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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 Non-right angled trigonometry</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 40 minutes"><i class="fa fa-clock-o"></i> 40'</span> </ol> <article id="main-article"> <h2><img alt="" src="../../files/GT/NRAT/Nrat-small-1.jpg" style="width: 100px; height: 100px; float: left;">Trigonometry for all triangles</h2> <p>Here we take trigonometry to the next level by working with triangles that do not have a right angle. We will work on three key rules. The Sine rule, the Cosine rule and the formula for th area of a triangle. All of these involve linking four bits of information about a given triangle. When you know three of them you can figure out the fourth. These problems are then extended in to three dimensions.</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"> <div> <p><span id="docs-internal-guid-b48816f2-7fff-8d79-0168-22143302ba52"></span></p> <p dir="ltr"><span id="docs-internal-guid-b48816f2-7fff-8d79-0168-22143302ba52">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-b48816f2-7fff-8d79-0168-22143302ba52">Derive, understand and use the sine rule for solving problems in non right angled trigonometry</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation"><span id="docs-internal-guid-b48816f2-7fff-8d79-0168-22143302ba52"><span id="docs-internal-guid-b48816f2-7fff-8d79-0168-22143302ba52">Derive, understand and use the cosine rule for solving problems in non right angled trigonometry</span></span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Work out the area of a non right angled triangle using trigonometry</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Solve problems involving the above</p> </li> </ul> </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>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>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-251" class="dynamic-gallery carousel slide" data-id="251"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos002.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos003.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos004.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos005.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos005.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos006.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos006.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos007.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos007.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos009.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos009.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-251/nrat-nologos001.jpeg" data-fancybox="gallery-251" title="" data-caption=""><img alt="" src="../../../std-galleries/15-251/nrat-nologos001.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-251" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-251" 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/15-251/nrat-nologos002-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos003-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos004-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos005-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos006-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos007-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos009-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-251/nrat-nologos001-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. Labelling conventions</p> </div> </div> <div class="panel-body"> <div> <p>Successfully using trigonometry with non right angled triangles depends on sticking to these important conventions. The key is recognising pairs of opposite angles and sides.</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/146365527"></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>2. The Sine rule</p> </div> </div> <div class="panel-body"> <div> <p>What is it and how do we use it to solve problems in non right angled triangles.</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. The Cosine rule</p> </div> </div> <div class="panel-body"> <div> <p>What is it and how do we use it to solve problems in non right angled triangles.</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/593194702"></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. Area of a Triangle</p> </div> </div> <div class="panel-body"> <div> <p>How to find the area of a triangle with no right angle and thus, without the perpendicular height.</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/593195630"></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>5. Angles between lines and planes</p> </div> </div> <div class="panel-body"> <div> <p>A brief visual to help you understand what is meant by this idea.</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/593196194"></iframe></div> </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> <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 Cards</h4> <p>Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.</p> <div id="carousel-252" class="dynamic-gallery carousel slide" data-id="252"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-252/nrat-nologos008.jpeg" data-fancybox="gallery-252" title="" data-caption=""><img alt="" src="../../../std-galleries/15-252/nrat-nologos008.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-252" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-252" 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/15-252/nrat-nologos008-thumb128.jpg"><li></li></ol> </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.</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="15-327-2261"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve the following equations, giving your answers correct to 3sf</p><p>a) <span class="math-tex">\(\frac { a }{ Sin20 } =\frac { 10 }{ sin30 } \)</span> b) <span class="math-tex">\(\frac { 18 }{ sinA } =\frac { 27 }{ sin40 } \)</span> c) <span class="math-tex">\({ a }^{ 2 }={ 7 }^{ 2 }+{ 10 }^{ 2 }-2\times 7\times 10\times cos48\)</span> d) <span class="math-tex">\(cosA\quad =\quad \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \)</span></p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="6.84"> <span class="review"></span> b) b = <input type="text" style="height: auto;" data-c="25.4"> <span class="review"></span> ° </p><p>c) a = <input type="text" style="height: auto;" data-c="7.44"> <span class="review"></span> d) A = <input type="text" style="height: auto;" data-c="13.8"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { a }{ sin20 } =\frac { 10 }{ sin30 } \\ a\quad =\quad \frac { 10 }{ sin30 } \times sin20\\ a\quad =\quad 6.84\quad (3sf)\)</span> </p><p>b) <span class="math-tex">\(\frac { 18 }{ sinA } =\frac { 27 }{ sin40 } \\ \frac { sinA }{ 18 } =\frac { sin40 }{ 27 } \\ sinA\quad =\quad \frac { sin40 }{ 27 } \times 18\\ A\quad =\quad { sin }^{ -1 }\left( \frac { sin40 }{ 27 } \times 18 \right) \\ A\quad =\quad 25.4°\quad (3sf)\)</span> </p><p>c) <span class="math-tex">\({ a }^{ 2 }={ 7 }^{ 2 }+{ 10 }^{ 2 }-2\times 7\times 10\times cos\quad 48\\ a=\sqrt { 149\quad -\quad 140cos48 } \\ a\quad =\quad 7.44\quad (3sf)\)</span> </p><p>d) <span class="math-tex">\(cosA\quad =\quad \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \\ A\quad =\quad { cos }^{ -1 }\left( \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \right) \\ A\quad =\quad 13.8° (3sf)\)</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">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q3.jpg" style="width: 600px; height: 272px;"></p><p>Mark each of the following statements about the diagram as either true (T) or false (F)</p></div><div class="q-answer"><p>Enter either 'T' for true or 'F' for false</p><p>a) <span class="math-tex">\(\frac { sinA }{ a } =\frac { sinB }{ b } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p><p>b) <span class="math-tex">\({ b }^{ 2 }={ a }^{ 2 }+{ c }^{ 2 }-2ac\times cosQ\)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </p><p>c) <span class="math-tex">\(cosB\quad =\quad \frac { { a }^{ 2 }+{ c }^{ 2 }-{ b }^{ 2 } }{ 2ac } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p><p>d) <span class="math-tex">\(\frac { c }{ sin(P+Q) } =\frac { a }{ sinA } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) T - Angles and sides in each fraction are opposite eachother</p><p>b) F - Q is not the angle between sides a and c</p><p>c) T - B is the angle between a and c</p><p>d) T - P + Q is the angle opposite side c and likewise for A and a</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 diagram below.</p><p>a) What is the exact value of z?</p><p>b) Find the values of x and y (give your answer to 3sf)</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q3a.jpg" style="width: 400px; height: 165px;"></p></div><div class="q-answer"><p>z = <input type="text" style="height: auto;" data-c="100"> <span class="review"></span> °</p><p>x = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> cm</p><p>y = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>a) Since angles in a triangle add up to 180°, z = 100</p><p>b) <span class="math-tex">\(\frac { x }{ sin50 } =\frac { 10 }{ sin30 } \\ x\quad =\quad \frac { 10 }{ sin30 } \times sin50\\ x\quad =\quad 15.3\quad cm\quad (3sf)\)</span></p><p>c) <span class="math-tex">\(\frac { y }{ sin100 } =\frac { 10 }{ sin30 } \\ y\quad =\quad \frac { 10 }{ sin30 } \times sin100\\ y\quad =\quad 19.7\quad cm\quad (3sf)\)</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">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>For the triangle below</p><p>a) Calculate the value of z to 3sf</p><p>b) Use that rounded value for z and the sine rule to work out the value of x</p><p>c) Again, use the rounded value of z to calculate the area of the triangle</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q4.jpg" style="width: 400px; height: 336px;"></p></div><div class="q-answer"><p>a) z = <input type="text" style="height: auto;" data-c="67.0"> <span class="review"></span> ° , b) x = <input type="text" style="height: auto;" data-c="24.2"> <span class="review"></span> m, c) Area = <input type="text" style="height: auto;" data-c="479"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { sinz }{ 57 } =\frac { sin44 }{ 43 } \\ sinz\quad =\quad \frac { sin44 }{ 43 } \times 57\\ z\quad =\quad { sin }^{ -1 }\left( \frac { sin44 }{ 43 } \times 57 \right) \\ z\quad =\quad 67.0°\quad (3sf)\)</span></p><p>b) Since z = 67.0° to 3sf, we can calculate that the thrid angle (opposite side x) is 23°</p><p> <span class="math-tex">\(\frac { x }{ sin23 } =\frac { 43 }{ sin44 } \\ x\quad =\quad \frac { 43 }{ sin44 } \times sin23\\ y\quad =\quad 24.2\quad cm\quad (3sf)\)</span></p><p>c) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 43\times 57\times sin23\\ Area\quad =\quad 479{ cm }^{ 2 }\)</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">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q5.jpg" style="width: 400px; height: 330px;"></p><p>a) Work out the value of x</p><p>b) Work out the area of the triangle</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="85.0"> <span class="review"></span> cm</p><p>b) Area = <input type="text" style="height: auto;" data-c="1680"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\({ x }^{ 2 }={ 52 }^{ 2 }+{ 65 }^{ 2 }-2\times 52\times 65\times cos95\\ a=\sqrt { 6929\quad -\quad 3380cos95 } \\ a\quad =\quad 85.0cm\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 52\times 65\times sin95\\ Area\quad =\quad 1683.56904\\ Area\quad =\quad 1680\quad { cm }^{ 2 }\quad (3sf)\)</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>Answer questions referring to the diagram below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q6.jpg" style="width: 400px; height: 187px;"></p><p>a) Calculate the value of x to 3sf.</p><p>b) Calculate the value of y to 3sf.</p><p>c) Use the rounded value of x to work out the area of the triangle</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="91.1"> <span class="review"></span> °</p><p>b) y = <input type="text" style="height: auto;" data-c="54.0"> <span class="review"></span> °</p><p>c) Area = <input type="text" style="height: auto;" data-c="102"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cosx\quad =\quad \frac { { 12 }^{ 2 }+{ 17 }^{ 2 }-{ 21 }^{ 2 } }{ 2\times 12\times 17 } \\ x\quad =\quad { cos }^{ -1 }\left( \frac { { 12 }^{ 2 }+{ 17 }^{ 2 }-{ 21 }^{ 2 } }{ 2\times 12\times 17 } \right) \\ x\quad =\quad 91.1°\)</span></p><p>b) <span class="math-tex">\(cosy\quad =\quad \frac { { 12 }^{ 2 }+{ 21 }^{ 2 }-{ 17 }^{ 2 } }{ 2\times 12\times 21 } \\ y\quad =\quad { cos }^{ -1 }\left( \frac { { 12 }^{ 2 }+{ 21 }^{ 2 }-{ 17 }^{ 2 } }{ 2\times 12\times 21 } \right) \\ y\quad =\quad 54.0°\)</span></p><p>c) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 12\times 17\times sin91.1\\ Area\quad =\quad 101.9812026\\ Area\quad =\quad 102\quad { cm }^{ 2 }\quad (3sf)\)</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">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the square based pyramid ABCDE</p><p><img alt="" src="../../files/GT/NRAT/q7.jpg" style="width: 400px; height: 285px;"> </p><p>EF = 4cm, GF = 3cm, CD = 6cm, EC = <span class="math-tex">\(\sqrt { 34 } cm\)</span></p><p>a) What is the angle <span class="math-tex">\(\hat { DEC } \)</span> (to 3sf)</p><p>b) What is the angle, y, between FG and the plane ECD</p><p>c) What is the angle, z, between EC and the plane ABCD</p></div><div class="q-answer"><p>a) <span class="math-tex">\(\hat { DEC } \)</span> = <input type="text" style="height: auto;" data-c="61.9"> <span class="review"></span> °</p><p>b) Angle = <input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> °</p><p>c) Angle = <input type="text" style="height: auto;" data-c="43.3"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cos\hat { DEC } \quad =\quad \frac { { \left( \sqrt { 34 } \right) }^{ 2 }+{ \left( \sqrt { 34 } \right) }^{ 2 }-{ 6 }^{ 2 } }{ 2\times \sqrt { 34 } \times \sqrt { 34 } } \\ \hat { DEC } \quad =\quad { cos }^{ -1 }\left( \frac { 34+34-{ 6 }^{ 2 } }{ 2\times 34 } \right) \\ \hat { DEC } \quad =\quad 61.9°\quad (3sf)\)</span> </p><p>Could also be done by splitting the triangle in totwo congruent right angled triangles since triangle ECD is isosceles.</p><p>b) Extracting a 2D triangle from the 3D situation. Triangle EFG is right angled with sides EF and FG known.</p><p><img alt="" src="../../files/GT/NRAT/q7a1.jpg" style="width: 300px; height: 213px;"> <span class="math-tex">\(y\quad =\quad \hat { FGE } \\ tany\quad =\quad \frac { 4 }{ 3 } \\ y\quad =\quad { tan }^{ -1 }\left( \frac { 4 }{ 3 } \right) \\ y\quad =\quad 53.1°\quad (3sf)\)</span></p><p>c) Extracting a 2D triangle from the 3D situation. Triangle EFC is right angled with sides EF and EC known.</p><p><img alt="" src="../../files/GT/NRAT/q7a2.jpg" style="width: 300px; height: 204px;"> <span class="math-tex">\(z\quad =\quad \hat { FCE } \\ sinz\quad =\quad \frac { 4 }{ \sqrt { 34 } } \\ z\quad =\quad { sin }^{ -1 }\left( \frac { 4 }{ \sqrt { 34 } } \right) \\ z\quad =\quad 43.3°\quad (3sf)\)</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">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use the map below to answer the questions</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q8a.jpg" style="width: 600px; height: 337px;"></p><ul><li>The map shows the locations of four boats A, B, C and D</li><li>The shortest distance from A to B is 1km</li><li>The shortest distance from B to C is 800m</li><li>The angle ABC is 100°</li></ul><p>a) What is the shortest distance from A to C in km? (to 3sf)</p><p>b) Use the result to part a) and the sine rule to work out the angle BCA</p><ul><li>Angle CBD = 60°</li><li>The distance BD = 300m</li></ul><p>c) What is the distance AD in km? (to 3sf)</p><ul><li>Angle DAC is 30°</li><li>Using rounded answers to previous parts of the question</li></ul><p>d) What is the distance DC?</p></div><div class="q-answer"><p>a) AC = <input type="text" style="height: auto;" data-c="1.38"> <span class="review"></span> km</p><p>b) Angle BCA = <input type="text" style="height: auto;" data-c="45.5"> <span class="review"></span> °</p><p>c) DIstance AD = <input type="text" style="height: auto;" data-c="1.29"> <span class="review"></span> km</p><p>d) Distance DC = <input type="text" style="height: auto;" data-c="0.696"> <span class="review"></span> km</p></div><div class="q-explanation"><p>a) Looking at the the triangle ABC</p><p><img alt="" src="../../files/GT/NRAT/q8a1.jpg" style="width: 300px; height: 162px;"> <span class="math-tex">\({ AC }^{ 2 }={ 1 }^{ 2 }+{ 0.8 }^{ 2 }-2\times 1\times 0.8\times cos100\\ AC=\sqrt { { 1 }^{ 2 }+{ 0.8 }^{ 2 }-2\times 1\times 0.8\times cos100 } \\ AC\quad =\quad 1.38km\quad (3sf)\)</span></p><p>b) Using AC = 1.38 km </p><p><span class="math-tex">\(\frac { sin\hat { BCA } }{ 1 } =\frac { sin100 }{ 1.38 } \\ sin\hat { BCA } \quad =\quad \frac { sin100 }{ 1.38 } \times 1\\ \hat { BCA } \quad =\quad { sin }^{ -1 }\left( \frac { sin100 }{ 1.38 } \times 1 \right) \\ \hat { BCA } \quad =\quad 45.5°\quad (3sf)\)</span></p><p>c) Looking at Triangle ABD</p><p><img alt="" src="../../files/GT/NRAT/q8a2.jpg" style="width: 300px; height: 159px;"> <span class="math-tex">\({ AD }^{ 2 }={ 1 }^{ 2 }+0.3^{ 2 }-2\times 1\times 0.3\times cos100\\ AD=\sqrt { { 1 }^{ 2 }+{ 0.3 }^{ 2 }-2\times 1\times 0.3\times cos100 } \\ AD\quad =\quad 1.29km\quad (3sf)\)</span></p><p>d) Considering triangle ADC</p><p><img alt="" src="../../files/GT/NRAT/q8a3.jpg" style="width: 300px; height: 167px;"> <span class="math-tex">\({ DC }^{ 2 }={ 1 }.29^{ 2 }+1.38^{ 2 }-2\times 1.29\times 1.38\times cos30\\ DC=\sqrt { { 1 }.29^{ 2 }+1.38^{ 2 }-2\times 1.29\times 1.38\times cos30 } \\ DC\quad =\quad 0.696km\quad (3sf)\)</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>The diagram below shows a metal structure.</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q9a.jpg" style="width: 500px; height: 297px;"></p><ul><li>Give all answers correct to 3sf</li><li>Use rounded answers in subsequent calculations</li></ul><p>a) What is the length AB?</p><p>b) What is angle CDB?</p><p>c) Which is longer AD or AC?</p></div><div class="q-answer"><p>a) AB = <input type="text" style="height: auto;" data-c="4.77"> <span class="review"></span> m</p><p>b) Angle CDB = <input type="text" style="height: auto;" data-c="61.1"> <span class="review"></span> °</p><p>c) Which is longer AD or DC? <input type="text" style="height: auto;" data-c="DC"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { AB }{ sin50 } =\frac { 4 }{ sin40 } \\ AB\quad =\quad \frac { 4 }{ sin40 } \times sin50\\ AB\quad =\quad 4.77\quad m\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(\frac { sin\hat { CDB } }{ 4 } =\frac { sin50 }{ 3.5 } \\ sin\hat { CDB } \quad =\quad \frac { sin50 }{ 3.5 } \times 4\\ \hat { CDB } \quad =\quad { sin }^{ -1 }\left( \frac { sin50 }{ 3.5 } \times 4 \right) \\ \hat { CDB } \quad =\quad 61.1°\quad (3sf)\)</span></p><p>c) For the length AD, since angle CDB = 61.1° then ADB must equal 118.9° and so angle DBA must be 21.1°</p><p><span class="math-tex">\(\frac { AD }{ sin21.1 } =\frac { 3.5 }{ sin40 } \\ AD\quad =\quad \frac { 3.5 }{ sin40 } \times sin21.1\\ AD\quad =\quad 1.96\quad m\quad (3sf)\)</span></p><p>For the length AC, since angle CDB is 61.1°, then DBC must be 68.9°</p><p><span class="math-tex">\(\frac { DC }{ sin68.9 } =\frac { 3.5 }{ sin50 } \\ DC\quad =\quad \frac { 3.5 }{ sin50 } \times sin68.9\\ DC\quad =\quad 4.26\quad m\quad (3sf)\)</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>Triangle ABC is the base of a triangular prism as shown in the diagram below</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q10.jpg" style="width: 700px; height: 251px;"></p><ul><li>The length AB is 10cm</li><li>The length AC is 13cm</li><li>The Length BC is 7cm</li><li>The prism has a depth of 5cm</li></ul><p>a) Calculate the angle BAC</p><p>b) What is the area of triangle ABC</p><p>A line is drawn from the point E down to the mid point of AC, M.</p><p>c) How long is that line EM?</p><p>d) What angle does the line EM make with the plane ABC</p></div><div class="q-answer"><p>a) Angle BAC = <input type="text" style="height: auto;" data-c="32.2"> <span class="review"></span> °</p><p>b) Area = <input type="text" style="height: auto;" data-c="34.6"> <span class="review"></span> cm<sup>2</sup></p><p>c) The line EM has length <input type="text" style="height: auto;" data-c="5.68"> <span class="review"></span> cm</p><p>d) The angle EM makes with the plane ABC = <input type="text" style="height: auto;" data-c="41.4"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cos\hat { BAC } \quad =\quad \frac { { 10 }^{ 2 }+{ 13 }^{ 2 }-{ 7 }^{ 2 } }{ 2\times 10\times 13 } \\ \hat { BAC } \quad =\quad { cos }^{ -1 }\left( \frac { { 10 }^{ 2 }+{ 13 }^{ 2 }-{ 7 }^{ 2 } }{ 2\times 10\times 13 } \right) \\ \hat { BAC } \quad =\quad 32.2°\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 10\times 13\times sin32.2\\ Area\quad =\quad 34.6\quad (3sf)\)</span></p><p>c) and d) Consider the triangle EBM</p><p><img alt="" src="../../files/GT/NRAT/q10a.jpg" style="width: 600px; height: 251px;"></p><p>The angle required is the angle EMB (The angle the line EM makes with the plane ABC)</p><p>EB is known as 5cm (depth of prism)</p><p>BM can be calculated using the cosine rule as follows.....</p><p><span class="math-tex">\({ BM }^{ 2 }=10^{ 2 }+6.5^{ 2 }-2\times 10\times 6.5\times cos32.2\\ BM=\sqrt { 10^{ 2 }+6.5^{ 2 }-2\times 10\times 6.5\times cos32.2 } \\ BM\quad =\quad 5.68cm\quad (3sf)\)</span></p><p>Now using triangle EBM and BM = 5.68</p><p><span class="math-tex">\(tan\hat { EMB } \quad =\quad \frac { 5 }{ 5.68 } \\ EMB\quad =\quad { tan }^{ -1 }\left( \frac { 5 }{ 5.68 } \right) \\ EMB\quad =\quad 41.4°\quad (3sf)\)</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.</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="../../files/GT/NRAT/NRAT1.jpg" style="width: 658px; height: 407px;"></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"> <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/593197102"></iframe></div> </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" 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