)</span> radians</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>22.5° = <span class="math-tex">\(\large \frac{\pi}{a}\)</span>radians</p><p>What is the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>180° = <span class="math-tex">\(\large \pi\)</span> radians</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>20° = <span class="math-tex">\(\large \frac{\pi}{a}\)</span>radians</p><p>What is the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>180° = <span class="math-tex">\(\large \pi\)</span> radians</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>72° = <span class="math-tex">\(\large \frac{a}{b}\pi\)</span> radians, where <span class="math-tex">\(\large \frac{a}{b}\)</span> is a fraction in its simplest form</p><p>What are the values of <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="5"> <span class="review"></span></p></div><div class="q-explanation"><p>180° = <span class="math-tex">\(\large \pi\)</span> radians</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>240° = <span class="math-tex">\(\large \frac{a}{b}\pi\)</span> radians, where <span class="math-tex">\(\large \frac{a}{b}\)</span> is a fraction in its simplest form</p><p>What are the values of <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="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>180° = <span class="math-tex">\(\large \pi\)</span> radians</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> <hr class="hidden-separator"> <p>Here is a quiz that practises calculations with arc lengths, sector areas and segment areas</p> <p> </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#21c2012b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="21c2012b"> <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%;"> Arc length and Sectors areas 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="11-911-2798" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The following diagram shows a circle centre O and radius 5cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 2 radians.</p><p>Find the length of the arc AB</p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q1.png" style="width: 300px; height: 311px;"></p></div><div class="q-answer"><p>length = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> cm</p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large l=r\theta=5\times 2= 10\)</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>The following diagram shows a circle centre O and radius 5cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 2 radians.</p><p>Find the area of the sector OAB</p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q1.png" style="width: 300px; height: 311px;"></p></div><div class="q-answer"><p>area = <input type="text" style="height: auto;" data-c="25 "> <span class="review"></span> cm²</p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large A=\frac{1}{2}r^2\theta=\frac{1}{2}\times5^2\times 2=25\)</span></p><p style="margin-left: 40px;"> </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>The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 80°.</p><p>The area of the sector OAB = <span class="math-tex">\(\large \frac{a\pi}{9}\)</span>cm²</p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q3.png" style="width: 300px; height: 304px;"></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="200"> <span class="review"></span></p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large A=\frac{\theta}{360}\pi r^2=\frac{80}{360}\pi \times10^2=\frac{200\pi}{9}\)</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>The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 80°</p><p>The length of the arc AB =<span class="math-tex">\(\large \frac{a\pi}{9}\)</span>cm</p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q3.png" style="width: 300px; height: 304px;"></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="40"> <span class="review"></span></p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large l=\frac{\theta}{360}\times2\pi r=\frac{80}{360}\times2\pi \times10=\frac{40}{9}\pi\)</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>The following diagram shows a circle centre O and radius 4cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 1 radian.</p><p>Find the area of the shaded region</p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q5.png" style="width: 300px; height: 302px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 8 (1-2\pi)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 2\pi-8\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 8\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 8 (2\pi-1)\)</span></label></p></div><div class="q-explanation"><p>The angle subtended at the centre of the circle is <span class="math-tex">\(\large 2\pi -1\)</span></p><p><span class="math-tex">\(\large A=\frac{1}{2}r^2\theta=\frac{1}{2}\times4^2\times (2\pi-1)=8 (2\pi-1)\)</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>The following diagram shows a circle centre O and radius <em><strong>r</strong></em>. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B\)</span> = 60°.</p><p>The length of the arc is <span class="math-tex">\(\large 4\pi\)</span> cm.</p><p>Find the radius, <strong><em>r</em></strong></p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q6.png" style="width: 300px; height: 297px;"></p></div><div class="q-answer"><p><em><strong>r</strong></em> = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large l=\frac{\theta}{360}\times 2\pi r\\ \large 4\pi=\frac{60}{360}\times 2\pi r\\ \large r=12\)</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>The following diagram shows a circle centre O and radius <em><strong>r</strong></em>. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=\theta\)</span> radians.</p><p>The length of the arc AB = r.</p><p>Find <span class="math-tex">\(\large \theta\)</span></p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q7.png" style="width: 300px; height: 280px;"></p></div><div class="q-answer"><p><span class="math-tex">\(\large \theta\)</span> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>This is the definition of 1 radian.</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>The following diagram shows a circle centre O and radius 8cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=\theta\)</span> radians.</p><p>The area of the sector OAB is 48cm².</p><p>Find the value of <span class="math-tex">\(\large \theta\)</span></p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q8.png" style="width: 300px; height: 279px;"></p></div><div class="q-answer"><p><span class="math-tex">\(\large \theta\)</span> = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></p></div><div class="q-explanation"><p style="margin-left: 40px;"><span class="math-tex">\(\large A=\frac{1}{2}r^2\theta\\ \large 48=\frac{1}{2}\times 8^2\times\theta\\ \large \theta = \frac{48}{36}=1.5\)</span></p><p style="margin-left: 40px;"> </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>The following diagram shows a circle centre O and radius 10cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=90°\)</span></p><p>Find the shaded area</p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q9a.png" style="width: 300px; height: 323px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 50-25\pi\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 25\pi-50\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 50-5\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large 25\pi\)</span></label></p></div><div class="q-explanation"><p>Area of segment = area of sector - area of triangle</p><p><span class="math-tex">\(\large A=\frac{\pi \times10^2}{4}-\frac{10\times10}{2}\\ \large A=25\pi-50\)</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>The following diagram shows a circle centre O and radius <em><strong>r</strong></em>. A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=\theta\)</span> radians.</p><p>Find the shaded area</p><p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/quiz/q10.png" style="width: 300px; height: 298px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{1}{2}r^2\theta-\frac{1}{2}r^2\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large r^2\theta-r^2\sin\theta\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large r\theta-\frac{r^2}{2}\)</span></label></p></div><div class="q-explanation"><p>Area of segment = area of sector - area of triangle</p><p><span class="math-tex">\(\large A= \frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta\)</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="927"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq1a.png" style="float: right; width: 300px; height: 292px;"><img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The following diagram shows a circle with centre O and radius 12cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large AÔB=50°\)</span></p> <p>a) Find the area of the minor sector OAB</p> <p>b) Find the area of the triangle AOB</p> <p>c) <strong>Hence</strong><em>, </em>find the area of the shaded segment</p> <hr class="hidden-separator"> <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">c) the area of the segment can be found from subtracting the area of the triangle from the area of the sector<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>a) The angle is given in degrees</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{sector}= \frac{\theta}{360}\pi r^2\\ \large A_{sector}= \frac{50}{360}\pi \times 12^2\\ \large A_{sector}=20\pi\\ \large A_{sector}\approx62.831...\\ \large A_{sector}\approx62.8cm^2 \)</span></p> <p>b) The area of the triangle is</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{triangle}= \frac{1}{2} ab \sin C\\ \large A_{triangle}= \frac{1}{2} r^2 \sin \theta\\ \large A_{triangle}= \frac{1}{2} \times12^2 \times\sin 50°\\ \large A_{triangle}\approx55.155...\\ \large A_{triangle}\approx55.2cm^2 \)</span></p> <p>c) The area of the segment = area of sector - area of triangle</p> <p><span style="color:#FF0000;">Be careful to use a higher degree of accuracy for parts a) and b) to give the final answer correct to 3 s.f.</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{sector}\approx62.831...-55.155...\\ \large A_{sector}\approx7.68cm^2\)</span></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="928"> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq2.png" style="width: 300px; height: 310px; float: right;"><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a circle with centre O and radius <strong><em>r</em></strong> cm</p> <p>The area of the shaded sector OAB is <span class="math-tex">\(\large \frac{40\pi}{3}\)</span> cm²</p> <p>The length of the minor arc AB is <span class="math-tex">\(\large \frac{10\pi}{3}\)</span> cm</p> <p>a) Find the radius of the circle</p> <p>b) Find the angle <span class="math-tex">\(\large \theta\)</span> , in radians</p> <hr class="hidden-separator"> <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 an equation for the arc length and another equation for the sector area.</p> <p><content>Solve the equations simultaneously.</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>a) The area of the shaded sector OAB is <span class="math-tex">\(\large \frac{40\pi}{3}\)</span> cm²</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A=\frac{1}{2}r^2\theta\\ \large \frac{40\pi}{3}=\frac{1}{2}r^2\theta\\\)</span></p> <p>The length of the minor arc AB is <span class="math-tex">\(\large \frac{10\pi}{3}\)</span> cm</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large l=r\theta\\ \large \frac{10\pi}{3}=r\theta\)</span></p> <p>Let's solve these equations by substituting for <span class="math-tex">\(\large r\theta\)</span> into the area equation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r^2\theta=\frac{40\pi}{3}\\\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r\times r\theta=\frac{40\pi}{3}\\\)</span><span style="color:#FF0000;"></span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r\times \frac{10\pi}{3}=\frac{40\pi}{3}\\\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r=4\)</span></p> <p style="margin-left: 40px;"><em>r</em> = 8 cm</p> <p>We can now find the angle by substituting this value into the length equation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{10\pi}{3}=8\times\theta\\ \large \theta=\frac{10\pi}{24}\\ \large \theta=\frac{5\pi}{12}\)</span></p> </section> </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="929"> <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 a circle with centre O and radius 5cm and another circle with centre P and radius <strong><em>r</em></strong>. The two circles overlap meeting at points A and B. <span class="math-tex">\(\large AÔP=45°\)</span> and <span class="math-tex">\(\large A\hat{P}O=30°\)</span></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3_1.png" style="width: 500px; height: 320px;"></p> <p>a) Show that <span class="math-tex">\(\large r=5\sqrt{2}\)</span> cm</p> <p>b) <strong><em>Hence</em></strong>, show that the shaded area bounded by the two circles is <span class="math-tex">\(\large \frac{25}{12}(7\pi-6-6\sqrt3)\)</span> cm²</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 Sine Rule to find <strong><em>r</em></strong>.</p> <p><content>b) You should divide the shaded area into two parts. Work out the green shaded area and blue shaded area separately.</content></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3a.png" style="width: 400px; height: 263px;"></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/radians%2C-arcs-and-sectors/esq3_ans_a.png" style="width: 600px; height: 341px;"></p> <p>b) Divide the shaded area into two parts.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3a.png" style="width: 400px; height: 263px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans2.png" style="width: 600px; height: 152px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans3.png" style="width: 600px; height: 207px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans4.png" style="width: 600px; height: 247px;"></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="930"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The following diagram shows a circle with centre O and radius <strong><em>r</em></strong>. A and B are points on the circumference of the circle and <span class="math-tex">\(\large A\hat{O} B =\theta\)</span> radians</p> <p>The area of the green shaded region is three times greater than the area of the blue region.</p> <p>a) Show that <span class="math-tex">\(\large \sin \theta=\frac{4\theta-2\pi}{3}\)</span></p> <p>b) Find the value of <span class="math-tex">\(\large \theta\)</span> , giving your answer correct to 3 significant figures.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4_1.png" style="width: 350px; height: 369px;"></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) The area of the blue segment = area of sector - area of triangle</p> <p><content>b) Use your graphical calculator to solve this equation.</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>a)</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4a.png" style="width: 300px; height: 316px; float: left;"></p> <p>area of the blue segment = area of sector - area of triangle</p> <p><span style="color:#0000CD;"><span class="math-tex">\(\large A_{segment}=\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta\)</span></span></p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4b.png" style="float: left; width: 300px; height: 296px;">The angle subtended by the the green sector is <span class="math-tex">\(\large 2\pi-\theta\)</span></p> <p>Area of the green sector is</p> <p><span style="color:#008000;"><span class="math-tex">\(\large A_{sector}=\frac{1}{2}r^2(2\pi-\theta)\)</span></span></p> <hr class="hidden-separator"> <p>The area of the green sector = three times the area of the blue segment</p> <p style="margin-left: 40px;"><span style="color:#008000;"><span class="math-tex">\(\large \frac{1}{2}r^2(2\pi-\theta)\)</span></span> = 3 <span style="color:#0000CD;"><span class="math-tex">\(\large (\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta)\)</span></span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r^2(2\pi-\theta)=\frac{3}{2}r^2(\theta-\sin\theta)\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 2\pi-\theta=3(\theta-\sin\theta)\\ \large 2\pi-\theta=3\theta-3\sin\theta\\ \large 3\sin\theta=4\theta-2\pi\\ \large \sin\theta=\frac{4\theta-2\pi}{3}\)</span></p> <p>b) We can solve this equation using our graphical equation.You can plot the graphs of <span class="math-tex">\(\large y=\sin\theta\\ \)</span> and <span class="math-tex">\(\large y=\frac{4\theta-2\pi}{3}\)</span> and find the point of intersection</p> <p style="margin-left: 40px;"><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4c.png" style="width: 300px; height: 173px;"></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \theta \approx2.18\)</span></p> <p>The diagram looks approaximately like the following</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4d.png" style="width: 300px; height: 283px;"></p> </section> </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="931"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <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 a circle with centre O and radius <strong><em>r</em></strong>. Points A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=\theta\)</span> radians. The tangents to the circle A and B intersect at C.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5.png" style="width: 450px; height: 376px;"></p> <p>a) Show that <span class="math-tex">\(\large AC=r\tan (\frac{\theta}{2})\)</span></p> <p>b)<strong><em> Hence</em></strong>, find the value of <span class="math-tex">\(\large \theta\)</span> when the two shaded regions have an equal area.</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>b) Area of red region = Area of kite OACB - Area of sector OAB</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5a.png" style="width: 350px; height: 292px;"></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) Triangle OAC is a right angled triangle, so we can use right-angled triangle trigonometry to work out the length AC (opposite to the angle <span class="math-tex">\(\large\frac{\theta}{2}\)</span></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5c.png" style="width: 300px; height: 204px;"><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5d.png" style="width: 250px; height: 149px;"></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \tan\frac{\theta}{2}=\frac{AC}{r}\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large AC=r\tan\frac{\theta}{2}\)</span></p> <p>b)</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5e.png" style="width: 600px; height: 211px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5f.png" style="width: 600px; height: 210px;"></p> <p><img alt="" 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