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class="page_title"> Geometry and Trigonometry Examination Questions SL <a href="#" class="mark-page-favorite pull-right" data-pid="2907" title="Mark as favorite" onclick="return 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="../2902/examination-questions.html">Examination Questions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Geometry and Trigonometry Examination Questions SL</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>On this page you can find examination questions from the topic of geometry and trigonometry</p> <div class="panel panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3-Dimensional Solids</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="921"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq1.png" style="width: 300px; height: 373px; float: right;"><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>A glass is made up of a hemisphere and a cone.</p> <p>Find the volume of the glass.</p> <p>Give your answer to 3 significant figures</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p style="margin:0in;font-family:Calibri;font-size:16.0pt">The radius of the cone and the hemisphere are both 5cm</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq1ans.png" style="width: 300px; height: 227px;"></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/volume_surface_area/esq1ans.png" style="width: 300px; height: 227px; float: left;">The radius of the cone and the hemisphere are both 5cm</p> <p>Volume of cone,</p> <p><span class="math-tex">\(\large V_{cone}=\pi r^2h\\ \large V_{cone}=\pi \times5^2\times 5\\ \large V_{cone}=125\pi \)</span></p> <p>Volume of hemisphere,</p> <p><span class="math-tex">\(\large V_{hemisphere}=2\pi r^2\\ \large V_{hemisphere}=2\times \pi \times 5^2\\ \large V_{hemisphere}=50\pi\)</span></p> <hr class="hidden-separator">Total Volume<span class="math-tex">\(\large=125\pi+50\pi\\ \large=175\pi\\ \large\approx549.779...cm^3\\ \large\approx550cm^3\)</span></section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="922"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q7.png" style="width: 300px; height: 180px; float: right;"> <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"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The <em><strong>total</strong></em> surface area of a hemisphere is 1360 cm²</p> <p>Find the radius.</p> <p>Give your answer to 3 significant figures.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p>The surface of a hemisphere is made of two parts</p> <ol> <li>the curved surface</li> <li>the circular base</li> </ol> <ol> </ol> </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>The surface area of a hemisphere,</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A=\pi r^2+\pi r^2\\ \large A=3\pi r^2\\ \large 1360=3\pi r^2\\ \large r^2=\frac{1360}{3\pi}\\ \large r=\sqrt{\frac{1360}{3\pi}}\\ \large r\approx12.0125...\\ \large r\approx12.0cm \)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="923"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) A sphere has a radius of 10cm. Find the volume, giving your answer in terms of <span class="math-tex">\(\large \pi\)</span>.</p> <p>b) A cone has the same volume and the same radius as the sphere. Find the height of the cone.</p> <p>c) Another sphere and cone have the same volume and the same radius, <strong><em>r</em></strong>. Find an equation for the height of the cone, <strong><em>h</em></strong> in terms of <strong><em>r</em></strong>.</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq3.png" style="width: 400px; height: 276px;"></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>c) Solve <span class="math-tex">\(\large \frac{4}{3}\pi r^3=\frac{1}{3}\pi r^2h\\\)</span></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><span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\\ \large V=\frac{4}{3}\pi \times10^3\\ \large V=\frac{4000}{3}\pi \ cm^3\)</span></p> <p>b)</p> <p><span class="math-tex">\(\large V=\frac{1}{3}\pi r^2h\\ \large \frac{4000}{3}\pi=\frac{1}{3}\pi \times 10^2\times h\\ \large h=40\ cm\)</span></p> <p>c)</p> <p><span class="math-tex">\(\large \frac{4}{3}\pi r^3=\frac{1}{3}\pi r^2h\\ \large \frac{4}{\rlap{/}3}\rlap{/}\pi r^3=\frac{1}{\rlap{/}3}\rlap{/}\pi r^2h\\ \large 4r^3=r^2h\\ \large 4r=h\\ \large h=4r\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="924"> <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>Three metal spheres have radii 1cm, 6cm and 8cm.</p> <p>The spheres are melted down and made into one bigger sphere.</p> <p>What is the radius of the single sphere?</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq4.png" style="width: 500px; height: 236px;"></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">The volume of a sphere is given by <span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\)</span><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>The volume of a sphere is given by <span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\)</span></p> <p>The Volume of the three spheres is</p> <p><span class="math-tex">\(\large V=\frac{4}{3}\pi \times1^3+\frac{4}{3}\pi \times 6^3+\frac{4}{3}\pi \times 8^3\\ \large V=\frac{4}{3}\pi \times(1^3+6^3+8^3)\\ \large V=\frac{4}{3}\pi \times(729)\\ \)</span></p> <p>The large sphere has the same volume as these three spheres</p> <p><span class="math-tex">\(\large \frac{4}{3}\pi \times R^3=\frac{4}{3}\pi \times(729)\\ \large R^3=729\\ \large R=9\ cm \)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="925"> <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>A cylindrical metal bar with height 12cm and diameter 12cm is melted down and made into spheres of diameter 3cm.</p> <p>How many spheres will it make?</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 volume of one sphere and the volume of the cylinder.<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>The volume of the cylindrical bar is</p> <p><span class="math-tex">\(\large V=\frac{1}{3}\pi\times 6^2\times12\\ \large V=144\pi \)</span></p> <p>The volume of one of the spheres is</p> <p><span class="math-tex">\(\large V=\frac{4}{3}\pi\times 1.5^3\\ \large V=4.5\pi \)</span></p> <p>Therefore, the number of spheres is <span class="math-tex">\(\large \frac{144\pi}{4.5\pi}\)</span></p> <p><span class="math-tex">\(\large 32\)</span> spheres</p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="926"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq6.png" style="width: 280px; height: 296px; float: right;"><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A solid is made up of a cone and a cylinder.</p> <p>The radius is 5cm, the height of the cone is 12cm and the height of the cylinder is 15cm.</p> <p>Show that the <strong>total</strong> surface area of the solid is <span class="math-tex">\(\large 240\pi\)</span></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>There are three surfaces to find: the curved surface of the cone, the curved surface of the cylinder and the circular base.</p> <p>To find the curved surface area of the cone, you need to find the slant height. Use Pythagoras' Theorem.<content></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/volume_surface_area/esq6a.png" style="width: 300px; height: 231px; float: left;">We need to find the slant height of the cone:</p> <p><span class="math-tex">\(\large l^2=5^2+12^2\\ \large l^2=169\\ \large l=13\)</span></p> <p>The surface area of the cone is</p> <p><span class="math-tex">\(\large A_{cone}=\pi rl\\ \large A_{cone}=\pi\times5\times 13\\ \large A_{cone}=65\pi\)</span></p> <hr class="hidden-separator"> <p>The surface area of the cylinder is</p> <p><span class="math-tex">\(\large A_{cylinder}=2\pi rh\\ \large A_{cylinder}=2\pi \times 5\times 15\\ \large A_{cylinder}=150\pi\)</span></p> <p>The area of the base is</p> <p><span class="math-tex">\(\large A_{base}=\pi r^2\\ \large A_{base}=\pi \times 5^2\\ \large A_{base}=25\pi \)</span></p> <hr class="hidden-separator"> <p>The total surface area is</p> <p><span class="math-tex">\(\large =65\pi+150\pi+25\pi\\ \large=240\pi\)</span></p> </section> <h4> </h4> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Radians, Arcs and Sectors</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="927"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq1a.png" style="float: right; width: 300px; height: 292px;"><img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The following diagram shows a circle with centre O and radius 12cm. A and B lie on the circumference of the circle and <span class="math-tex">\(\large AÔB=50°\)</span></p> <p>a) Find the area of the minor sector OAB</p> <p>b) Find the area of the triangle AOB</p> <p>c) <strong>Hence</strong><em>, </em>find the area of the shaded segment</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">c) the area of the segment can be found from subtracting the area of the triangle from the area of the sector<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) The angle is given in degrees</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{sector}= \frac{\theta}{360}\pi r^2\\ \large A_{sector}= \frac{50}{360}\pi \times 12^2\\ \large A_{sector}=20\pi\\ \large A_{sector}\approx62.831...\\ \large A_{sector}\approx62.8cm^2 \)</span></p> <p>b) The area of the triangle is</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{triangle}= \frac{1}{2} ab \sin C\\ \large A_{triangle}= \frac{1}{2} r^2 \sin \theta\\ \large A_{triangle}= \frac{1}{2} \times12^2 \times\sin 50°\\ \large A_{triangle}\approx55.155...\\ \large A_{triangle}\approx55.2cm^2 \)</span></p> <p>c) The area of the segment = area of sector - area of triangle</p> <p><span style="color:#FF0000;">Be careful to use a higher degree of accuracy for parts a) and b) to give the final answer correct to 3 s.f.</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A_{sector}\approx62.831...-55.155...\\ \large A_{sector}\approx7.68cm^2\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="928"> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq2.png" style="width: 300px; height: 310px; float: right;"><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a circle with centre O and radius <strong><em>r</em></strong> cm</p> <p>The area of the shaded sector OAB is <span class="math-tex">\(\large \frac{40\pi}{3}\)</span> cm²</p> <p>The length of the minor arc AB is <span class="math-tex">\(\large \frac{10\pi}{3}\)</span> cm</p> <p>a) Find the radius of the circle</p> <p>b) Find the angle <span class="math-tex">\(\large \theta\)</span> , in radians</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Write an equation for the arc length and another equation for the sector area.</p> <p><content>Solve the equations simultaneously.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) The area of the shaded sector OAB is <span class="math-tex">\(\large \frac{40\pi}{3}\)</span> cm²</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A=\frac{1}{2}r^2\theta\\ \large \frac{40\pi}{3}=\frac{1}{2}r^2\theta\\\)</span></p> <p>The length of the minor arc AB is <span class="math-tex">\(\large \frac{10\pi}{3}\)</span> cm</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large l=r\theta\\ \large \frac{10\pi}{3}=r\theta\)</span></p> <p>Let's solve these equations by substituting for <span class="math-tex">\(\large r\theta\)</span> into the area equation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r^2\theta=\frac{40\pi}{3}\\\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r\times r\theta=\frac{40\pi}{3}\\\)</span><span style="color:#FF0000;"></span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r\times \frac{10\pi}{3}=\frac{40\pi}{3}\\\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r=4\)</span></p> <p style="margin-left: 40px;"><em>r</em> = 8 cm</p> <p>We can now find the angle by substituting this value into the length equation</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{10\pi}{3}=8\times\theta\\ \large \theta=\frac{10\pi}{24}\\ \large \theta=\frac{5\pi}{12}\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="929"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a circle with centre O and radius 5cm and another circle with centre P and radius <strong><em>r</em></strong>. The two circles overlap meeting at points A and B. <span class="math-tex">\(\large AÔP=45°\)</span> and <span class="math-tex">\(\large A\hat{P}O=30°\)</span></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3_1.png" style="width: 500px; height: 320px;"></p> <p>a) Show that <span class="math-tex">\(\large r=5\sqrt{2}\)</span> cm</p> <p>b) <strong><em>Hence</em></strong>, show that the shaded area bounded by the two circles is <span class="math-tex">\(\large \frac{25}{12}(7\pi-6-6\sqrt3)\)</span> cm²</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) You can use the Sine Rule to find <strong><em>r</em></strong>.</p> <p><content>b) You should divide the shaded area into two parts. Work out the green shaded area and blue shaded area separately.</content></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3a.png" style="width: 400px; height: 263px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a)</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3_ans_a.png" style="width: 600px; height: 341px;"></p> <p>b) Divide the shaded area into two parts.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3a.png" style="width: 400px; height: 263px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans2.png" style="width: 600px; height: 152px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans3.png" style="width: 600px; height: 207px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq3ans4.png" style="width: 600px; height: 247px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="930"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The following diagram shows a circle with centre O and radius <strong><em>r</em></strong>. A and B are points on the circumference of the circle and <span class="math-tex">\(\large A\hat{O} B =\theta\)</span> radians</p> <p>The area of the green shaded region is three times greater than the area of the blue region.</p> <p>a) Show that <span class="math-tex">\(\large \sin \theta=\frac{4\theta-2\pi}{3}\)</span></p> <p>b) Find the value of <span class="math-tex">\(\large \theta\)</span> , giving your answer correct to 3 significant figures.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4_1.png" style="width: 350px; height: 369px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) The area of the blue segment = area of sector - area of triangle</p> <p><content>b) Use your graphical calculator to solve this equation.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a)</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4a.png" style="width: 300px; height: 316px; float: left;"></p> <p>area of the blue segment = area of sector - area of triangle</p> <p><span style="color:#0000CD;"><span class="math-tex">\(\large A_{segment}=\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta\)</span></span></p> <hr class="hidden-separator"> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4b.png" style="float: left; width: 300px; height: 296px;">The angle subtended by the the green sector is <span class="math-tex">\(\large 2\pi-\theta\)</span></p> <p>Area of the green sector is</p> <p><span style="color:#008000;"><span class="math-tex">\(\large A_{sector}=\frac{1}{2}r^2(2\pi-\theta)\)</span></span></p> <hr class="hidden-separator"> <p>The area of the green sector = three times the area of the blue segment</p> <p style="margin-left: 40px;"><span style="color:#008000;"><span class="math-tex">\(\large \frac{1}{2}r^2(2\pi-\theta)\)</span></span> = 3 <span style="color:#0000CD;"><span class="math-tex">\(\large (\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin\theta)\)</span></span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{1}{2}r^2(2\pi-\theta)=\frac{3}{2}r^2(\theta-\sin\theta)\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large 2\pi-\theta=3(\theta-\sin\theta)\\ \large 2\pi-\theta=3\theta-3\sin\theta\\ \large 3\sin\theta=4\theta-2\pi\\ \large \sin\theta=\frac{4\theta-2\pi}{3}\)</span></p> <p>b) We can solve this equation using our graphical equation.You can plot the graphs of <span class="math-tex">\(\large y=\sin\theta\\ \)</span> and <span class="math-tex">\(\large y=\frac{4\theta-2\pi}{3}\)</span> and find the point of intersection</p> <p style="margin-left: 40px;"><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4c.png" style="width: 300px; height: 173px;"></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \theta \approx2.18\)</span></p> <p>The diagram looks approaximately like the following</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq4d.png" style="width: 300px; height: 283px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="931"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a circle with centre O and radius <strong><em>r</em></strong>. Points A and B lie on the circumference of the circle and <span class="math-tex">\(\large A\hat{O}B=\theta\)</span> radians. The tangents to the circle A and B intersect at C.</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5.png" style="width: 450px; height: 376px;"></p> <p>a) Show that <span class="math-tex">\(\large AC=r\tan (\frac{\theta}{2})\)</span></p> <p>b)<strong><em> Hence</em></strong>, find the value of <span class="math-tex">\(\large \theta\)</span> when the two shaded regions have an equal area.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>b) Area of red region = Area of kite OACB - Area of sector OAB</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5a.png" style="width: 350px; height: 292px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Triangle OAC is a right angled triangle, so we can use right-angled triangle trigonometry to work out the length AC (opposite to the angle <span class="math-tex">\(\large\frac{\theta}{2}\)</span></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5c.png" style="width: 300px; height: 204px;"><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5d.png" style="width: 250px; height: 149px;"></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \tan\frac{\theta}{2}=\frac{AC}{r}\)</span></p> <p style="margin-left: 40px;"><span class="math-tex">\(\large AC=r\tan\frac{\theta}{2}\)</span></p> <p>b)</p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5e.png" style="width: 600px; height: 211px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5f.png" style="width: 600px; height: 210px;"></p> <p><img alt="" src="../../files/trigonometry/radians%2C-arcs-and-sectors/esq5g.png" style="width: 600px; height: 254px;"></p> </section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Right-angled Trigonometry</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="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sine and Cosine Rule</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="535"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The following diagram shows a quadrilateral ABCD.</p> <p><img alt="" src="../../files/trigonometry/triangle-geometry/esq1.jpg" style="width: 300px; height: 353px;"></p> <p>AB = 7cm , AD = 5cm ∠DAB=120° , ∠DBC=45° , ∠BCD=60°</p> <p>BD = <span class="math-tex">\(\sqrt{a}\)</span></p> <p>CD = <span class="math-tex">\(\sqrt{b}\)</span></p> <p><span class="math-tex">\(a,b \in \mathbb{Z}\)</span></p> <p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p> </div> <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">You will need to use both the Cosine Rule and the Sine Rule in this question.<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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine1.pdf" width="640"></iframe></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="536"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The following diagram shows a quadrilateral ABCD.</p> <p><img alt="" src="../../files/trigonometry/triangle-geometry/esq2.jpg" style="width: 400px; height: 201px;"></p> <p>AD = x – 1 , BD = x + 1 , DC = 2x and <span class="math-tex">\(\angle CDA\)</span> = 120°</p> <p>The sum of the area of triangle ADC and triangle BDC is <span class="math-tex">\(4 \sqrt{3}\)</span></p> <p>Find <strong><em>x</em></strong></p> </div> <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"><span class="math-tex">\(sin\theta = sin(180-\theta)\)</span><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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine2.pdf" width="640"></iframe></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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="534"> <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>In a triangle ABC, AB = 8cm, BC = a, AC = b and <span class="math-tex">\(\angle BAC\)</span> = 30°</p> <p>a) Show that <span class="math-tex">\(b^2-8\sqrt{3}b+64-a^2=0\)</span></p> <p>b) Hence find the possible values of <strong><em>a</em></strong> (in cm) for which the triangle has two possible solutions.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Use the cosine rule.</p> <p>The following applet might help you visualise the triangle. Drag the slider to see the different possible values that a can take</p> <p><content> </content></p> <p style="text-align: center"><iframe height="406px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/gctxceqp/width/762/height/406/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="cosine rule ESQ3" width="762px"></iframe></p> <p> </p> <p><content></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/triangle-geometry/esq_trigonometry_sine-cosine3.pdf" width="640"></iframe></p> </section> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Unit Circle</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="527"> <p><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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>Given that <span class="math-tex">\(cosx=-\frac{\sqrt{7}}{3}\)</span> and <span class="math-tex">\(\frac{\pi}{2}\le x\le \pi\)</span> , find the possible values of sinx and tanx</p> </div> <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>Draw a circle.</p> <p><content><strong><em>x</em></strong><em> </em>is in the second quadrant</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/unit-circle/esq_unit_circle1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/esq_unit_circle1.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="528"> <p><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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>If <span class="math-tex">\(tanx=\frac{12}{5}\)</span> and <span class="math-tex">\(\pi\le x\le \frac{3\pi}{2}\)</span> , find the value of cosx</p> </div> <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>Draw a circle.</p> <p><content><strong><em>x</em></strong><em> </em>is in the third quadrant</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/unit-circle/esq_unit_circle2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/esq_unit_circle2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <p></p> </div> </div> <div class="panel panel-default panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Trigonometric Graphs</p> </div> </div> <div class="panel-body"> <div> <div class="panel 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> <div class="smart-object center" data-id="943"> <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>The height <strong><em>h</em></strong> of water, in metres, in a habour is modelled by the function <span class="math-tex">\(\large h(t)=5.5\sin(0.5(t-1.5))+12\)</span> where <strong><em>t</em></strong> is time after midday in hours.</p> <p>a) Find the initial height of the water.</p> <p>b) At what time is it when the water reaches this height again?</p> <p>c) Find the maximum height of the water.</p> <p>d) How much time is there in between the first and second time that the water at 16 metres?</p> <p>Give heights to 3 significant figures and times to the nearest minute</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>This question gets you to use your calculator to solve problems.</p> <p>Ensure that your calculator is in radian mode.</p> <p>Adjust the view window so that you get a good view of the graph</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2h1.png" style="width: 268px; height: 154px;"></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) 8.25 metres</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_1.png" style="width: 268px; height: 154px;"></p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_2.png" style="width: 268px; height: 154px;"> <img alt="" src="../../files/trigonometry/trig-graphs/esq2_3.png" style="width: 268px; height: 154px;"></p> <p>b) 21:17 or 9.17 pm</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_4.png" style="width: 268px; height: 154px;"> 0.283x60 = 17 minutes</p> <p>c) 4.64 meters</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_5.png" style="width: 268px; height: 154px;"></p> <p>d) 3 hours 2 mins</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq2_6.png" style="width: 268px; height: 154px;"> <img alt="" src="../../files/trigonometry/trig-graphs/esq2_7.png" style="width: 268px; height: 154px;"></p> <p>6.155 - 3.129 = 3.026</p> <p>0.026 x 60 = 2 mins</p> </section> <h4> </h4> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="942"> <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>The following diagram shows a Ferris wheel.</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/ferris2.png" style="width: 250px; height: 257px;"></p> <p>The height, <strong><em>h</em></strong> metres of a seat above ground after <strong><em>t</em></strong> minutes is given by <span class="math-tex">\(\large h(t)=a\ \cos(bt)+c\)</span> , where <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c <span class="math-tex">\(\large \in \mathbf{R}\)</span></em></strong></p> <p>The following graph shows the height of the seat.</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq1.png" style="width: 500px; height: 383px;"></p> <p>Find the values of <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></strong></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>amplitude is <strong><em>|a|</em></strong></p> <p><content></content>period = <span class="math-tex">\(\large \frac{2\pi}{b}\)</span></p> <p>vertical shift = <strong><em>c</em></strong></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/trig-graphs/esq1ans.png" style="width: 400px; height: 317px;"></p> <p>amplitude is <strong><em>|a|</em></strong><em> = 6 , <strong>a = -6</strong></em></p> <p><content></content>period = <span class="math-tex">\(\large \frac{2\pi}{b}=5\\ \large b = \frac{2\pi}{5}\)</span></p> <p>vertical shift = <strong><em>c = 8</em></strong></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="944"> <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>Consider a function <strong><em>f</em></strong>, such that <span class="math-tex">\(\large f(x) = 5.6\cos(\frac{\pi}{a}(x-1))+b\)</span> , <span class="math-tex">\(\large 0\le x\le 15\)</span>, <span class="math-tex">\(\large a,b\in \mathbf{R}\)</span></p> <p>The function <strong><em>f</em></strong> has a local maximum at the point (1 , 8.8) and a local minimum at the point (10 , -2.4)</p> <p>a) Find the period of the function</p> <p>b) <strong>Hence</strong>, find the value of <strong><em>a</em></strong>.</p> <p>c) Find the value of <strong><em>b</em></strong>.</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 <span class="math-tex">\(\large f(x)=a \cos(b(x+c))+d\)</span></p> <p>period = <span class="math-tex">\(\large \frac{2\pi}{b}\)</span><content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) period = 18</p> <p><img alt="" src="../../files/trigonometry/trig-graphs/esq3.png" style="width: 500px; height: 223px;"></p> <p>b) period = <span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{a}}\)</span></p> <p><span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{a}}=18\\ \large 2a = 18\\ \large a = 9\)</span></p> <p>c) Use (1 , 8.8)</p> <p><span class="math-tex">\(\large5.6\cos(\frac{\pi}{9}(1-1))+b=8.8\\ \large5.6\cos(0)+b=8.8\\ \large5.6+b=8.8\\ \large b=3.2\)</span></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="945"> <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>Consider a function <strong><em>f</em></strong>, such that <span class="math-tex">\(\large f(x) = a\sin(\frac{\pi}{15}(x+2))+b\)</span> , <span class="math-tex">\(\large a,b\in \mathbf{R}\)</span></p> <p>The function <strong><em>f</em></strong> has passes through the points (10.5 , 5.5) and (15.5 , 2.5)</p> <p>Find the value of <em><strong>a</strong></em> and the value of <strong><em>b</em></strong></p> <p> </p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <p> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Substitute the values (10.5 , 5.5) and (15.5 , 2.5) into the function.</p> <p>This will give you two equations for the unknowns <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <p>Solve these equations</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><span class="math-tex">\(\large f(x) = a\cos(\frac{\pi}{15}(x+2))+b\)</span></p> <hr class="hidden-separator"> <p>(10.5 , 5.5)</p> <p><span class="math-tex">\(\large a\sin(\frac{\pi}{15}(10.5+2))+b=5.5\\ \large a\sin(\frac{12.5}{15}\pi)+b=5.5\\ \large a\sin(\frac{25}{30}\pi)+b=5.5\\ \large a\sin(\frac{5}{6}\pi)+b=5.5\\ \large a(0.5)+b=5.5\\ \large a+2b=11\)</span></p> <hr class="hidden-separator"> <p>(15.5 , 2.5)</p> <p><span class="math-tex">\(\large a\sin(\frac{\pi}{15}(15.5+2))+b=2.5\\ \large a\sin(\frac{17.5}{15}\pi)+b=2.5\\ \large a\sin(\frac{35}{30}\pi)+b=2.5\\ \large a\sin(\frac{7}{6}\pi)+b=2.5\\ \large a(-0.5)+b=2.5\\ \large -a+2b=5\)</span></p> <hr class="hidden-separator"> <p>a + 2b = 11</p> <p>-a + 2b = 5</p> <hr class="hidden-separator"> <p>4b = 16</p> <p><strong>b = 4</strong></p> <p><strong>a= 3</strong></p> <hr class="hidden-separator"></section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-expandable panel-has-border"> <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="946"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows a ball attached to the end of a spring.<br> <img alt="" src="../../files/trigonometry/trig-graphs/esq5.png" style="width: 350px; height: 392px;"></p> <p>The height, <strong><em>h</em></strong>, in mtres of the ball above the ground <strong><em>t</em></strong> seconds after being released can be modelled by the function</p> <p><span class="math-tex">\(\large h(t)=a\cos(\frac{\pi}{b}t)+c\)</span> , <span class="math-tex">\(\large a,b, c\in \mathbf{R}\)</span></p> <p>The ball is release from an initial height of 4 metres.</p> <p>After <span class="math-tex">\(\large \frac{4}{3}\)</span> seconds, the ball is at a height of 1.6 metres.</p> <p>It takes the ball 4 seconds to return to its initial height.</p> <p>Find the values of <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></strong>.</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>If it takes 4 seconds to return to the initial height, then the period = 4</p> <p><content>Use this information to work out <em><strong>b</strong></em></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>If it takes 4 seconds to return to the initial height, then the period = 4</p> <p><span class="math-tex">\(\large \frac{2\pi}{\frac{\pi}{b}}=4\\ \large b=2\)</span></p> <hr class="hidden-separator"> <p><span class="math-tex">\(\large h(t)=a\cos(\frac{\pi}{2}t)+c\)</span></p> <p>When t = 0, h = 4</p> <p><span class="math-tex">\(\large a\cos(0)+c=4\\ \large a+c=4\)</span></p> <hr class="hidden-separator"> <p>When t = <span class="math-tex">\(\large \frac{4}{3}\)</span>, h = 1.6</p> <p><span class="math-tex">\(\large a\cos(\frac{\pi}{2}(\frac{4}{3}))+c=1.6\\ \large a\cos(\frac{2\pi}{3})+c=1.6\\ \large a(-0.5)+c=1.6\\ \large -a+2c=3.2\)</span></p> <hr class="hidden-separator"> <p>a + c = 4</p> <p>-a + 2c = 3.2</p> <p>3c = 7.2</p> <p><strong>c = 2.4</strong></p> <p><strong>a = 1.6</strong></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Pythagorean Identities</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="panel-body"> <div class="smart-object center" data-id="910"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Show that the equation <span class="math-tex">\(\large 2 \sin^2x=3 \cos x\)</span> may be written in the form</p> <p><span class="math-tex">\(\large 2 \cos^2x+3 \cos x-2=0\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large 2 \sin^2x=3 \cos x\)</span> , for <span class="math-tex">\(\large 0\le x\le2\pi\)</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>a) Use <span class="math-tex">\(\large \cos^2 x+\sin ^2x\equiv1\)</span><content></content></p> <p>b) Use the answer from part a). This is a quadratic equation.</p> <p>Don't forget that <span class="math-tex">\(\large \cos^2 x\)</span> means <span class="math-tex">\(\large (\cos x)^2\)</span></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/trig-identities/pythag-identities-sl/esq1a.png" style="width: 600px; height: 229px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq1b_1.png" style="width: 600px; height: 295px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq1b_2.png" style="width: 600px; height: 142px;"></p> </section> <h4> </h4> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="panel-body"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Given that <span class="math-tex">\(x=\frac{2}{cos\theta}\)</span> and <span class="math-tex">\(y=3tan\theta\)</span></p> <p>show that <span class="math-tex">\(\frac{x^2}{4}-\frac{y^2}{9}=1\)</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"><content> </content><span class="math-tex">\(\tan^2\theta=\frac{sin²\theta}{cos²\theta}\)</span></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-identities/esq_trig_identies1.pdf" target="_blank">here</a></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-expandable panel-has-border"> <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="909"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The following diagram shows triangle ABC with AB = 4 and AC = 5</p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/q3-question.png" style="width: 350px; height: 209px;"><strong>DIAGRAM NOT TO SCALE</strong></p> <p>a) Given that <span class="math-tex">\(\large \sin \hat A=\frac{3}{4}\)</span>, find the value of <span class="math-tex">\(\large \cos \hat A\)</span></p> <p>b) <strong>Hence</strong>, show that the length of <span class="math-tex">\(\large BC=\sqrt{41-10\sqrt{7}}\)</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>a) You can use the Pythagorean identity <span class="math-tex">\(\large \cos^2\theta+\sin^2\theta\equiv1\)</span><content></content>, to find <span class="math-tex">\(\large \cos \hat A\)</span></p> <p>b) Use the Cosine Rule, <em>c</em>² = a² + <em>b</em>² - 2<em>ab</em> cos<em>C</em></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/trig-identities/pythag-identities-sl/esq3a.png" style="width: 600px; height: 293px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq3b.png" style="width: 600px; height: 360px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="911"> <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>Prove that <span class="math-tex">\(\large \frac{\sin ^3\theta}{\tan \theta}+\cos^3\theta\equiv\cos\theta\)</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>This is a very difficult proof for SL students!</p> <p>There is always more than one way to carry out this proof.</p> <p><content>The easiest, is to start with the left hand side and consider that <span class="math-tex">\(\large \tan\theta\equiv\frac{\sin\theta}{\cos\theta}\)</span></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/trig-identities/pythag-identities-sl/esq4_1.png" style="width: 600px; height: 319px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq4_2.png" style="width: 600px; height: 313px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq4_3.png" style="width: 600px; height: 246px;"></p> </section> <h4> </h4> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Double Angle Formula</p> </div> </div> <div class="panel-body"> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>Let f(x) = (cos2x - sin2x)²</p> <p>a) Show that f(x) can be expressed as 1 - sin4x</p> <p>b) Let f(x) = 1 - sin4x. Sketch the graph of <strong><em>f</em></strong> for <span class="math-tex">\(0\le x\le \pi \)</span></p> <p><img alt="" src="../../files/trigonometry/trig-identities/esq2.png" style="width: 600px; height: 388px;"></p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Expand the brackets</p> <p>Consider the following identities <span class="math-tex">\(cos^{ 2 }\theta +sin^{ 2 }\theta \equiv 1\\ sin2\theta \equiv 2sin\theta cos\theta \)</span></p> <p>b) In order to sketch this function, consider the transformations from the graph of y = sinx</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-identities/esq_trig_identies2.pdf" target="_blank">here</a></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <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>Solve <span class="math-tex">\(cos2θ=sinθ\)</span> for <span class="math-tex">\(0\le \theta \le 2\pi \)</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">Use the identity <span class="math-tex">\(cos2\theta \equiv 1 -2sin^2\theta\)</span><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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-identities/esq_trig_identies4.pdf" target="_blank">here</a></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div> <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>a) Show that <span class="math-tex">\(cos2\theta-3cos\theta+2\equiv 2{ cos }^{ 2 }\theta -3cos\theta +1\)</span></p> <p>b) <strong>Hence</strong><em>, solve <span class="math-tex">\(cos2\theta-3cos\theta+2=0\)</span> for <span class="math-tex">\(0\le \theta \le 2\pi \)</span></em></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"><content> </content>a) Use the following identity <span class="math-tex">\(cos2\theta \equiv 2cos^2\theta -1\)</span></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-identities/esq_trig_identies5.pdf" target="_blank">here</a></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <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>Let <span class="math-tex">\(cos\theta=\frac{2}{3}\)</span>, where <span class="math-tex">\(0\le \theta \le \frac { \pi }{ 2 } \)</span></p> <p>Find the value of</p> <p>a) <span class="math-tex">\(sin\theta\)</span></p> <p>b) <span class="math-tex">\(sin2\theta\)</span></p> <p>c) <span class="math-tex">\(sin4\theta\)</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>a) Draw a triangle and work out <span class="math-tex">\(sin\theta\)</span></p> <p><content>b) Use the identity <span class="math-tex">\(sin2\theta \equiv 2sin\theta cos\theta \)</span></content></p> <p>c) Work out <span class="math-tex">\(cos2\theta\)</span> first. <span class="math-tex">\(2\theta\)</span> is obtuse</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-identities/esq_trig_identies3.pdf" target="_blank">here</a></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </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>Solving Trigonometric Equations</p> </div> </div> <div class="panel-body"> <div> <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> <div class="smart-object center" data-id="532"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let f(x)= cosx and g(x) = <span class="math-tex">\(\frac{2x^2}{1-x}\)</span></p> <p>a) Show that g∘f(x) = 1 can be written as 2cos²x + cosx - 1 = 0</p> <p>b) Hence solve g∘f(x)=1 for <span class="math-tex">\(-\pi\le x\le \pi\)</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>a) Start by finding the compostive function g∘f(x).</p> <p>b) This is a quadratic equation. Some students find it easier to factorise by substituting y = cosx and solving for y first.</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-equations/esq_esq_trig_equ1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_esq_trig_equ1.pdf" width="640"></iframe></p> </section> </div> </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> <div class="smart-object center" data-id="531"> <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>Solve <span class="math-tex">\(\log _{ 3 }{ sinx-\log _{ 3 }{ cosx=0.5 } } \)</span> for <span class="math-tex">\(0\le x\le 2\pi\)</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>Use log laws to simlify equation: <span class="math-tex">\(\log _{ a }{ \frac { x }{ y } =\log _{ a }{ x } -\log _{ b }{ y } } \)</span></p> <p><content>and convert log equation into index equation <span class="math-tex">\(a^x=b⇔x=log_ab\)</span></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-equations/esq_trig_equ2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_trig_equ2.pdf" width="640"></iframe></p> </section> </div> </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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="533"> <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>1 + cosx + cos²x + cos<sup>3</sup>x + ... = <strong><span class="math-tex">\(2 + \sqrt2\)</span></strong></p> <p>Find <strong><em>x</em></strong> given that <span class="math-tex">\(-\frac {\pi}{2}\le x\le \frac {\pi}{2}\)</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>1 + cosx + cos²x + cos<sup>3</sup>x + ... is an infinite geometric series. Can you find the sum?</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/trigonometry/trig-equations/esq_esq_trig_equ3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/trig-equations/esq_esq_trig_equ3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2907"> <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 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