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class="mark-page-favorite pull-right" data-pid="2196" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1070/geometry-and-trigonometry.html">Geometry and Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">3.3 Applications of trigonometry</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"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/images/compass.jpg');"> </div> <img src="../../images/compass-1.jpg" style="display: none"> <div class="content"> <h3 class="theading">Getting our Bearings</h3> <p class="text">Given the focus on applications in this course, we will look more specifically at real world problems which trigonometry can help solve. Examples of this include bearings and angles of elevation/depression, which are likely to form a large part of the trigonometric questions in your final examinations. </p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <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><span id="docs-internal-guid-6189e667-7fff-a9c5-e6be-ba243a20f549"></span></p> <p dir="ltr"><span id="docs-internal-guid-6189e667-7fff-a9c5-e6be-ba243a20f549">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Know what an angle of elevation and depression is.</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find angles within a bearings problem. </p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Construct a diagram based on a worded bearings problem.</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <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"> <div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Angles of Elevation and Depression</p> </div> </div> <div class="panel-body"> <div> <p>What are angles of elevation and depression?<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/462621361" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Bearings</p> </div> </div> <div class="panel-body"> <div> <p>What are bearings?<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/462622997" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Word Problems </p> </div> </div> <div class="panel-body"> <div> <p>How can I construct diagrams from word problems?<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/462623971" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div id="carousel-270" class="dynamic-gallery carousel slide" data-id="270"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-270/flashcard1.png" data-fancybox="gallery-270" title="" data-caption=""><img alt="" src="../../../std-galleries/15-270/flashcard1.png"></a></div></div><a class="left carousel-control" href="#carousel-270" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-270" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-270/flashcard1-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self checking quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-572-2196"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify the angle of depression in the figure below: </p><p><img alt="" src="../../images/depression-1.png" style="width: 300px; height: 135px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p></div><div class="q-explanation"><p>The angle of depression is the angle between the horizontal line of sight and the line of sight <em>down to</em> an object. It is therefore angle C. </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the angle of elevation, <span class="math-tex">\(x\)</span>, from the persons head to the top of the tree. </p><p><img alt="" src="../../images/elevation-1.png" style="width: 300px; height: 186px;"></p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="48.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using right-angle triangle trigonometry, we can find the angle of elevation: </p><p><span class="math-tex">\(cos(\theta)=\frac{10}{15}\)</span></p><p>So, <span class="math-tex">\(\theta=\cos^{-1}\left(\frac{10}{15}\right)\)</span></p><p><span class="math-tex">\(\theta=48.2^{o}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>A person is standing near a tree. If the angle of elevation is given by 43 degrees, and the person is half the height of the tree, find the height of the tree. </p><p><img alt="" src="../../images/elevation-2.png" style="width: 300px; height: 168px;"></p></div><div class="q-answer"><p>Height of Tree =<input type="text" style="height: auto;" data-c="18.7"> <span class="review"></span> </p></div><div class="q-explanation"><p>First finding the height of the right-angled triangle: </p><p><span class="math-tex">\(\tan\left(43^{o}\right)=\frac{h}{10}\)</span></p><p><span class="math-tex">\(h=10\cdot\tan\left(43^{o}\right)\)</span></p><p><span class="math-tex">\(h=9.3251...m\)</span></p><p>Given that the person is half the height of the tree, we can now double the previous answer to obtain the height of the tree. </p><p>Height of Tree <span class="math-tex">\(=18.7 \space m\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A lighthouse keeper measures the angle of depression between himself and a ship as <span class="math-tex">\(3^{o}\)</span>. If the keeper is 20 metres above the water level, how far is the ship from the lighthouse?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>381 m</span></label> </p><p><label class="radio"> <input type="radio"> <span>103 m</span></label> </p><p><label class="radio"> <input type="radio"> <span>20 m</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>382 m</span></label> </p></div><div class="q-explanation"><p>Here is the worded statement translated into a picture: </p><p><img alt="" src="../../images/q4.png" style="width: 300px; height: 96px;"></p><p>Using right-triangle trigonometry: SOH CAH TOA</p><p><span class="math-tex">\(\tan\left(87\right)=\frac{x}{20}\)</span></p><p>So <span class="math-tex">\(x=20\cdot\tan\left(87\right)\)</span></p><p><span class="math-tex">\(x= 382 \space{m} (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write the bearing of T from P in each diagram below:</p><p><img alt="" src="../../images/q5a.png" style="width: 180px; height: 249px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(30^{o}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(030^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(150^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(330^{o}\)</span></span></label> </p></div><div class="q-explanation"><p>A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is <span class="math-tex">\(030^{o}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write the bearing of T from P in each diagram below:</p><p><img alt="" src="../../images/q5b.png" style="width: 180px; height: 187px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(290^{o}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(110^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(070^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(220^{o}\)</span></span></label> </p></div><div class="q-explanation"><p>A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is <span class="math-tex">\(180^{o}+110^{o}=290^{o}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The bearing of Ship A to Ship B is <span class="math-tex">\(080^{o}\)</span>. Give the bearing of Ship A from Ship B. You do not need to put the degree sign in the textbox provided. </p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="260"> <span class="review"></span> </p></div><div class="q-explanation"><p>By drawing a sketch, we first see that finding the supplementary angle (<span class="math-tex">\(100^{o}\)</span>) then allows us to calculate the bearing of A from B (or B to A). </p><p><img alt="" src="../../images/q6.png" style="width: 200px; height: 102px;"></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Ships A and B leave port at the same time. Ship A travels at 20 kmh on a bearing of <span class="math-tex">\(045^{o}\)</span> and Ship B travels at 15 kmh on a bearing of <span class="math-tex">\(110^{o}\)</span>. How far are the ships away from each other after 3 hours of travelling?</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="57.8"> <span class="review"></span> </p></div><div class="q-explanation"><p>Starting with a sketch: </p><p><img alt="" src="../../images/q8.png" style="width: 300px; height: 321px;"></p><p>Once the sketch has been drawn, use the cosine rule to find the missing length. </p><p><span class="math-tex">\(x^{2}=60^{2}+45^{2}-2\cdot60\cdot45\cdot\cos\left(65\right)\)</span></p><p><span class="math-tex">\(x = 57.8 \space{km}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>From a lamp post, a skateboarder travels 200 m east before turning on a bearing of <span class="math-tex">\(162^{o}\)</span> for 160 m. She then turns on a bearing of <span class="math-tex">\(242^{o}\)</span> for 380 m, and finally in a straight line back to the original lamp post. </p><p>Which of sketches below represent this information?</p></div><div class="q-answer"><p><label class="radio"> </label></p><p><label class="radio"> <input type="radio"> <span><img alt="" src="../../images/q9b.png" style="width: 200px; height: 163px;"></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><img alt="" src="../../images/q9a.png" style="width: 200px; height: 189px;"></span></label> </p></div><div class="q-explanation"><p>The bearings in the correct sketch both turn clockwise from North, whereas the second bearing in the incorrect skecth turns anti-clockwise. </p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the dotted line distance in the diagram below: </p><p><img alt="" src="../../images/q10.png" style="width: 200px; height: 182px;"></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="292"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by finding the included angle between 200 m and 160 m. This is <span class="math-tex">\(360^{o}-90^{o}-162^{o}=108^{o}\)</span>. Then use the cosine rule to find the missing length: </p><p><span class="math-tex">\(x^{2}=200^{2}+160^{2}-2\cdot200\cdot160\cdot\cos\left(108\right)\)</span></p><p><span class="math-tex">\(x = 292 \space{m}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel-footer"> <div> </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> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <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>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/examq1.png" style="width: 600px; height: 200px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/471347614" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../images/examq2.png" style="width: 600px; height: 145px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/471348399" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p><img alt="" src="../../images/examq3.png" style="width: 600px; height: 295px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" 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