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false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Right-angled 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"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/looking-down-1.png" style="float: left; width: 100px; height: 100px;"> In this page, we will learn about right-angled trigonometry. For many, this will not be a new topic, nor will it be a difficult one. However, it deserves some time, as it forms the important step to understanding most of the other work in the geometry and trigonometry topic. The challenging part is when we look at trigonometry in 3 dimensions, as it can be difficult to visualise the triangles needed. The emphasis of this page is looking at these 3-D problems.</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>Using sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.</li> <li>Angles of elevation and depression..</li> <li>The size of an angle between between a line and a plane.</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/rt_angled_trigonometry/revision-notes_rt-angled-trigonometry.pdf" width="100%"></iframe></p> <p>Print from <a href="../../files/trigonometry/rt_angled_trigonometry/revision-notes_rt-angled-trigonometry.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 using the three trigonometric ratios</p> <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#f930adb0"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="f930adb0"> <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%;"> Right-angled Trigonometry 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="11-918-2820" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p> <img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q1.png" style="width: 186px; height: 132px;"></p><p>Which trigometric ratio would you use to calculate <span class="math-tex">\(\large \theta\)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> cos</label></p><p><label class="radio"><input type="radio"> tan</label></p><p><label class="radio"><input class="c" type="radio"> sin</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \sin\theta=\frac{opp}{hyp}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q2.png" style="width: 188px; height: 157px;"></p><p>Which trigometric ratio would you use to calculate <span class="math-tex">\(\large \theta\)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> sin</label></p><p><label class="radio"><input type="radio"> cos</label></p><p><label class="radio"><input class="c" type="radio"> tan</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \tan\theta=\frac{opp}{adj}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q3.png" style="width: 240px; height: 153px;"></p><p>Which trigometric ratio would you use to calculate <span class="math-tex">\(\large \theta\)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> tan</label></p><p><label class="radio"><input type="radio"> cos</label></p><p><label class="radio"><input class="c" type="radio"> sin</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \sin\theta=\frac{opp}{hyp}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q4.png" style="width: 177px; height: 138px;"></p><p>Which trigometric ratio would you use to calculate <span class="math-tex">\(\large \theta\)</span> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> tan</label></p><p><label class="radio"><input class="c" type="radio"> cos</label></p><p><label class="radio"><input type="radio"> sin</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \cos\theta=\frac{adj}{hyp}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q5.png" style="width: 171px; height: 147px;"></p><p>Which trigometric ratio would you use to calculate <em>x</em> ?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> tan</label></p><p><label class="radio"><input type="radio"> cos</label></p><p><label class="radio"><input type="radio"> sin</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \tan\theta=\frac{opp}{adj}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q6.png" style="width: 217px; height: 137px;"></p><p>Which trigometric ratio would you use to calculate <em>x</em> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> sin</label></p><p><label class="radio"><input class="c" type="radio"> cos</label></p><p><label class="radio"><input type="radio"> tan</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \cos\theta=\frac{adj}{hyp}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q7.png" style="width: 277px; height: 145px;"></p><p>Which trigometric ratio would you use to calculate <em>x</em> ?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> tan</label></p><p><label class="radio"><input type="radio"> cos</label></p><p><label class="radio"><input class="c" type="radio"> sin</label></p></div><div class="q-explanation"><p> <span class="math-tex">\(\large \sin\theta=\frac{opp}{hyp}\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q8.png" style="width: 176px; height: 133px;"></p><p>Calculate <em>x</em> to 3 significant figures</p></div><div class="q-answer"><p><em>x</em> = <input type="text" style="height: auto;" data-c="3.08"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \sin38°=\frac{x}{5}\\
\large x=\sin38°\times5\\
\large x\approx3.08\)</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q9.png" style="width: 201px; height: 137px;"></p><p>Calculate <em>x</em> to 3 significant figures</p></div><div class="q-answer"><p><em>x</em> = <input type="text" style="height: auto;" data-c="14.3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \cos56°=\frac{8}{x}\\
\large x=\frac{8}{\cos56°}\\
\large x\approx14.3\)</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><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q10.png" style="width: 247px; height: 128px;"></p><p>Calculate <em>x</em> to 3 significant figures</p></div><div class="q-answer"><p><em>x</em> = <input type="text" style="height: auto;" data-c="6.10"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz1/q10a.png" style="width: 246px; height: 127px; float: left;">Calculate the height of the triangle first.</p><p><span class="math-tex">\(\large \sin30°=\frac{h}{10}\\
\large h=\sin30°\times10\\
\large h=5\)</span></p> <p> </p><p><span class="math-tex">\(\large \sin55°=\frac{5}{x}\\
\large x=\frac{5}{\sin55°}\\
\large x\approx6.10\)</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> </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">Here is a quiz that gets you to think about trigonometry in 3 dimensions <hr class="hidden-separator"> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#63a22953"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="63a22953"> <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%;"> 3D Trigonometry 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="11-917-2820" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p> <img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/q1.png" style="width: 350px; height: 332px;"></p><p>ABCDE is a rectangular-based pyramid. O is the centre of the base.</p><p>Which of the following is the angle between AE and the plane ABCD?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle EAO</label></p><p><label class="radio"><input type="radio"> angle EAD</label></p><p><label class="radio"><input type="radio"> angle EAB</label></p><p><label class="radio"><input type="radio"> angle EBO</label></p></div><div class="q-explanation"><p>We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans1.png" style="width: 350px; height: 321px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/q2.png" style="width: 300px; height: 308px;"></p><p>ABCDEF is a triangular prism</p><p>Which of the following is the angle between AF and the plane ABCD?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> FAC</label></p><p><label class="radio"><input type="radio"> FAD</label></p><p><label class="radio"><input type="radio"> EAD</label></p><p><label class="radio"><input type="radio"> FAB</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans2.png" style="width: 250px; height: 253px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/q3.png" style="width: 350px; height: 367px;"></p><p>ABCDEFGH is a cuboid.</p><p>Angle HBA is the angle between HB and which plane?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> ADHE</label></p><p><label class="radio"><input type="radio"> ABFE</label></p><p><label class="radio"><input type="radio"> ABCD</label></p><p><label class="radio"><input type="radio"> EFGH</label></p></div><div class="q-explanation"><p>Line AH lies in the plane ADHE</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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid1.png" style="width: 300px; height: 289px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between EC and the plane ABCD?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle ECA</label></p><p><label class="radio"><input type="radio"> angle ECD</label></p><p><label class="radio"><input type="radio"> angle ECF</label></p><p><label class="radio"><input type="radio"> angle ECB</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans4.png" style="width: 300px; height: 320px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid1.png" style="width: 300px; height: 289px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between EC and the plane CDHG?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle ECH</label></p><p><label class="radio"><input type="radio"> angle ECG</label></p><p><label class="radio"><input type="radio"> angle ECD</label></p><p><label class="radio"><input type="radio"> angle ECF</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans5.png" style="width: 300px; height: 328px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid1a.png" style="width: 350px; height: 382px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between EC and the plane BCGF?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle ECF</label></p><p><label class="radio"><input type="radio"> angle ECB</label></p><p><label class="radio"><input type="radio"> angle ECD</label></p><p><label class="radio"><input type="radio"> angle ECB</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans6.png" style="width: 350px; height: 378px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/q7.png" style="width: 350px; height: 288px;"></p><p>ABCDEF is a triangular prism</p><p>Angle EDB is the angle between ED and which plane?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> ADFE</label></p><p><label class="radio"><input type="radio"> ABE</label></p><p><label class="radio"><input type="radio"> BCFE</label></p><p><label class="radio"><input class="c" type="radio"> ABCD</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans7.png" style="width: 350px; height: 301px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid3.png" style="width: 400px; height: 368px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between HB and the plane ABCD?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle HBD</label></p><p><label class="radio"><input type="radio"> angle HBA</label></p><p><label class="radio"><input type="radio"> angle HBF</label></p><p><label class="radio"><input type="radio"> angle HBC</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans8.png" style="width: 400px; height: 377px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid3.png" style="width: 400px; height: 368px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between HB and the plane ADHE?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle BHA</label></p><p><label class="radio"><input type="radio"> angle BHE</label></p><p><label class="radio"><input type="radio"> angle BHD</label></p><p><label class="radio"><input type="radio"> angle BHF</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans9.png" style="width: 350px; height: 338px;"></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 alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/cuboid3.png" style="width: 400px; height: 368px;"></p><p>ABCDEFGH is a cuboid.</p><p>Which of the following is the angle between HB and the plane ABEF?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> angle HBE</label></p><p><label class="radio"><input type="radio"> angle HBF</label></p><p><label class="radio"><input type="radio"> angle HBA</label></p><p><label class="radio"><input type="radio"> angle HBD</label></p></div><div class="q-explanation"><p> We can see the angle required in this diagram</p><p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/quiz2/ans10.png" style="width: 350px; height: 324px;"></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 class="smart-object center" data-id="934"> <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-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><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq1q.png" style="width: 450px; height: 293px;"></p> <p>A, B and C are points on horizontal ground.</p> <p>C is due West of B. A is due South of B. AB = 60m</p> <p>A flagpole stands vertically at B.</p> <p>From A, the angle of elevation of the top of the flagpole is 11°.</p> <p>From B, the angle of elevation of the top of the flagpole is 15°.</p> <p>Calculate the distance AC giving your answer to 3 significant figures.</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">Work out the height of the flagpole, then find length BC<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><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq1ans1.png" style="width: 250px; height: 239px; float: left;">We can find the height of the flagpole</p> <p><span class="math-tex">\(\large\tan11°=\frac{h}{60}\\ \large h=60\times\tan11°\\ \large h\approx 11.7\)</span></p> <p><span style="color:#FF0000;">use calculator memory to store the exact value of this length</span></p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq1ans2.png" style="width: 250px; height: 253px; float: left;">Now we can find the length BC</p> <p><span class="math-tex">\(\large\tan15°=\frac{h}{BC}\\ \large BC=\frac{h}{tan15°}\\ \large BC=\frac{60\times \tan11°}{tan15°}\\\\ \large BC\approx 43.5\)</span><span style="color:#FF0000;">use the exact value of h</span></p> <p><span style="color:#FF0000;">use calculator memory to store the exact value of this length</span></p> <hr class="hidden-separator">ABC is a right-angled triangle <p><span class="math-tex">\(\large AC^2=AB^2+BC^2\\ \large AC^2=60^2+(43.5...)^2\\ \large AC\approx74.1m\)</span></p> </section> </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 class="smart-object center" data-id="932"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq2q.png" style="width: 500px; height: 331px;"></p> <p>The diagram shows a cuboid ABCDEFGH. AB = 8 cm, AE = 6 cm and BC = 15 cm.</p> <p>a) Find the length of AC.</p> <p style="margin-left: 40px;">Give your answer correct to 3 significant figures</p> <p>b) Find the size of the angle between the line EC and the plane ABCD.</p> <p style="margin-left: 40px;">Give your answer correct to 1 decimal place.</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) Here is the right-angled triangled you should consider for this question</p> <p><content></content><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq2h.png" style="width: 400px; height: 320px;"></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/rt_angled_trigonometry/esq2a.png" style="width: 400px; height: 316px; float: left;"></p> <p>ABC is a right-angled triangle.</p> <p>Use Pythagoras' Theorem to work out AC.</p> <p>AC² = 8² + 15²</p> <p>AC² = 289</p> <p>AC = 17cm</p> <hr class="hidden-separator"> <p>b)</p> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq2b.png" style="width: 400px; height: 338px; float: left;">The angle between the line EC and the plane ABCD can be visualised in the diagram.</p> <p>EAC is a right-anghled triangle.</p> <p>We can use trigonometry to work out the angle</p> <p><span class="math-tex">\(\large \tan\theta=\frac{6}{17}\\ \large\theta=\tan^{-1}(\frac{6}{17})\\ \large\theta\approx19.4°\)</span></p> </section> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="933"> <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-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><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq3q.png" style="width: 500px; height: 336px;"></p> <p>ABCDEF is a prism in which the triangle BCF is the cross section.</p> <p>BC = 12cm, EF = 16cm, angle CBF = 30° and angle FCB = 90°</p> <p>The angle AF makes with the plane ABCD is <span class="math-tex">\(\large \theta\)</span>.</p> <p>Show that <span class="math-tex">\(\large \tan\theta=\frac{\sqrt{3}}{5}\)</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>The following triangle helps you to see the angle that is required.</p> <p><content><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq3h.png" style="width: 400px; height: 377px;"> </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>We can work out the lengths CF and CA</p> <p>Use the triangle CBF to work out CF</p> <p><span class="math-tex">\(\large \tan30°=\frac{CF}{12} \qquad \qquad \tan30°=\frac{\sqrt{3}}{3}\\ \large \frac{\sqrt{3}}{3}=\frac{CF}{12}\\ \large \frac{12\sqrt{3}}{3}=CF\\ \large CF=4\sqrt{3}\)</span></p> <hr class="hidden-separator"> <p>Use Pythagoras' Theorem to work out CA</p> <p>CA² = 12² + 16²</p> <p>CA² = 400</p> <p>CA = 20</p> <hr class="hidden-separator"> <p>Now put this information in the diagram</p> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq3ans.png" style="width: 450px; height: 386px; float: left;"></p> <p><span class="math-tex">\(\large \tan\theta=\frac{4\sqrt{3}}{20}\\ \large \tan\theta=\frac{\sqrt{3}}{5}\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="935"> <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-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq4q.png" style="width: 500px; height: 356px;"></p> <p>ABCDEFG is a triangular prism.</p> <p>AB = 12cm, AE = 8cm, EF = 18cm.</p> <p>Angle BAE = 90°</p> <p>G is the midpoint of BC.</p> <p>Calculate the angle between EG and the plane ABCD.</p> <p>Give your answer correct to 1 decimal place.</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>Here is the triangle that contains the angle required</p> <p><content><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq4h.png" style="width: 400px; height: 377px;"> </content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq4ans1.png" style="width: 400px; height: 380px; float: left;">We shoud calculate the length AG using Pythagoras' Theorem.</p> <p>G is the midpoint of BC. Therefore BG is 9cm.</p> <p>AG² = 9² + 12²</p> <p>AG² = 225</p> <p>AG = 15</p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq4ans2.png" style="width: 400px; height: 360px; float: left;">Here is the angle that we are required to find.</p> <p>It is a right-angled triangle.</p> <p><span class="math-tex">\(\large \tan\theta =\frac{8}{15}\\ \large\theta=\tan^{-1}(\frac{8}{15})\\ \large\theta\approx24.0°\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel 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="936"> <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><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5q.png" style="width: 550px; height: 337px;"></p> <p>ABCDEF is a prism.</p> <p>AB = AE = BE = 6cm. BC = 10cm</p> <p>Calculate</p> <p>a) the length EC</p> <p>b) the angle AEC</p> <p>c) the angle between EC and the plane ABCD</p> <p>Give lengths to 3 significant figures and angles to 1 decimal place.</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>For parts b) and c), the challenge is to visualise the triangles required to find the angles. The question aims to show you that the angles in part b) and c) are not the same!</p> <p>b)</p> <p><content><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5h1.png" style="width: 400px; height: 323px;"></content></p> <p>c)</p> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5h2.png" style="width: 400px; height: 346px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5a1.png" style="width: 250px; height: 159px; float: left;">a) WE can find the length EC using Pythagoras' Theorem</p> <p>EC² = 10² + 6²</p> <p>EC² = 136</p> <p><span class="math-tex">\(\large EC = \sqrt{136}\)</span></p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5a2.png" style="width: 250px; height: 257px; float: left;">b) The length AC is the same as EC</p> <p>We have an isoceles triangle. Use just the top half of the triangle and right-angled trigonometry to find the angle ACE (or you can use the whole triangle and the cosine rule).</p> <p><span class="math-tex">\(\large \cos\theta=\frac{3}{\sqrt{136}}\\ \large \theta=\cos^{-1}(\frac{3}{\sqrt{136}})\\ \large \theta\approx75.1°\)</span></p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/rt_angled_trigonometry/esq5a3.png" style="width: 400px; height: 338px; float: left;">c) The angle between EC and the plane ABCD can be visualised in this diagram.</p> <p>We know the length EC =<span class="math-tex">\(\large \sqrt{136}\)</span></p> <p>We need to find another legnth in this right-anghled triangle. Find the height of the triangle:</p> <p><span class="math-tex">\(\large h^2=6^2-3^2\\ h=\sqrt{27}\)</span></p> <hr class="hidden-separator"> <p>Let <span class="math-tex">\(\large \alpha\)</span> be the angle between EC and the plane ABCD</p> <p><span class="math-tex">\(\large\sin\alpha=\frac{\sqrt{27}}{\sqrt{136}}\\ \large\alpha\approx26.5°\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2820"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Right-angled Trigonometry</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My 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