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id="main-column"> <h1 class="page_title"> Geometry and Trigonometry Examination Questions HL <a href="#" class="mark-page-favorite pull-right" data-pid="2908" 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 HL</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="530"> <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/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 cotx</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_circle1hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/esq_unit_circle1hl.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </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 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="529"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <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 cosecx</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>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_circle2hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/trigonometry/unit-circle/esq_unit_circle2hl.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </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-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="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 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="912"> <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/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">\(\large \text{cosec}^2x-\cot ^2x\equiv1\)</span></p> <p>b)<strong> Henc</strong>e, prove that <span class="math-tex">\(\large \text{cosec}^4x-\cot^4x\equiv \text{cosec}^2x+\cot ^2x\)</span></p> <p>c) Given that <span class="math-tex">\(\large \text{arctan}(2)\approx63.4°\)</span>, solve</p> <p style="margin-left: 120px;"><span class="math-tex">\(\large \text{cosec}^4x-\cot^4x=2-\cot x\)</span> , for <span class="math-tex">\(\large 0 \le x \le360°\)</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 the identity <span></span><span class="math-tex">\(\large 1+\cot^{ 2 }\theta \equiv \text{cosec}^{2}\theta\)</span><span></span></p> <p><content>b) Notice that the left-hand side of the identity is the difference of two squares</content></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large a^2-b^2\equiv (a-b)(a+b)\\ \)</span></p> <p style="margin-left: 80px;"><span class="math-tex">\(\large a^4-b^4\equiv (a²-b²)(a²+b²)\\ \)</span></p> <p>c) Use your answer in part b) to write a quadratic equation for cot x. Again, you will need the identity <span class="math-tex">\(\large 1+\cot^{ 2 }\theta \equiv \text{cosec}^{2}\theta\)</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/esq5a.png" style="width: 600px; height: 97px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq5b.png" style="width: 600px; height: 220px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq5c.png" style="width: 600px; height: 234px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq5c_2.png" style="width: 600px; height: 182px;"></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="913"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Prove that <span class="math-tex">\(\large \frac{1-\tan^2x}{1+\tan^2x}\equiv\cos2x\)</span></p> <p>b) <strong>Hence</strong>, show that</p> <p style="margin-left: 80px;"><span class="math-tex">\(\large \tan\frac{\pi}{8}=\sqrt{3-2\sqrt{2}}\)</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>a) Start with the left-hand side of the identity.</p> <p style="margin-left: 40px;">Use <span class="math-tex">\(\tan x\equiv\frac{\sin x}{\cos x}\)</span><content></content></p> <p>b) The term <strong>hence</strong> is important here. You need to use the result proved in part a)</p> <p style="margin-left: 40px;">Try substituting <span class="math-tex">\(\large x=\frac{\pi}{8}\)</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/esq6a_1.png" style="width: 600px; height: 293px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq6a_2.png" style="width: 600px; height: 130px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq6b_1.png" style="width: 600px; height: 326px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/pythag-identities-sl/esq6b_2.png" style="width: 600px; height: 311px;"></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>Compound Angle Formulae</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="915"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>If <span class="math-tex">\(\large \sin A=\frac{4}{5}\)</span> , where <span class="math-tex">\(\large 0\le A\le\frac{\pi}{2}\)</span></p> <p>and <span class="math-tex">\(\large \cos B=-\frac{12}{13}\)</span> , where <span class="math-tex">\(\large \pi \le B\le\frac{3\pi}{2}\)</span></p> <p>work out <span class="math-tex">\(\large \cos(B-A)\)</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>You will need to work out cosA and sinB</p> <p><content>You will need to use the compound formula <span class="math-tex">\(\large\cos(B-A)\equiv\cos B\cos A+ \sin B \sin A\)</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/compound-angle/esq1a.png" style="width: 600px; height: 312px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq1b.png" style="width: 600px; height: 188px;"></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="916"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>If <span class="math-tex">\(\large \sin(x+30°)=2\cos(x+60°)\)</span>, then show that <span class="math-tex">\(\large \tan x=\frac{\sqrt{3}}{9}\)</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">You will need to remember the exact values for sin30°, cos30°, sin60° and cos60° <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/trig-identities/compound-angle/esq2a.png" style="width: 600px; height: 274px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq2b.png" style="width: 600px; height: 296px;"></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="917"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) By writing 15° as 45° - 30° , find the value of sin15°</p> <p>b) <strong>Hence</strong>, show that the area of this triangle <span class="math-tex">\(\large =4(\sqrt{3}-1)\)</span></p> <p style="margin-left: 40px;"><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq3q.png" style="width: 350px; height: 173px;"></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) We need to use the compound angle formula to evaluate sin(45°-30°)</p> <p><content>b) We need to use the area of the triangle formula. <span class="math-tex">\(\large A= \frac{1}{2}ab\sin C\)</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/compound-angle/esq3a.png" style="width: 600px; height: 180px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq3b.png" style="width: 600px; height: 311px;"></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="918"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large \frac{\sin(A+B)+\sin(A-B)}{\cos(A+B)+\cos(A-B)}=\tan A\)</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"><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq4a.png" style="width: 600px; height: 157px;"><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/trig-identities/compound-angle/esq4a.png" style="width: 600px; height: 157px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq4b.png" style="width: 600px; height: 254px;"></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="919"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that <span class="math-tex">\(\large \tan 3x\equiv \frac{3\tan x-\tan^3x}{1-3\tan^2x}\)</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 the compound angle formula to work out <span class="math-tex">\(\large \tan(2x + x)\)</span><content></content></p> <p>Very careful manipulation is required in this question. Take your time to make sure that you do not make a mistake.</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/compound-angle/esq5a.png" style="width: 600px; height: 261px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5b.png" style="width: 600px; height: 136px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5c.png" style="width: 600px; height: 266px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq5d.png" style="width: 600px; height: 44px;"></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="920"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Write <span class="math-tex">\(\large \cos4x\)</span> in terms of <span class="math-tex">\(\large\cos x\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large 8\cos^4x-8\cos^2x+1=\sin4x\)</span> , for <span class="math-tex">\(\large 0\le x\le\pi\)</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) Write <span class="math-tex">\(\large\cos4x\equiv \cos(2x+2x)\)</span> and use the double angle formula for <span class="math-tex">\(\large \cos2x\)</span><content></content> and <span class="math-tex">\(\large \sin2x\)</span></p> <p>b) Remember that <span class="math-tex">\(\large \tan x\equiv\frac{\sin x}{\cos x}\)</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><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6a.png" style="width: 600px; height: 230px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6b.png" style="width: 600px; height: 108px;"></p> <p>b)</p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6c.png" style="width: 600px; height: 235px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/compound-angle/esq6d.png" style="width: 600px; height: 234px;"></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 Formulae</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 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="914"> <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/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">\(\large \tan 2x \cot 2x\equiv \frac{2}{1-\tan^2x}\)</span></p> <p>b) <strong>Hence</strong>, solve <span class="math-tex">\(\large \tan 2x \cot 2x=3\)</span> , for <span class="math-tex">\(\large -\frac{\pi}{2}<x<\frac{\pi}{2}\)< span=""></x<\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">a) Use the double angle formula for tan2x<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/trig-identities/double-angle-sl/esq5a.png" style="width: 600px; height: 188px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/esq5b_1.png" style="width: 600px; height: 313px;"></p> <p><img alt="" src="../../files/trigonometry/trig-identities/double-angle-sl/esq5b_2.png" style="width: 600px; height: 151px;"></p> </section> <h4> </h4> </div> </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> <p> </p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Vectors - Scalar Product and Angles</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="337"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A line <span class="math-tex">\({ L }_{ 1 }\)</span> passes through A(2,0,-3) and B(4,3,2).</p> <p>a) Find the equation of the line <span class="math-tex">\({ L }_{ 1 }\)</span></p> <p>A second line <span class="math-tex">\({ L }_{ 2}\)</span> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 3 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 1 \\ -4 \\ k \end{matrix} \right) \)</span></p> <p>b) Given that <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span>are perpendicular, find k.</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><content> </content>a) Line <span class="math-tex">\({ L }_{ 1 }\)</span> is parallel to <span class="math-tex">\(\overrightarrow { AB }\)</span></p> <p>b) If <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span> are perpendicular, then the scalar product of the direction vectors = 0</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/vectors/vectors_angles/esqs/esq_vectors_angles3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles3.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="336"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span class="math-tex">\(\overrightarrow { AB }\)</span> and <span class="math-tex">\(\overrightarrow { AC }\)</span> are two vectors such that <span class="math-tex">\(\overrightarrow { AB } =\left( \begin{matrix} 3 \\ -1 \\ 2 \end{matrix} \right) \)</span> and <span class="math-tex">\(\overrightarrow { AC } =\left( \begin{matrix} 2 \\ 0 \\ 1 \end{matrix} \right) \)</span></p> <p>Find <span class="math-tex">\(\hat { BAC } \)</span> to the nearest degree.</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><content> </content><img alt="" src="../../files/vectors/vectors_angles/esqs/eqs2.jpg" style="width: 200px; height: 121px;"></p> <p><span class="math-tex">\(cos\theta =\frac { a\cdot b }{ \left| a \right| \left| b \right| } \)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="341"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The angle between the line <span class="math-tex">\({ L }_{ 1 }\)</span> and <span class="math-tex">\({ L }_{ 2 }\)</span> is <span class="math-tex">\(\frac{\pi}{2}\)</span>.</p> <p><span class="math-tex">\( { L }_{ 1 }:\quad \frac { x+2 }{ 3 } =2y+1=\frac { 5-z }{ 2 } \)</span></p> <p><span class="math-tex">\( { L }_{ 2 }: \quad x =\frac { y-2}{ 3} =kz \)</span></p> <p>Find k.</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><content> </content>It is a good idea to write the equations of the lines in the standard form</p> <p><span class="math-tex">\(\frac { x-{ x }_{ 0 } }{ l } =\frac { y-{ y }_{ 0 } }{ m } =\frac { z-{ z }_{ 0 } }{ n } \)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vectors_angles/esqs/esq_vectors_angles7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles7.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="339"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Find the angle between the planes <span class="math-tex">\({ \Pi }_{ 1 }\)</span> and <span class="math-tex">\({ \Pi }_{ 2 }\)</span> to the nearest degree.</p> <p><span class="math-tex">\({ \Pi }_{ 1 }: 2x-3y+z=0\)</span></p> <p><span class="math-tex">\({ \Pi }_{ 2 }: x+2y+5z=-4\)</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><content> </content>angle between two planes = angle between normals</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/vectors/vectors_angles/esqs/esq_vectors_angles5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="342"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value of x for which the vectors <span class="math-tex">\(\left( \begin{matrix} sinx \\ \sqrt{3} \\ 0 \end{matrix} \right) \)</span> and <span class="math-tex">\(\left( \begin{matrix} 4cosx \\-1\\ 2 \end{matrix} \right) \)</span>are perpendicular, <span class="math-tex">\(0\le x\le \frac { \pi }{ 2 } \)</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"><content> </content>sin2x=2sinxcosx</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/vectors/vectors_angles/esqs/esq_vectors_angles8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles8.pdf" width="640"></iframe></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-expandable panel-has-border"> <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="343"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>OABC is a parallelogram.</p> <p><span class="math-tex">\(\overrightarrow { OA } =\textbf{a}\)</span> <strong><em> </em></strong><span class="math-tex">\(\overrightarrow { OB } =\textbf{b}\)</span> <span class="math-tex">\(\overrightarrow { OC } =\textbf{a}+\textbf{b}\)</span></p> <p>Given that <span class="math-tex">\((\textbf{ a }+\textbf{ b })\cdot (\textbf{ a }-\textbf{ b })=0\)</span> what can you conclude</p> <h4> </h4> <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"> <ul> <li><content> </content>Draw a diagram!</li> <li>The distributive law holds for the scalar product</li> <li><span class="math-tex">\(\textbf{ a }\cdot \textbf{ a }={ \left| \textbf{a} \right| }^{ 2 }\)</span></li> </ul> </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/vectors/vectors_angles/esqs/esq_vectors_angles9.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles9.pdf" width="640"></iframe></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-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="344"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><img alt="" src="../../files/vectors/vectors_angles/esqs/pythagoras.jpg" style="float: right; width: 200px; height: 227px;"><a aria-labelledby="cke_468_label" href="javascript:void(0)" id="cke_469_uiElement" role="button" style="-moz-user-select: none;" title="OK"><span id="cke_468_label"></span></a>ACB is a right-angled triangle</p> <p><span class="math-tex">\(\overrightarrow { CB } = \textbf {a }\)</span> <span class="math-tex">\(\overrightarrow { AC } = \textbf {b }\)</span></p> <p>a) Write <span class="math-tex">\(\overrightarrow { AB } \)</span> in terms of <strong><em>a </em></strong>and <strong><em>b</em></strong></p> <p>b) Find <span class="math-tex">\(\textbf{ a }\cdot \textbf{ b }\)</span></p> <p>c) Show that <span class="math-tex">\({ \left| \textbf { a+b } \right| }^{ 2 }={ \left| \textbf { a } \right| }^{ 2 }+{ \left| \textbf{ b } \right| }^{ 2 }\)</span> and hence prove Pythagoras' Theorem.</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><span class="math-tex">\(\textbf{ a }\cdot \textbf{ a }={ \left| \textbf{a} \right| }^{ 2 }\)</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/vectors/vectors_angles/esqs/esq_vectors_angles10.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vectors_angles/esqs/esq_vectors_angles10.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </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>Vector Equation of a Line</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> <div class="smart-object center" data-id="326"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A line L passes through the points A(1,-1,3) and B(3,-4,4)</p> <p>Point C (x,y,1) also lies on the line L. Find x and y.</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">If the equation of the straight line is given by <span class="math-tex">\(\textbf{r}= \textbf{a}+\lambda \textbf{b}\)</span> then a certain value of <span class="math-tex">\(\lambda \)</span> will define the position of the position vector <span class="math-tex">\(\overrightarrow { OC } \)</span>. Find this value and use it to find x and y.</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/vectors/lines/esq_eqofline2hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline2hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="328"> <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-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>A line L passes through the points A(0,2,-4) and B(3,-3,2)</p> <p>Point C also lies on the line L. Find the coordinates of C given that <span class="math-tex">\(\left| \overrightarrow { AC } \right| =\left| \overrightarrow { AB } \right| \)</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>Draw a diagram!</p> <p>Find the equation of the straight line. What is the value of <span class="math-tex">\(\lambda \)</span> that defines the position of B? Think about what this value should be for C.</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/vectors/lines/esq_eqofline3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline3.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 class="smart-object center" data-id="329"> <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>A line L passes through the points A(0,2,-4) and B(3,-3,2)</p> <p>Point C also lies on the line L. Find the possible coordinates of C given that <span class="math-tex">\(\left| \overrightarrow { AC } \right| =2\left| \overrightarrow { AB } \right| \)</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>Draw a diagram!</p> <p>Is there just one answer?</p> <p>Find the equation of the straight line. What is the value of <span class="math-tex">\(\lambda \)</span> that defines the position of B? Think about what this value should be for C.</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/vectors/lines/esq_eqofline4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/esq_eqofline4.pdf" width="640"></iframe></p> </section> <h4> </h4> </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>Vectors - Intersection of Lines</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="453"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A line L<sub>1</sub> passes through the points P(-13,-6,1) and Q(3,2,-3).</p> <p>A second line L<sub>2</sub> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 9\\12 \\ 2 \end{matrix} \right) +s\left( \begin{matrix} -3 \\ 2\\4 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Show that <span class="math-tex">\(\overrightarrow{PQ}=\left( \begin{matrix} 16 \\ 8\\-4 \end{matrix} \right) \)</span></li> <li>Hence, write down the equation L<sub>1</sub> in the form <span class="math-tex">\(\textbf{r}=\textbf{a}+t \textbf{b}\)</span></li> <li>The lines L<sub>1</sub> and L<sub>2</sub> intersect at the point R. Find the coordinates of R.</li> </ol> <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">\(\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}\)</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/vectors/intersection_lines/esq_intersections1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections1.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> <div class="smart-object center" data-id="454"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div>The diagram shows quadrilateral ABCD with vertices A(6,0) , B(3,5) , C(-10,4) and D(1,-3).</div> <div><img alt="" src="../../files/vectors/intersection_lines/quadrilateral" style="width: 600px; height: 403px;"></div> <div> <ol style="list-style-type:lower-alpha;"> <li>Find <span class="math-tex">\(\overrightarrow{AC}\)</span></li> <li>Show that <span class="math-tex">\(\overrightarrow{BD}\)</span> is perpendicular to <span class="math-tex">\(\overrightarrow{AC}\)</span></li> <li>Write down the equation of the line (AC) in the form</li> <li>Write down the equation of the line (BD)</li> <li>The lines (AC) and (BD) intersect at E. Find the coordinates of E</li> </ol> </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>b. Use the scalar product to show that the lines are perpendicular</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/vectors/intersection_lines/esq_intersections2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-border panel-has-colored-body 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="455"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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 key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <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/vectors/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> <div class="smart-object center" data-id="455"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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 key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <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/vectors/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </div> <div class="smart-object center" data-id="455"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Two boats A and B, move so that a time <em><strong>t</strong></em> hours, their positions, in kilometres, are given by</p> <p><span></span><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} -2 \\ -12 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -4 \end{matrix} \right) \)</span><span></span></p> <p><span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 11 \\ -11 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 3 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the position where the two boats cross.</li> <li>Show that the boats do not collide.</li> </ol> <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 key to answering this question correctly is to make the assumption that they will have the same position at some time, but that <strong>the time is not the same for the two boats</strong>.</p> <p>Use t<sub>A</sub> and t<sub>B</sub></p> <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/vectors/intersection_lines/esq_intersections3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections3.pdf" width="640"></iframe></p> </section> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="456"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The line L<sub>1</sub> has equation <span></span><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\-1 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ -2\\k \end{matrix} \right) \)</span><span></span></p> <p>The line L<sub>2</sub> has equation <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2\\1 \\ -3 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -1\\-1 \end{matrix} \right) \)</span></p> <ol style="list-style-type:lower-alpha;"> <li>The point A(3,1,-1) lies on the line L<sub>1</sub>. Show that k = 4.</li> <li>Show that the lines and L<sub>1</sub> are L<sub>2</sub> perpendicular.</li> <li>Show that the lines and L<sub>1</sub> do not L<sub>2</sub> intersect.</li> <li>The point B lies on the line The point C has coordinates (2,1,-3). ABC forms an isosceles triangles with AC=BC. Find the coordinates of B.</li> </ol> <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>d. You should be able to work out the length of AC</p> <p>Consider that B lies on the line L<sub>1</sub> . Find the vector <span class="math-tex">\(\overrightarrow{BC}\)</span></p> <p><img alt="" src="../../files/vectors/intersection_lines/esq4.jpg" style="width: 435px; height: 258px;"><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/vectors/intersection_lines/esq_intersections4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection_lines/esq_intersections4.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>Vectors - Kinematics</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="352"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/vectors/kinematics/nearmiss.jpg" style="width: 400px; height: 209px;"></p> <p>During an air show, two planes, A and B, perform a manoeuvre in which their paths cross in a <em>near miss</em>. The two planes are flying at the same altitude.</p> <p><span class="math-tex">\(\textbf{ r }_{ A }=\left( \begin{matrix} 150 \\ 320 \end{matrix} \right) +t\left( \begin{matrix} 200 \\ 300 \end{matrix} \right) \)</span></p> <p><span class="math-tex">\(\textbf{ r }_{ B }=\left( \begin{matrix} 875 \\ 110 \end{matrix} \right) +t\left( \begin{matrix} -100 \\ 400 \end{matrix} \right) \)</span></p> <p>t = time in seconds. Distances are given in metres.</p> <p>a) Show that the two planes cross paths, but the planes do not collide</p> <p>b) Find the distance between the planes when t = 0.</p> <p>c) Show that the distance <strong>d </strong>between A and B at any time t can be given by the expression</p> <p style="margin-left: 40px;">d = <span class="math-tex">\(\sqrt { 100000{ t }^{ 2 }-477000t+569725 } \)</span></p> <p>d) To the nearest metre, find the closest distance that the two planes get to one another.</p> <h4> </h4> <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><img alt="" class="gifffer" data-gifffer="/media/mathsanalysis/files/vectors/kinematics/closest_distance.gif" style="width: 625px; height: 445px;"></p> <p>a) To decide if the planes collide, find time, t<sub>1</sub> when x<sub>A</sub>=x<sub>B</sub> . Then find time, t<sub>2</sub> when y<sub>A</sub>=y<sub>B</sub>. Show t<sub>1</sub> <span class="math-tex">\(\neq \)</span> t<sub>2</sub></p> <p>b) Use Pythagoras' Theorem to find the distance between the points</p> <p>d) Use you calculator!</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/vectors/kinematics/esq_vectors_kinematics1hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/kinematics/esq_vectors_kinematics1hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="353"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><em>Distances in this question are given in metres</em></p> <p>Two brothers, Orville and Wilbur are testing their model airplanes. The position of Orville's airplane <strong><em>t</em></strong> seconds after taking off from ground level is given by</p> <p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 12 \\ -19\\0 \end{matrix} \right) +t\left( \begin{matrix} -4 \\ 4\\3 \end{matrix} \right) \)</span></p> <p>a) Find the height of the plane after 4 seconds.</p> <p>b) Wilbur's airplane takes off <strong>after</strong> Orville's airplane <strong><em>s</em></strong> seconds after taking off is given by</p> <p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} -26 \\ 25\\0 \end{matrix} \right) +s\left( \begin{matrix} 2 \\ -4\\8 \end{matrix} \right) \)</span></p> <p>Find the angle between the two paths.</p> <p>c) The two airplanes collide at (-20,13,24). How long after Orville’s airplane takes off does Wilbur’s airplane take off?</p> <p>d) Find the speed of the two airplanes at the moment of the collision.</p> <h4> </h4> <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">d) the speed of the two airplanes does not change. Use the velocity vectors.</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/vectors/kinematics/esq_vectors_kinematics2hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/kinematics/esq_vectors_kinematics2hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="350"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><em>All distances in this question are in km.</em></p> <p>An interceptor missile, M<sub>1 </sub>is positioned at the origin. A missile, M<sub>2</sub> is launched from (-20,7) with velocity <span class="math-tex">\(\left( \begin{matrix} 3 \\ 1 \end{matrix} \right) \)</span> kms<sup>-1</sup> . M<sub>1</sub> is capable of twice the speed of M<sub>2</sub> . How many seconds later must the interceptor missile, M<sub>1</sub> be launched if it is to travel the <strong>shortest possible distance</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">To travel the shortest possible distance, M<sub>1</sub> must travel in a direction perpendicular to M<sub>2</sub> .</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/vectors/kinematics/esq_vectors_kinematics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/kinematics/esq_vectors_kinematics3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </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>Vector Product</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> <div class="smart-object center" data-id="362"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><em><strong>a</strong> </em>= <span class="math-tex">\(\left( \begin{matrix} 2 \\ 3 \\ -5 \end{matrix} \right) \)</span> and <strong><em>b </em></strong>= <span class="math-tex">\(\left( \begin{matrix} 3 \\ -2 \\ 4 \end{matrix} \right) \)</span></p> <p>Find <span class="math-tex">\(\textbf{a}\times \textbf{b}\)</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><span></span><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span><span></span></p> <p><strong><em>a </em></strong>x <strong><em>b </em></strong>is perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong>. Check your answer by showing <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{a}=0\)</span> and <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{b}=0\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vector_product/esq_vector_vproduct1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct1.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="363"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><em><strong>a</strong> </em>= 3<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + <strong><em>k</em></strong> and <strong><em>b </em></strong>= 2<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 2<strong><em>k</em></strong></p> <p>Find a unit vector that is perpendicular to <strong><em>a</em></strong> and <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><span></span>Use the vector product</p> <p><span class="math-tex">\(\textbf{v}\times \textbf{w}=\left( \begin{matrix} { \textbf{v} }_{ 1 } \\ { \textbf{v} }_{ 2 } \\ { \textbf{v} }_{ 3 } \end{matrix} \right) \times \left( \begin{matrix} { \textbf{w} }_{ 1 } \\ { \textbf{w} }_{ 2 } \\ { \textbf{w} }_{ 3 } \end{matrix} \right) =\left( \begin{matrix} { \textbf{v} }_{ 2 }{ \textbf{w} }_{ 3 }-{ \textbf{v} }_{ 3 }{ \textbf{w} }_{ 2 } \\ { \textbf{v} }_{ 3 }{ \textbf{w} }_{ 1 }-{ \textbf{v} }_{ 1 }{ \textbf{w} }_{ 3 } \\ { \textbf{v} }_{ 1 }{ \textbf{w} }_{ 2 }-{ \textbf{v} }_{ 2 }{ \textbf{w} }_{ 1 } \end{matrix} \right)\)</span></p> <p><span></span></p> <p>A unit vector is a vector with magnitude = 1</p> <p><strong><em>a </em></strong>x <strong><em>b </em></strong>is perpendicular to <strong><em>a</em></strong> and <strong><em>b</em></strong>. Check your answer by showing <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{a}=0\)</span> and <span class="math-tex">\((\textbf{a}\times \textbf{b})\cdot \textbf{b}=0\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vector_product/esq_vector_vproduct2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="364"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The area of a parallelogram formed by two adjacent vectors <strong><em>a </em></strong>and <strong><em>b</em></strong> is 7 square units.</p> <p><strong><em>a </em></strong>= <span class="math-tex">\(\left( \begin{matrix} -3 \\ 4 \\ k \end{matrix} \right) \)</span> <strong><em>b</em></strong> = <span class="math-tex">\(\left( \begin{matrix} 3 \\ -2 \\ -2 \end{matrix} \right) \)</span></p> <p>Find k</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><span>Area of parallelogram = </span> <span></span><span class="math-tex">\(\left| \textbf{a}\times \textbf{b} \right| \)</span><span></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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vector_product/esq_vector_vproduct3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="365"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Given that <span class="math-tex">\(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\)</span> show that <span class="math-tex">\(\textbf{v}- \textbf{w} \)</span> is parallel to <span class="math-tex">\(\textbf{u}\)</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><span>If <em><strong><span class="math-tex">\(\textbf{a}\times \textbf{b}=0\)</span> </strong></em>then <strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> <span class="math-tex">\(\textbf{u}\times \textbf{v} = \textbf{u}\times \textbf{w}\)</span></p> <p><span class="math-tex">\(\textbf{u}\times \textbf{v} - \textbf{u}\times \textbf{w}=0\)</span></p> <p><span class="math-tex">\(\textbf{u}\times ( \textbf{v} - \textbf{w})=0\)</span></p> <p>Since the vector product is zero, the vectors are parallel</p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="366"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) For any two vectors <strong><em>v </em></strong> and <strong><em>w</em></strong> prove Lagrange's Identity</p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{v}\times \textbf{w} \right| }^{ 2 }+{ \left( \textbf{v}\cdot \textbf{w} \right) }^{ 2 }={ \left| \textbf{v} \right| }^{ 2 }{ \left| \textbf{w} \right| }^{ 2 }\)</span></p> <p>b) <strong>Hence</strong>, find <span class="math-tex">\(\textbf{v}\cdot \textbf{w}\)</span> if</p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{v} \right| }=3\)</span></p> <p style="text-align: center;"><span class="math-tex">\({ \left| \textbf{w} \right| }=4\)</span></p> <p style="text-align: center;"><span class="math-tex">\(\textbf{v}\times \textbf{w} =\left( \begin{matrix} -1 \\ 2 \\ 3 \end{matrix} \right) \)</span><strong> </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>Look in formula booklet for formulae for <span class="math-tex">\({ \left| \textbf{v}\times \textbf{w} \right| }\)</span> and <span class="math-tex">\(\textbf{v}\cdot \textbf{w}\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/vector_product/esq_vector_vproduct6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct6.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="367"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The points A, B and C are given by the position vectors <strong><em>a</em></strong>, <strong><em>b</em></strong> and <strong><em>c</em></strong>.</p> <p>If A, B and C are collinear, show that</p> <p style="text-align: center;"><span class="math-tex">\(\textbf{b}\times \textbf{c}=\textbf{a}\times (\textbf{c}-\textbf{b})\)</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>If A, B and C are collinear, then <strong><em><span class="math-tex">\(\overrightarrow { AB } \)</span></em></strong><strong><em> </em></strong> is parallel to <strong><em><span class="math-tex">\(\overrightarrow { BC } \)</span></em></strong></p> <p><img alt="" src="../../files/vectors/vector_product/esqcollinear1.jpg" style="width: 400px; height: 309px;"></p> <p>What do you know about the vector product of parallel vectors?</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/vectors/vector_product/esq_vector_vproduct7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct7.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="452"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><strong><em>a</em></strong> and <strong><em>b</em></strong> are vectors</p> <p>Show that <span class="math-tex">\(|\textbf{a}×\textbf{b}|^{ 2 }+|\textbf{a} ∙\textbf{b}|^{ 2 }=(\textbf{a}\textbf{b})^{ 2 }\)</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"><span class="math-tex">\(|\textbf{a}×\textbf{b}|=\textbf{a}\textbf{b}sin\theta \)</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/vectors/vector_product/esq_vector_vproduct8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/vector_product/esq_vector_vproduct8.pdf" width="640"></iframe></p> </section> </div> </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>Vectors - Equations of Planes</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> <div class="smart-object center" data-id="317"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A plane has vector equation <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right) +\lambda \left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</span></p> <p>Show that the Cartesian equation of the plane is x - 5y + 3z + 9 = 0</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><content> </content>Find the normal to the plane by finding the vector product of <span class="math-tex">\(\left( \begin{matrix} 2 \\ 1 \\ 1 \end{matrix} \right)\)</span> and <span class="math-tex">\(\left( \begin{matrix} 3 \\ 0 \\ -1 \end{matrix} \right) \)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane4.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="314"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The coordinates of points A, B and C are given as (5,4,1) , (5,1,-2) and (1,-1,2) respectively.</p> <p>a) Find the equation of the plane that passes through A, B and C</p> <p>b) Find the equation of the plane that is perpendicular to AB and passes through C</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><content> </content>a) The normal to the plane can be found from <span class="math-tex">\(\overrightarrow { AB } \times \overrightarrow { AC } \)</span></p> <p>b) The normal to the plane is vector <span class="math-tex">\(\overrightarrow { AB }\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="315"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A plane has vector equation <span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 1 \\ 2 \\ 0 \end{matrix} \right) +\mu \left( \begin{matrix} -2 \\ 0 \\ 5 \end{matrix} \right) +\lambda \left( \begin{matrix} 0 \\ -4 \\ 5 \end{matrix} \right) \)</span></p> <p>a) Find the Cartesian equation of the plane</p> <p>b) The plane meets the x, y and z axes at A, B and C respectively. OABC forms a pyramid. Find the volume of the pyramid.</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><content> </content>a) The normal to the plane can be found from the vector product of the two directions and the point (1 , 2 , 0).</p> <p>b) The plane crosses the x axis when y = z = 0. Find the corrdinates of A. Similarly, find coordinates of B and C.</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/vectors/planes/eqplane2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane2.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="316"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the Cartesian equation of the plane that is perpendicular to the plane 2x - y + z = 8 and contains the points A(4,2,-3) and B(6,1,-1).</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><content> </content>The plane is perpendicular to the normal and the vector <span class="math-tex">\(\overrightarrow { AB } \)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/planes/eqplane3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/planes/eqplane3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </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>Vectors - Intersection of Planes</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> <div class="smart-object center" data-id="448"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the intersection of the planes <span class="math-tex">\({ \Pi }_{ 1 } \)</span> and <span class="math-tex">\({ \Pi }_{ 2 }\)</span> in the form <span class="math-tex">\(\textbf {r}=\textbf {a}+\lambda\textbf {b}\)</span> where the components of <strong><em>b</em></strong> are integers</p> <p style="margin-left: 80px;"><span class="math-tex">\({ \Pi }_{ 1 }:\quad \quad x+2y−z=5\\ { \Pi }_{ 2 }:\quad \quad 2x−y+3z=−4\)</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. Eliminate z and write y in terms of x</p> <p><content>2. Eliminate y and write z in terms of x</content></p> <p>3. Write parametric equations</p> <p style="margin-left: 120px;">x = ...</p> <p style="margin-left: 120px;">y = ...</p> <p style="margin-left: 120px;">z = ...</p> <p>4. Convert to vector form</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/vectors/intersection-of-planes/esq_vectors_intersectingplanes1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </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="449"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p style="margin-left: 80px;"><span class="math-tex">\(\ \ \ x + \ \ y +\ \ z =8\\ \ \ ax − \ y \ \ \quad \ \ \ =3\\ −x+3y+4z=b\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>There is no unique solution solution to the system of equations. Find the value of <em><strong>a</strong></em>.</li> <li>Given that the system can be solved, find the value of <em><strong>b</strong></em>.</li> </ol> <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><content> </content>1. Eliminate z by combining equation 1 and 3</p> <p>2. Equate the coefficients of y with equation from above and equation 2</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/vectors/intersection-of-planes/esq_vectors_intersectingplanes2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes2.pdf" width="640"></iframe></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="450"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The three planes <span class="math-tex">\({ \Pi }_{ 1 }\)</span> , <span class="math-tex">\({ \Pi }_{ 2 }\)</span> and <span class="math-tex">\({ \Pi }_{ 3 }\)</span> meet at a line</p> <p><span class="math-tex">\({ \Pi }_{ 1 }:\quad 2x+\ y+3z=a\\ { \Pi }_{ 2 }:\quad \ x−2y+2z=−9 \\ { \Pi }_{ 3 }:\quad 3x+4y+4z=−1\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find a</li> <li>Find the equation of the straight line in the form <span class="math-tex">\(\textbf {r}=\textbf {a}+\lambda\textbf {b}\)</span> where the components of <strong><em>b</em></strong> are integers</li> </ol> <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"> <ul> <li>Combine the equations to eliminate x</li> <li>Equate the coefficients of y and z</li> </ul> <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/vectors/intersection-of-planes/esq_vectors_intersectingplanes3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes3.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="451"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find the value of k which makes the following system of equations inconsistent:</p> <p style="margin-left: 80px;"><span class="math-tex">\(x +2y+kz=−1\\ 2x+ \ y− \ z=3\\ kx−2y+ z=1\)</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"> <p>Combine the equations to eliminate <strong><em>y</em></strong></p> <p><content>Equate coefficients of <strong><em>z</em></strong></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/vectors/intersection-of-planes/esq_vectors_intersectingplanes4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/intersection-of-planes/esq_vectors_intersectingplanes4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </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>Vectors - Intersection of Lines and Planes</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> <div class="smart-object center" data-id="458"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The points A and B are given by A(-8,1,-2) and B(-2,-1,2).</p> <p>A plane Π is defined by the equation <span class="math-tex">\(2x−y−3z=−8\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find a vector equation of the line L passing through the points A and B.</li> <li>Find the coordinates of the point of intersection of the line and the plane.</li> </ol> <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>To find the equation of the line, you will need to find <span class="math-tex">\(\overrightarrow{AB}\)</span></p> <p><span class="math-tex">\(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}\)</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><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/line-and-plane/esq_line_and_plane1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane1.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="459"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the plane <span class="math-tex">\(x−2y+4z=−15\)</span> and the line</p> <p><span class="math-tex">\(x=3+kλ\\ y=−2+λ\\ z=(2k+6)−2λ\)</span></p> <p>The line and the plane are perpendicular. Find</p> <ol style="list-style-type:lower-alpha;"> <li>The value of <strong><em>k</em></strong></li> <li>The coordinates of the point of intersection of the line and the plane.</li> </ol> <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 lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">If line and plane are <span style="font-weight:bold">perpendicular</span>, then line is <span style="font-weight:bold">parallel</span> to normal to the plane</p> <p><content><img alt="" src="../../files/vectors/line-and-plane/esq2.jpg" style="width: 300px; height: 199px;"> </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/vectors/line-and-plane/esq_line_and_plane2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane2.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-has-border panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="460"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span class="math-tex">\(Π_{ 1 }\)</span>and <span class="math-tex">\(Π_{ 2 }\)</span> are planes such that</p> <p><span class="math-tex">\(Π_{ 1 }:2x−y−2z=0\)</span></p> <p>and</p> <p><span class="math-tex">\(Π_{ 2 }:−2x+3y+3z=4 \)</span></p> <p><em><strong>L</strong></em> is the intersection of planes <span class="math-tex">\(Π_{ 1 }\)</span> and <span class="math-tex">\(Π_{ 2 }\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Find the equation of the line</li> </ol> <p>A third plane <span class="math-tex">\(Π_{ 3 }\)</span> is defined by the equation <span class="math-tex">\(kx+(k−1)y−z=5\)</span></p> <ol style="list-style-type:lower-alpha;"> <li value="2">Find the value of <strong><em>k </em></strong>such that the line <strong><em>L</em></strong> does <strong>not</strong> intersect with <span class="math-tex">\(Π_{ 3 }\)</span></li> </ol> <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 the line <strong><em>L</em></strong> does not intersect with <span class="math-tex">\(Π_{ 3 }\)</span> then they must be parallel</p> <p><img alt="" src="../../files/vectors/line-and-plane/esq3.jpg" style="width: 252px; height: 247px;"></p> <p>Here is a graph of the three planes and the line</p> <p><content> </content><img alt="" src="../../files/vectors/line-and-plane/esq3a.jpg" style="width: 500px; height: 390px;"></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/vectors/line-and-plane/esq_line_and_plane3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane3.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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="461"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The point A (3, 1, –2) is on the line L, which is perpendicular to the plane <span class="math-tex">\(2x−3y−z+9=0\)</span>.</p> <ol style="list-style-type:lower-alpha;"> <li>Find the Cartesian equation of the line L.</li> <li>Find the point R which is the intersection of the line L and the plane.</li> <li>The point A is reflected in the plane. Find the coordinates of the image of A.</li> </ol> <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 points A , R and A' lie on the straight line:</p> <p><content> </content><img alt="" src="../../files/vectors/line-and-plane/esq4.jpg" style="width: 400px; height: 401px;"></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/vectors/line-and-plane/esq_line_and_plane4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/line-and-plane/esq_line_and_plane4.pdf" 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