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pull-right" data-pid="2260" 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">3.2 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"> <h2><img alt="" src="../../files/GT/RAT/Rat2-small-1.jpg" style="width: 100px; height: 100px; float: left;">SOHCAHTOA</h2> <p>This part of trigonometry can be seen in the recurring relationships between the sides of right angles triangles. The relationship means that we can use two bits of information about a triangle to work out a third and so on until we know everything we need to know. As such it is a really useful tool both for further mathematics an in a number of very practical applications.</p> <hr> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-c843e278-7fff-eec9-456e-d9eeb9a13e12"></span></p> <p dir="ltr"><span id="docs-internal-guid-c843e278-7fff-eec9-456e-d9eeb9a13e12">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"><span id="docs-internal-guid-c843e278-7fff-eec9-456e-d9eeb9a13e12">Understand trigonometric ratios in right angled triangles</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Use these ratios to find unknown angles and sides in right angled triangles and associated problems</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 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>Slides Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-23" class="dynamic-gallery carousel slide" data-id="23"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/3-23/1.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/1.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/2.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/2.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/3.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/3.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/4.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/4.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/5.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/5.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/6.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/6.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/7.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/7.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/8.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/8.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/9.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/9.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/10.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/10.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-23/11.jpg" data-fancybox="gallery-23" title="" data-caption=""><img alt="" src="../../../std-galleries/3-23/11.jpg"></a></div></div><a class="left carousel-control" href="#carousel-23" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-23" 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/3-23/1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/3-23/2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/3-23/3-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/3-23/4-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/3-23/5-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/3-23/6-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/3-23/7-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/3-23/8-thumb128.jpg"><li><li data-index="8"><img title="Click to view" src="../../../std-galleries/3-23/9-thumb128.jpg"><li><li data-index="9"><img title="Click to view" src="../../../std-galleries/3-23/10-thumb128.jpg"><li><li data-index="10"><img title="Click to view" src="../../../std-galleries/3-23/11-thumb128.jpg"><li></li></ol> </div> </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>1. Understanding trig ratios</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;">This video explains the fundamental concept behind trigonometrical relationships.<iframe allowfullscreen="" frameborder="0" height="316" mozallowfullscreen="" src="https://player.vimeo.com/video/146220215" webkitallowfullscreen="" width="500"></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. Making Statements</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;">Solving problems with trigonometry rests heavily on being able to take a given scenario and make correct statements of equality about it. These statements can then be used to find unknowns.<iframe allowfullscreen="" frameborder="0" height="340" mozallowfullscreen="" src="https://player.vimeo.com/video/146221571" webkitallowfullscreen="" width="500"></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. Rearranging equations</p> </div> </div> <div class="panel-body"> <div> <p>Often the thing that students find most difficult when solving trig problems is successfully rearranging the equations. This is a really important bit of algebraic manipulation that is useful lots of places.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/146221909"></iframe></div> </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>4. Solving for angles</p> </div> </div> <div class="panel-body"> <p>Here we look at solving right angled trig problems where the unknown is an angle.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/701259845"></iframe></div> <p> </p> </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>5. Solving for side lengths</p> </div> </div> <div class="panel-body"> <p>Here we look at solving right angled trig problems where the unknown is an side length.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/701275061"></iframe></div> <p> </p> </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>6. Angles of elevation and depression</p> </div> </div> <div class="panel-body"> <div> <p>These are important definitions to be aware of so you don't take a wrong turn at the beginning of a question.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/593177344"></iframe></div> </div> </div> <div class="panel-footer"> <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> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <h4>Flash Card</h4> <p>Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.</p> <p style="text-align: center;"><br> <img alt="" src="../../files/GT/RAT/FC.jpg" style="width: 700px; height: 534px;"></p> </div> </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 hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above. NOTE - Make sure that you have set your GDC to be in degrees. See the <a href="../1054/your-graphical-display-calculator.html" title="Your Graphical Display Calculator">Your Graphical Display Calculator</a> page for help with this.</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="15-326-2260"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your calculator to work out solve the following equations. Give your answers as decimals correct to 3sf.</p><p><em><span style="font-family:times new roman,times,serif;">a) x = sin 35</span></em></p><p><em><span style="font-family:times new roman,times,serif;">b) y = tan 75</span></em></p><p><em><span style="font-family:times new roman,times,serif;">c) z = cos 26</span></em></p><p><em><span style="font-family:times new roman,times,serif;">d) Sinp = 0.7</span></em></p><p><em><span style="font-family:times new roman,times,serif;">e) Cosq = 0.1</span></em></p><p><em><span style="font-family:times new roman,times,serif;">f) Tanr = 2.5</span></em></p></div><div class="q-answer"><p><em><span style="font-family:times new roman,times,serif;">a) x = <input type="text" style="height: auto;" data-c="0.574"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">b) y = <input type="text" style="height: auto;" data-c="3.73"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">c) z = <input type="text" style="height: auto;" data-c="0.899"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">d) p = <input type="text" style="height: auto;" data-c="44.4"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">e) q = <input type="text" style="height: auto;" data-c="84.3"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">f) r = <input type="text" style="height: auto;" data-c="68.2"> <span class="review"></span> </span></em></p></div><div class="q-explanation"><p>This is really just about correctly using the buttons on your calculator. For the last three parts when you have the sine of an angle, you need the inverse sine of that number to get the angle. For example, </p><p>part d) <span class="math-tex">\(sin\quad p\quad =\quad 0.7\\ so\quad p\quad =\quad { sin }^{ -1 }(0.7)\quad =\quad 44.4\\ \\ \)</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">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your calculator to work out solve the following equations. Give your answers as decimals correct to 3sf</p></div><div class="q-answer"><p><span style="font-family:times new roman,times,serif;"><em>a) <span class="math-tex">\(sin\quad 30\quad =\quad \frac { x }{ 10 } \)</span> , so x = <input type="text" style="height: auto;" data-c="5.00"> <span class="review"></span> </em></span></p><p><span style="font-family:times new roman,times,serif;"><em>b) <span class="math-tex">\(Cos\quad 40\quad =\quad \frac { 5 }{ x } \)</span> , so x <input type="text" style="height: auto;" data-c="6.53"> <span class="review"></span> </em></span></p><p><span style="font-family:times new roman,times,serif;"><em>c) <span class="math-tex">\(Tan\quad x\quad =\quad \frac { 7 }{ 14 } \)</span>, so x = <input type="text" style="height: auto;" data-c="26.6"> <span class="review"></span> </em></span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(sin\quad 30\quad =\quad \frac { x }{ 10 } \\ so\quad x\quad =\quad 10\quad \times \quad sin\quad 30\quad =\quad 5.00\)</span></p><p>b) <span class="math-tex">\(Cos\quad 40\quad =\quad \frac { 5 }{ x } \\ so\quad x\quad =\quad \frac { 5 }{ cos\quad 40 } =\quad 6.53\)</span></p><p>c) <span class="math-tex">\(Tan\quad x\quad =\quad \frac { 7 }{ 14 } \\ so\quad x\quad =\quad { tan }^{ -1 }\left( \frac { 7 }{ 14 } \right) =\quad 26.6\)</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>Consider the folloiwng diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q3.jpg" style="width: 600px; height: 272px;"></p><p>Capital letters (in green) are angles. Lower case letters (in blue) are side lengths.</p><p>Next to each of the statements below, enter either 'T' for True of 'F' for False</p></div><div class="q-answer"><ol><li><span class="math-tex">\(Sin\quad A\quad =\quad \frac { f }{ b } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Cos\quad B\quad =\quad \frac { f }{ a } \)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad P\quad =\quad \frac { f }{ d } \)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li><li><span class="math-tex">\(Cos\quad Q\quad =\quad \frac { e }{ a } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad B\quad =\quad \frac { f }{ e } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Sin\quad Q\quad =\quad \frac { e }{ a } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Sin\quad A\quad =\quad Cos\quad P\)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad A\quad =\quad Tan\quad B\)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li></ol></div><div class="q-explanation"><p>This question is really about encouraging the making of statements that are true about a given situation. In practice you can then pick which ones will be mist hepful to you. Clearly it depends on remmebering the trig ratios - SOHCAHTOA and identifying the Hypotenuse as the longest side and the positions of the Opposite and adject sides relative to the angle in question.</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>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q4.jpg" style="width: 300px; height: 224px;"></p><p>Use trigonometry to work out the lengths of sides a and b in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="6.43"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="7.66"> <span class="review"></span> </p></div><div class="q-explanation"><p>a, <span class="math-tex">\(Sin40\quad =\quad \frac { a }{ 10 } \\ so\quad a\quad =\quad 10\times sin40\\ a\quad =\quad 6.43\quad (3sf)\)</span></p><p>b, <span class="math-tex">\(Cos40\quad =\quad \frac { b }{ 10 } \\ so\quad b\quad =\quad 10\times cos40\\ a\quad =\quad 7.66\quad (3sf)\)</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>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q5.jpg" style="width: 300px; height: 220px;"></p><p>Use trigonometry to work out the lengths of sides a and b in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="32.0"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> </p></div><div class="q-explanation"><p>a, <span class="math-tex">\(sin28\quad =\quad \frac { 15 }{ a } \\ so\quad a\quad =\quad \frac { 15 }{ sin28 } \\ a\quad =\quad 32.0\quad (3sf)\)</span></p><p>b, <span class="math-tex">\(tan28\quad =\quad \frac { b }{ 15 } \\ so\quad b\quad =\quad 15\times tan28\\ a\quad =\quad 7.98\quad (3sf)\)</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">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q6.jpg" style="width: 300px; height: 225px;"></p><p>Use trigonometry to work out the lengths of angles A and B in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>A = <input type="text" style="height: auto;" data-c="42.9"> <span class="review"></span> </p><p>B = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> </p></div><div class="q-explanation"><p>A, <span class="math-tex">\(cos\quad A\quad =\quad \frac { 41 }{ 56 } \\ so,\quad A\quad =\quad { cos }^{ -1 }\left( \frac { 41 }{ 56 } \right) \\ A\quad =\quad 42.9\quad (3sf)\)</span></p><p>B, <span class="math-tex">\(sin\quad B\quad =\quad \frac { 41 }{ 56 } \\ so,\quad B\quad =\quad { sin }^{ -1 }\left( \frac { 41 }{ 56 } \right) \\ A\quad =\quad 47.1\quad (3sf)\)</span></p><p>NOTE, A + B = 90 as they should!</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 diagram below shows a ladder put against a house for cleaning the gutters. The yop of the ladder is 5m above the ground and the angle the ladder makes with the horizontal must be 65°.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q7.jpg" style="width: 500px; height: 361px;"></p><p>a) How far away (<em>x</em>) from the wall must the base of the ladder be put to make the angle 65°</p><p>b) How long does the ladder need to be? (Use trigonometry and unrounded numbers to answer this question)</p><p>c) If the length of the ladder is fixed at 6m and the height against the house remains at 5m, what would the angle between the ladder and the gorund become?</p><p>d) What is the precentage error in the new angle from the required 65°? (Use the rounded answer to part c)</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="2.33"> <span class="review"></span> m</p><p>b) length of ladder = <input type="text" style="height: auto;" data-c="5.52"> <span class="review"></span> m</p><p>c) New angle betwen the ladder and the ground = <input type="text" style="height: auto;" data-c="56.4"> <span class="review"></span> °</p><p>d) Percentage error = <input type="text" style="height: auto;" data-c="13.2"> <span class="review"></span> %</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(tan\quad 65\quad =\quad \frac { 5 }{ x } \\ so,\quad x\quad =\quad \frac { 5 }{ tan\quad 65 } \\ A\quad =\quad 2.33\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(sin\quad 65\quad =\quad \frac { 5 }{ l } \\ so,\quad l\quad =\quad \frac { 5 }{ sin\quad 65 } \\ A\quad =\quad 5.52\quad (3sf)\\ \\ OR\quad { l }^{ 2 }\quad =\quad { 5 }^{ 2 }+{ x }^{ 2 }\\ Using\quad the\quad unrounded\quad answer\quad for\quad x\quad gives\quad the\quad same\quad result\quad for\quad l\)</span></p><p>NOTE, you could have used cosine with x and 65°. Again if you used the unrounded anser for x you would have got the same result.</p><p>c) <img alt="" src="../../files/GT/RAT/q7a.jpg" style="width: 200px; height: 259px;"> <span class="math-tex">\(sin\quad A\quad =\quad \frac { 5 }{ 6 } \\ so,\quad A\quad =\quad { sin }^{ -1 }\left( \frac { 5 }{ 6 } \right) \\ A\quad =\quad 56.4\quad (3sf)\)</span></p><p>d) = <span class="math-tex">\(\frac { 65\quad -\quad 56.4 }{ 65 } \times \quad 100\quad =\quad 13.2\)</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">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows part of a metal structure and lists some its geometrical properties.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q8.jpg" style="width: 500px; height: 319px;"></p></div><div class="q-answer"><p>Find the folllowing giving all answers correct to 3sf</p><p>a) The length AC = <input type="text" style="height: auto;" data-c="5.00"> <span class="review"></span> m</p><p>b) <span class="math-tex">\(\hat { CAD } \)</span> = <input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> °</p><p>c) <span class="math-tex">\(\hat { mAD } \)</span> = <input type="text" style="height: auto;" data-c="45.0"> <span class="review"></span> </p><p>In changes to the design it is decided that <span class="math-tex">\(\hat { mAD } \)</span> needs to be exactly halved </p><p>d) What is the new length AM? <input type="text" style="height: auto;" data-c="3.25"> <span class="review"></span> m</p></div><div class="q-explanation"><p>a) <span class="math-tex">\({ AC }^{ 2 }\quad =\quad { 4 }^{ 2 }+{ 5 }^{ 2 }\\ AC\quad =\sqrt { 25 } =5.00\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(tan\hat { CAD } =\frac { 4 }{ 3 } \\ so,\quad \hat { CAD } \quad =\quad { tan }^{ -1 }\left( \frac { 4 }{ 3 } \right) \\ \hat { CAD } \quad =\quad 53.1°\)</span></p><p>c) <span class="math-tex">\(tan\hat { mAD } =\frac { 2 }{ 3 } \\ so,\quad \hat { mAD } \quad =\quad { tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) \\ \hat { mAD } \quad =\quad 45.0°\)</span></p><p>d) <img alt="" src="../../files/GT/RAT/q8a.jpg" style="width: 200px; height: 134px;"> <span class="math-tex">\(cos22.5\quad =\quad \frac { 3 }{ mA } \\ so,\quad mA\quad =\quad \frac { 3 }{ cos\quad 22.5 } \\ mA\quad =\quad 3.25m\quad (3sf)\)</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">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>USe the diagram below for the following questions.</p><p>The angle of elevation from Point A to some looking out of the window at point B is 60°</p><p>a) What is the height, h, of point B above the ground?</p><p>A person looking at point A from point C is 35m above the ground.</p><p>b) What is the angle of depression, D, they are looking at?</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q9.jpg" style="width: 500px; height: 390px;"></p></div><div class="q-answer"><p>a) Height h = <input type="text" style="height: auto;" data-c="20.9"> <span class="review"></span> </p><p>b) Angle of depression, a = <input type="text" style="height: auto;" data-c="71.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <img alt="" src="../../files/GT/RAT/q9-answer.jpg" style="width: 250px; height: 186px;"> <span class="math-tex">\(tan60\quad =\quad \frac { h }{ 12 } \\ so,\quad h\quad =\quad 12\times tan60\\ h\quad =\quad 20.9m\quad (3sf)\)</span></p><p>b) <img alt="" src="../../files/GT/RAT/q9a2.jpg" style="width: 300px; height: 233px;"> <span class="math-tex">\(tanD\quad =\quad \frac { 35 }{ 12 } \\ so,\quad D\quad =\quad { tan }^{ -1 }\left( \frac { 35 }{ 12 } \right) \\ h\quad =\quad 71.1°\quad (3sf)\)</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">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows ta narrow chanel of water between the cliff coasts of two islands. Some distances are shown.</p><p style="text-align: center;"><img alt="" src="../../files/LSP/PB/q10.jpg" style="width: 600px; height: 445px;"></p><p>The diagram also shows.....</p><ul><li>The top of a lighthouse at point <strong>A</strong></li><li>The view from the base of the light house at point <strong>B</strong></li><li>A boat at point <strong>C</strong></li><li>The view from the top of the cliff at point <strong>D</strong></li></ul><p>The angle of elevation of point D from point C is 65°. The angle of depression of point C from point D is 40°.</p><p>a) Find the width of the water chanel by finding the respective distances x and y to 3sf. (Use rounded answers to get the total width)</p><p>The angle of elevation from point C to point A is 75° .</p><p>b) What is the height of the lighthouse above the the ground? (Use rounded ansers to any distances you have calculated)</p><p>c) What is the angle, p, of depression from the top of the light house to point D? (Use rounded ansers to any distances you have calculated)</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="11.9"> <span class="review"></span> m to (3sf), y = <input type="text" style="height: auto;" data-c="8.39"> <span class="review"></span> m (3sf),</p><p>Total distance = <input type="text" style="height: auto;" data-c="20.3"> <span class="review"></span> m</p><p>b) h = <input type="text" style="height: auto;" data-c="34.4"> <span class="review"></span> m</p></div><div class="q-explanation"><p>a) <img alt="" src="../../files/LSP/PB/q10a2.jpg" style="width: 200px; height: 136px;"> <span class="math-tex">\(tan40\quad =\quad \frac { 10 }{ x } \\ x\quad =\quad \frac { 10 }{ tan40 } \\ x\quad =\quad 11.9\quad (3sf)\)</span></p><p> <img alt="" src="../../files/LSP/PB/q10a1.jpg" style="width: 200px; height: 193px;"> <span class="math-tex">\(tan65\quad =\quad \frac { 18 }{ y } \\ y\quad =\quad \frac { 18 }{ tan65 } \\ y\quad =\quad 8.39\quad (3sf)\)</span></p><p>Total distance = 8.39 + 11.9 = 20.29 , 20.3 (3sf)</p><p>b) <img alt="" src="../../files/LSP/PB/q10a3.jpg" style="width: 200px; height: 395px;"> <span class="math-tex">\(tan75\quad =\quad \frac { h+10 }{ 11.9 } \\ h\quad +\quad 10\quad =\quad 11.9\times tan75\\ h\quad =\quad 11.9\times tan75\quad -\quad 10\\ h\quad =\quad 34.4\quad m\quad (3sf)\)</span></p><p>c) <img alt="" src="../../files/LSP/PB/q10a4.jpg" style="width: 200px; height: 100px;"> <span class="math-tex">\(tanp\quad =\quad \frac { 10\quad +\quad 34.4\quad -\quad 18 }{ 11.9 } =\frac { 26.4 }{ 20.3 } \\ p\quad =\quad { tan }^{ -1 }\left( \frac { 26.4 }{ 20.3 } \right) \\ p\quad =\quad 52.4\quad (3sf)\)</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="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. NOTE - Make sure that you have set your GDC to be in degrees. 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