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href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1061/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">2.5 Trigonometric Models </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"> <div class="intro-card" readonly="false"><img class="intro-image" readonly="true" src="../../images/maarten-klumper-wq-lcy1wogs-unsplash-1.jpg"> <div class="content" readonly="true"> <p><strong>The Mathematics of Periodic Phenomena</strong></p> <p>Phenomena which display periodic nature, such as the motion of tides, the orbits of planets, or the spinning of a ferris wheel, that we can understand mathematically using trigonometic models. Once we develop a model, we can stride forwards to predict when something may happpen again in the future. Hence, in this unit we will learn about sine and cosine functions and how we can use these to develop models and make predictions. </p> </div> </div> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-17457ac1-7fff-e9b8-06b5-d76956ef1b5a"></span></p> <p dir="ltr"><span id="docs-internal-guid-17457ac1-7fff-e9b8-06b5-d76956ef1b5a">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li>Recognise real world phenomena which are periodic.</li> <li>Understand what sine and cosine are and where the sine and cosine graphs come from. </li> <li>Know about the key features of trigonometric graphs, such as the amplotude and period. Know how to transform trigonometric graphs. </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Be able to use a general trigonometric equation to solve applications problems. </p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Introducing Trigonometric Models </p> </div> </div> <div class="panel-body"> <div> <p>Why are trigonometric models so helpful for describing things that continuously repeat? An introductory video which looks at some real world phenomena using trigonometry.<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/531171651" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. The Unit Circle and Trigonometric Graphs</p> </div> </div> <div class="panel-body"> <div> <p>What exactly is sine and cosine? We'll use the unit circle to understand what they are, and where their graphs come from. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/531204613" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Intuitive Understanding of a Trigonometric Model </p> </div> </div> <div class="panel-body"> <div> <p>This video presents an intuitive understanding of transformations of the sine and cosine curve in relation to amplitude and period of a wave. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/532675947" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. General Form of a Trigonometric Model</p> </div> </div> <div class="panel-body"> <div> <p>A more technical analysis of transformations which provides the formulae you can use to find a trigonometriy model<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/532676346" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sumary</p> </div> </div> <div class="panel-body"> <div id="carousel-269" class="dynamic-gallery carousel slide" data-id="269"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-269/flashcard1.png" data-fancybox="gallery-269" title="" data-caption=""><img alt="" src="../../../std-galleries/15-269/flashcard1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-269/flashcard2.png" data-fancybox="gallery-269" title="" data-caption=""><img alt="" src="../../../std-galleries/15-269/flashcard2.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-269/flashcard3.png" data-fancybox="gallery-269" title="" data-caption=""><img alt="" src="../../../std-galleries/15-269/flashcard3.png"></a></div></div><a class="left carousel-control" href="#carousel-269" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-269" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-269/flashcard1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-269/flashcard2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-269/flashcard3-thumb128.jpg"><li></li></ol> <p> </p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <p>Quiz</p> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos and above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-575-2186"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>On a unit circle, <span class="math-tex">\(\sin\left(30\right)\)</span> represents...</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> The length of a point on the circumference at 30 degrees from (0, 0). </label></p><p><label class="radio"><input class="c" type="radio"> The height of a point on the circumference at 30 degrees from (0,0).</label></p></div><div class="q-explanation"><p>See answer given. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=\cos\left(x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>180 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>360 degrees</span></label> </p></div><div class="q-explanation"><p><img alt="" src="../../images/period.png" style="width: 400px; height: 166px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=\cos\left(2x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>360 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>180 degrees</span></label> </p></div><div class="q-explanation"><p>Utilising the formula, <span class="math-tex">\(Period=\frac{360}{b}\)</span>;</p><p><span class="math-tex">\(Period=\frac{360}{2}=180 ^{\circ}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=2\cos\left(x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>360 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>180 degrees</span></label> </p></div><div class="q-explanation"><p>In this function the amplitude has doubled but the period has not changed from <span class="math-tex">\(f\left(x\right)=\cos\left(x\right)\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function is given by <span class="math-tex">\(f\left(x\right)=4\cos\left(kx\right)\)</span>. If the period of the function is 120 degrees, find the value of <span class="math-tex">\(k\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1</span></label> </p><p><label class="radio"> <input type="radio"> <span>4</span></label> </p><p><label class="radio"> <input type="radio"> <span>3</span></label> </p></div><div class="q-explanation"><p>Utilising the formula, <span class="math-tex">\(b=\frac{360}{Period}\)</span>;</p><p><span class="math-tex">\(b=\frac{360}{120}=3\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=3\sin\left(4x\right)+2\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>Amplitude: <span class="math-tex">\(3\)</span></span></label>, Period: <span class="math-tex">\(90\)</span> degrees; Principal Axis: <span class="math-tex">\(y=2\)</span> </p><p><label class="radio"> <input type="radio"> <span>Amplitude: <span class="math-tex">\(3\)</span></span></label>, Period: <span class="math-tex">\(4\)</span>, Principal Axis: <span class="math-tex">\(y=2\)</span><label class="radio"></label><br> </p></div><div class="q-explanation"><p>The point of this question is to show that the <span class="math-tex">\(4\)</span> inside the brackets is not itself the period of the wave, but it does of course affect the period. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=\sin\left(10x\right)-1\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p>Amplitude: <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p><p>Period: <input type="text" style="height: auto;" data-c="36"> <span class="review"></span> </p><p>Principal Axis: <input type="text" style="height: auto;" data-c="y=-1"> <span class="review"></span> </p></div><div class="q-explanation"><p>See answers given.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=5\sin(\frac{x}{2})\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p>Amplitude: <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> </p><p>Period: <input type="text" style="height: auto;" data-c="720"> <span class="review"></span> </p><p>Principal Axis: <input type="text" style="height: auto;" data-c="y=0 or x-axis"> <span class="review"></span> </p></div><div class="q-explanation"><p>In this case for the period, rewrite <span class="math-tex">\(\frac{x}{2}\)</span> as <span class="math-tex">\(\frac{1}{2}x\)</span> and utilise the formula, <span class="math-tex">\(Period=\frac{360}{b}\)</span>;</p><p><span class="math-tex">\(Period=\frac{360}{0.5}=720 ^{\circ}\)</span>. Other answers are given and more easily understood. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the curve below is given as, <span class="math-tex">\(f\left(x\right)=a\cos\left(bx\right)+c\)</span>. </p><p>Find the values of <span class="math-tex">\(a\)</span>, <span class="math-tex">\(b\)</span> and <span class="math-tex">\(c\)</span>.</p><p><img alt="" src="../../images/period2.png" style="width: 400px; height: 215px;"></p></div><div class="q-answer"><p>a: <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>b: <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p><p>c: <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> </p></div><div class="q-explanation"><p>You can use the equations given in video 4, or find these values more intuitively.</p><p>The principal axis (or centre line) or the curve is the most logical to identify first. This corresponds to the value <span class="math-tex">\(c\)</span> which is <span class="math-tex">\(-1\)</span>.</p><p>The amplitude is the height between the principal axis and the maximum (or minimum points. This corresponds to the value <span class="math-tex">\(a\)</span> which is <span class="math-tex">\(2\)</span>. </p><p>The period of the wave is <span class="math-tex">\(90\)</span> degrees. Intuitively this means that <span class="math-tex">\(4\)</span> waves fit into a <span class="math-tex">\(360\)</span> degree domain which means that the value <span class="math-tex">\(b\)</span> is <span class="math-tex">\(4\)</span>. You might also find this using <span class="math-tex">\(b=\frac{360}{Period}\)</span>;</p><p><span class="math-tex">\(b=\frac{360}{90}=4\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the curve below is given as, <span class="math-tex">\(f\left(x\right)=a\sin\left(bx\right)+c\)</span>. </p><p>Find the values of <span class="math-tex">\(a\)</span>, <span class="math-tex">\(b\)</span> and <span class="math-tex">\(c\)</span>.</p><p><img alt="" src="../../images/period3.png" style="width: 400px; height: 196px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{x}{3}\right)+1\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{x}{3}\right)\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{2x}{3}\right)+1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{2x}{3}\right)\)</span></span></label> </p></div><div class="q-explanation"><p>Utilising the formulae in this case: </p><p><span class="math-tex">\(a=\frac{\max-\min}{2}=\frac{1.5--1.5}{2}=\frac{3}{2}=1.5\)</span></p><p>To find <span class="math-tex">\(b\)</span>, first identify the period of the wave. Half of the wave is shown with the co-ordinates given, so half the period is <span class="math-tex">\(810-270=540\)</span>. The period is therefore <span class="math-tex">\(540\cdot2=1080\)</span>. Hence, <span class="math-tex">\(b=\frac{360}{1080}=\frac{1}{3}\)</span>.</p><p><span class="math-tex">\(c=\frac{\max+\min}{2}=\frac{1.5+-1.5}{2}=\frac{0}{2}=0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <div> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/q1(3).png" style="width: 500px; height: 158px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/533222956" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/q2(1).png" style="width: 680px; height: 210px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/533228530" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/q3a.png" style="width: 650px; height: 285px;"></p> <p><img alt="" src="../../images/q3b.png" style="width: 650px; height: 230px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw 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