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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="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Sine and Cosine Rule</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><img alt="" src="../../files/trigonometry/triangle-geometry/main-1.jpg" style="float: left; width: 100px; height: 100px;"> In this page, we will look at the Sine Rule, the Cosine Rule and the formula for the Area of a Triangle. The formula for these three are fairly straight forward to use, but don't take this topic too lightly, sometimes the exam questions can be quite challenging. Questions on this topic can even come up on the non-calculator paper when we use the angles 30°, 45°, 60°, ...</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>the sine rule: <span class="math-tex">\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)</span></li> <li>the cosine rule: <span class="math-tex">\(c^2=a^2+b^2-2ab\cos C\)</span></li> <li>the area of a triangle =<span class="math-tex">\({ 1\over 2}ab\sin C\)</span></li> </ul> </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 pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/trigonometry/triangle-geometry/revision-notes_trianglegeometry.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/trigonometry/triangle-geometry/revision-notes_trianglegeometry.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#f377184a"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="f377184a"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Triangle Geometry <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-214-686" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Using the following diagram, complete the following equation</p><p><img alt="" src="../../files/trigonometry/triangle-geometry/quiz/q1.jpg" style="width: 300px; height: 217px;"></p><p><span class="math-tex">\(\frac{5}{sinA°}=\frac{b}{sin57°}\)</span></p></div><div class="q-answer"><p>A = <input type="text" style="height: auto;" data-c="44"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the Sine Rule, a is opposite A and b is opposite B</p><p><span class="math-tex">\(\frac{a}{sinA}=\frac{b}{sinB}\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Using the following diagram, complete the following equation</p><p><img alt="" src="../../files/trigonometry/triangle-geometry/quiz/q2.jpg" style="width: 300px; height: 140px;"></p><p><span class="math-tex">\(\frac{5}{sinA°}=\frac{8}{sinB°}\)</span></p></div><div class="q-answer"><p>A = <input type="text" style="height: auto;" data-c="30"> <span class="review"></span></p><p>B = <input type="text" style="height: auto;" data-c="52"> <span class="review"></span></p></div><div class="q-explanation"><p>The side with length 8 is opposite the unmarked angle. We calculate 180-30-98 = 52</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In the triangle ABC, BC = 7cm , AC = 7.5 cm and <span class="math-tex">\(\hat { BAC } \)</span>= 66°</p><p>If we use the Sine Rule to calculate <span class="math-tex">\(\hat { BAC } \)</span> , one solution is 78°.</p><p>Find another possible solution</p></div><div class="q-answer"><p><span class="math-tex">\(\hat { BAC } \)</span> = <input type="text" style="height: auto;" data-c="102"> <span class="review"></span> °</p></div><div class="q-explanation"><p>This is the ambiguous case for the Sine Rule since sin78° = sin(180°-78°)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Using the following diagram, complete the following equation</p><p><img alt="" src="../../files/trigonometry/triangle-geometry/quiz/q4.jpg" style="width: 300px; height: 148px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="11"> <span class="review"></span> ² = 5² + <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> ² - 2 x <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> x 8 x cos 105°</p></div><div class="q-explanation"><p>Using the Cosine Rule</p><p>c² = a² + b² - 2abcosC</p><p>The side c is opposite the angle C</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Using the following diagram, complete the following equation</p><p><img alt="" src="../../files/trigonometry/triangle-geometry/quiz/q5.jpg" style="width: 300px; height: 123px;"></p><p>cos A° = <span class="math-tex">\(\frac{3^2+b^2-6.2^2}{2\times3\times c}\)</span></p></div><div class="q-answer"><p>A = <input type="text" style="height: auto;" data-c="124"> <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="4"> <span class="review"></span></p></div><div class="q-explanation"><p>cos 124° = <span class="math-tex">\(\frac{3^2+4^2-6.2^2}{2\times3\times 4}\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In the triangle ABC, AC = 4cm , BC = 6cm and angle <span class="math-tex">\(\hat { B } \)</span> = 30°</p><p>a) Work out sin <span class="math-tex">\(\hat { A } \)</span></p><p>There are two possible answers for angle A, A<sub>1</sub> and A<sub>2</sub> where A<sub>1</sub> < A<sub>2</sub></p><p>b) Find A<sub>1</sub> and A<sub>2 </sub>to 3 significant figures</p></div><div class="q-answer"><p>a) sin <span class="math-tex">\(\hat { A } \)</span> = <input type="text" style="height: auto;" data-c="0.75"> <span class="review"></span></p><p>b) A<sub>1</sub> = <input type="text" style="height: auto;" data-c="48.6"> <span class="review"></span> ° , A<sub>2</sub> = <input type="text" style="height: auto;" data-c="131"> <span class="review"></span> °</p></div><div class="q-explanation"><p>Using the Sine Rule</p><p><span class="math-tex">\(\frac{4}{sin30°}=\frac{6}{sinA}\)</span></p><p><span class="math-tex">\(sinA=\frac{6\times sin30°}{4}\)</span></p><p><span class="math-tex">\(sinA=\frac{6\times 0.5}{4}\)</span></p><p>sinA = 0.75</p><p>b) Ambiguous Case</p><p>A<sub>1</sub> = arcsin(0.75)=48.6°</p><p>A<sub>2</sub> = 180°- 48.6°=131.4°</p><p>Rounded to 3 s.f. A<sub>2</sub> = 131°</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> In the triangle ABC, AB = 8cm, AC = <span class="math-tex">\(2 \sqrt{3} \)</span> cm and <span class="math-tex">\(\hat { BAC } \)</span> = 30°.</p><p>If BC = <span class="math-tex">\(a \sqrt{b} \)</span> , work out <strong>a</strong> and <strong>b</strong></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="7"> <span class="review"></span></p></div><div class="q-explanation"><p>Using Cosine Rule</p><p>BC² = 8² + (<span class="math-tex">\(2 \sqrt{3} \)</span>)² - 2x <span class="math-tex">\(2 \sqrt{3} \)</span> x 8 x cos 30°</p><p>BC² = 64 + 4x3 - 2 x <span class="math-tex">\(2 \sqrt{3} \)</span> x 8 x <span class="math-tex">\(\frac{\sqrt{3}}{2}\)</span></p><p>BC² = 64 + 12 - 2 x 3 x 8</p><p>BC² = 28</p><p>BC = <span class="math-tex">\( \sqrt{28}\)</span></p><p>BC = <span class="math-tex">\( 2\sqrt{7}\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> In the triangle ABC, AC = 12 cm , <span class="math-tex">\(\hat{ABC}\)</span> = 45° and <span class="math-tex">\(\hat{ACB}\)</span> = 75°</p><p>If BC = <span class="math-tex">\(a \sqrt{b} \)</span> , work out <strong>a</strong> and <strong>b</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the Sine Rule</p><p><span class="math-tex">\(\frac{BC}{sin60°}=\frac{12}{sin45°}\)</span></p><p><span class="math-tex">\(BC=\frac{12\times sin60°}{sin45°}\)</span></p><p><span class="math-tex">\(BC=\frac{12\times\frac { \sqrt { 3 } }{ 2 } }{\frac { 1 }{ \sqrt { 2 } } }\)</span></p><p><span class="math-tex">\(BC=\frac { 6\sqrt { 3 } }{ \frac { 1 }{ \sqrt { 2 } } } \)</span></p><p><span class="math-tex">\(BC = 6\sqrt { 3 } \times \frac { \sqrt { 2 } }{ 1 } \)</span></p><p><span class="math-tex">\(BC = 6\sqrt { 6 }\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> In the triangle ABC, AB = <span class="math-tex">\(3{\sqrt{2}}\)</span> cm , AC = 8 cm and <span class="math-tex">\(\hat{BAC}\)</span> = 135°</p><p>Find the area of the triangle ABC</p></div><div class="q-answer"><p>Area = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span> cm²</p></div><div class="q-explanation"><p>Area = <span class="math-tex">\(\frac { 1 }{ 2 } \times 8\times 3\sqrt { 2 } \times sin135°\)</span></p><p>= <span class="math-tex">\(\frac { 1 }{ 2 } \times 8\times 3\sqrt { 2 } \times \frac { 1 }{ \sqrt { 2 } } \)</span></p><p>= 12</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In the triangle ABC, the area = 52 cm², AB = 12cm and AC = 13 cm.</p><p>There are two possible values for angle A, A<sub>1</sub> and A<sub>2</sub> where A<sub>1</sub> < A<sub>2</sub></p><p>Find A<sub>1</sub> and A<sub>2 </sub>to 3 significant figures</p></div><div class="q-answer"><p>A<sub>1</sub> = <input type="text" style="height: auto;" data-c="41.8"> <span class="review"></span> °</p><p>A<sub>2</sub> = <input type="text" style="height: auto;" data-c="138"> <span class="review"></span> °</p></div><div class="q-explanation"><p>Area = 0.5 x 12 x 13 x sinA</p><p>52 = 78 x sin A</p><p>sinA = <span class="math-tex">\(\frac{52}{78}\)</span></p><p>A = <span class="math-tex">\(arcsin(\frac{52}{78})\)</span> = 41.8°</p><p>A<sub>2</sub> = 180° - 41.8° = 138.2°</p><p>To 3 s.f. A<sub>2</sub> = 138°</p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </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 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" 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