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314 lines
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HTML
314 lines
14 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/trig-sine-law.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:02:57 GMT -->
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<title>The Law of Sines</title>
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<div id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">The Law of Sines</h1>
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<p><b>The Law of Sines </b>(or<b> Sine Rule</b>) is very useful for solving triangles:</p>
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<p class="center large"><span class="intbl">
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<em>a</em>
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<strong>sin A</strong>
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</span> = <span class="intbl">
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<em>b</em>
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<strong>sin B</strong>
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</span> = <span class="intbl">
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<em>c</em>
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<strong>sin C</strong>
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</span></p>
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<p>It works for any triangle:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr align="center">
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<td><img src="images/triangle-sides-angles.svg" alt="triangle"></td>
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<td>
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<p><b>a</b>, <b>b</b> and <b>c</b> are sides.</p>
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<p><b>A</b>, <b>B</b> and <b>C</b> are angles.</p>
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<p><i>(Side a faces angle A,<br>
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side b
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faces angle B and<br>
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side c faces angle C).</i></p>
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</td>
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</tr>
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</tbody></table>
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<p>And it says that:</p>
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<p class="center">When we <b>divide side a by the sine of angle A</b><br>
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it is equal to <b>side b divided by the sine of angle B</b>,<br>
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and also equal to <b>side c divided by the sine of angle C</b></p>
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<h2>Sure ... ?</h2>
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<p>Well, let's do the calculations for a triangle I prepared earlier:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/trig-5-8-9.gif" alt="5,8,9 Triangle" height="126" width="212"></td>
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<td>
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<p class="center larger"><span class="intbl">
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<em>a</em>
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<strong>sin A</strong>
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</span> = <span class="intbl">
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<em>8</em>
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<strong>sin(62.2°)</strong>
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</span> = <span class="intbl">
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<em>8</em>
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<strong>0.885...</strong>
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</span> = <b>9.04...</b></p>
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<p class="center larger"><span class="intbl">
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<em>b</em>
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<strong>sin B</strong>
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</span> = <span class="intbl">
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<em>5</em>
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<strong>sin(33.5°)</strong>
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</span> = <span class="intbl">
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<em>5</em>
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<strong>0.552...</strong>
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</span> = <b>9.06...</b></p>
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<p class="center larger"><span class="intbl">
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<em>c</em>
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<strong>sin C</strong>
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</span> = <span class="intbl">
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<em>9</em>
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<strong>sin(84.3°)</strong>
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</span> = <span class="intbl">
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<em>9</em>
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<strong>0.995...</strong>
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</span> = <b>9.04...</b></p>
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</td>
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</tr>
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</tbody></table>
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<p class="center">The answers are <b>almost the same!</b><br>
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<i>(They would be <b>exactly</b> the same if we used perfect accuracy).</i></p>
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<p>So now you can see that:</p>
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<p class="center large"><span class="intbl">
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<em>a</em>
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<strong>sin A</strong>
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</span> = <span class="intbl">
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<em>b</em>
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<strong>sin B</strong>
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</span> = <span class="intbl">
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<em>c</em>
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<strong>sin C</strong>
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</span></p>
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<div class="fun">
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<h3>Is This Magic?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/triangle-law-sines.svg" alt="triangle a b c"></p>
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<p>Not really, look at this general triangle and imagine it is two right-angled triangles sharing the side <b>h</b>:</p>
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<p> </p>
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<p>The <a href="../sine-cosine-tangent.html">sine of an angle</a> is the opposite divided by the hypotenuse, so:</p>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/sin-divide.svg" alt="triangle a b c"></p>
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<table style="border: 0;">
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<tbody>
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<tr valign="top">
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<td align="right">sin(A) = h/b</td>
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<td><span style="float:right; margin: 0 0 5px 10px;"><img src="../images/style/so.svg" alt="so symbol"></span></td>
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<td align="left"> </td>
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<td align="left"> b sin(A) = h</td>
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</tr>
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<tr valign="top">
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<td align="right">sin(B) = h/a</td>
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<td><span style="float:right; margin: 0 0 5px 10px;"><img src="../images/style/so.svg" alt="so symbol"></span></td>
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<td align="left"> </td>
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<td align="left">a sin(B) = h</td>
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</tr>
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</tbody></table>
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<div style="clear:both"></div>
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<p><b>a sin(B)</b> and <b>b sin(A)</b> both equal <b>h</b>, so we get:</p>
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<p class="center larger">a sin(B) = b sin(A)</p>
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<p>Which can be rearranged to:</p>
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<p class="center larger"><span class="intbl">
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<em>a</em>
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<strong>sin A</strong>
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</span> = <span class="intbl">
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<em>b</em>
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<strong>sin B</strong>
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</span></p>
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<p>We can follow similar steps to include c/sin(C)</p>
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</div>
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<h2>How Do We Use It?</h2>
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<p>Let us see an example:</p>
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<div class="example">
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<h3>Example: Calculate side "c"</h3>
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<p class="center"><img src="images/trig-sineruleex1.gif" alt="triangle 35 degrees, 105 degrees, 7"></p>
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<div class="tbl">
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<div class="row"><span class="left">Law of Sines:</span><span class="right">a/sin A = b/sin B = c/sin C</span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">a/sin A = 7/sin(35°) = c/sin(105°)</span></div>
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<div class="row"><span class="left">Ignore a/sin A (not useful to us):</span><span class="right">7/sin(35°) = c/sin(105°)</span></div>
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<p>Now we use our algebra skills to rearrange and solve:</p>
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<div class="row"><span class="left">Swap sides:</span><span class="right">c/sin(105°) = 7/sin(35°)</span></div>
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<div class="row"><span class="left">Multiply both sides by sin(105°):</span><span class="right">c = ( 7 / sin(35°) ) × sin(105°)</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">c = ( 7 / 0.574... ) × 0.966...</span></div>
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<div class="row"><span class="left"> </span><span class="right">c = <b>11.8</b> (to 1 decimal place)</span></div>
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</div>
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</div>
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<h2>Finding an Unknown Angle</h2>
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<p>In the previous example we found an unknown side ...</p>
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<p class="center">... but we can also use the Law of Sines to find an <b>unknown angle</b>.</p>
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<p>In this case it is best to turn the fractions upside down (<b>sin A/a</b> instead of <b>a/sin A</b>, etc):</p>
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<p class="center large"><span class="intbl">
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<em>sin A</em>
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<strong>a</strong>
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</span> = <span class="intbl">
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<em>sin B</em>
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<strong>b</strong>
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</span> = <span class="intbl">
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<em>sin C</em>
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<strong>c</strong>
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</span></p>
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<div class="example">
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<h3>Example: Calculate angle B</h3>
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<p class="center"><img src="images/trig-sineruleex2.gif" alt="triangle 63 degrees, 4.7, 5.5"></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">sin A / a = sin B / b = sin C / c</span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">sin A / a = sin B / 4.7 = sin(63°) / 5.5</span></div>
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<div class="row"><span class="left">Ignore "sin A / a":</span><span class="right">sin B / 4.7 = sin(63°) / 5.5</span></div>
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<div class="row"><span class="left">Multiply both sides by 4.7:</span><span class="right">sin B = (sin(63°)/5.5) × 4.7 </span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">sin B = 0.7614... </span></div>
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<div class="row"><span class="left">Inverse Sine:</span><span class="right">B = sin<sup>−1</sup>(0.7614...) </span></div>
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<div class="row"><span class="left"> </span><span class="right">B = <b>49.6°</b></span></div>
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</div>
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</div>
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<h2>Sometimes There Are Two Answers !</h2>
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<p>There is one <b>very</b> tricky thing we have to look out for:</p>
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<p class="center large">Two possible answers.</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/trig-sine-law-ambig.gif" alt="Sine Law Ambiguous Case" height="140" width="210"></td>
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<td>
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<p>Imagine we know angle <b>A</b>, and sides <b>a</b> and <b>b</b>.</p>
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<p>We can swing side <b>a</b> to left or right and come up with two possible results (a small triangle and a much wider triangle)</p>
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<p class="center">Both answers are right!</p>
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</td>
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<td> </td>
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</tr>
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</tbody></table>
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<p>This only happens in the "<a href="trig-solving-ssa-triangles.html">Two Sides and
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an Angle <b>not</b> between</a>" case, and even then not always, but we have to watch out for it.</p>
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<p>Just think "could I swing that side the other way to also make a correct answer?"</p>
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<p> </p>
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<div class="example">
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<h3>Example: Calculate angle R</h3>
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<p class="center"><img src="images/trig-sineruleex3a.gif" alt="triangle 39 degrees, 41, 28"></p>
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<p>The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that's OK. We just use P,Q and R instead of A, B and C in The Law of Sines.</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">sin R / r = sin Q / q</span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">sin R / 41 = sin(39°)/28</span></div>
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<div class="row"><span class="left">Multiply both sides by 41:</span><span class="right">sin R = (sin(39°)/28) × 41 </span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">sin R = 0.9215...</span></div>
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<div class="row"><span class="left">Inverse Sine:</span><span class="right">R = sin<sup>−1</sup>(0.9215...) </span></div>
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<div class="row"><span class="left"> </span><span class="right">R = <b>67.1°</b></span></div>
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</div>
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</div>
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<p>But wait! There's another angle that also has a sine equal to 0.9215...</p>
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<p><b>The calculator won't tell you this</b> but sin(112.9°) is also equal to 0.9215...</p>
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<p>So, how do we discover the value 112.9°?</p>
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<p>Easy ... take 67.1° away from 180°, like this:</p>
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<p class="center large">180° − 67.1° = 112.9°</p>
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<div class="example">
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<p>So there are two possible answers for R: <b>67.1°</b> and <b>112.9°</b>:</p>
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<p src="images/trig-sineruleex3b.gif" align="center"><img src="images/trig-sineruleex3b.gif" alt="trig sine rule two angles example"></p>
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<p class="center">Both are possible! Each one has the 39° angle, and sides of 41 and 28.</p>
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</div>
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<p class="center large">So, always check to see whether the alternative answer makes sense.</p>
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<ul>
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<li>... sometimes it will (like above) and there are<b> two solutions</b></li>
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<li>... sometimes it won't (see below) and there is <b>one solution</b></li>
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</ul>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/trig-sineruleex2a.gif" alt="trig sine rule one angle example" height="155" width="214"></td>
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<td>
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<p>We looked at this triangle before.</p>
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<p>As you can see, you can try swinging the "5.5" line around, but no other solution makes sense.</p>
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<p>So this has only one solution.</p>
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</td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="questions">
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<script>
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getQ(252, 1508, 1509, 1510, 1511, 253, 2382, 2383, 3936, 3937);
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</script> </div>
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<div class="related">
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<a href="trig-cosine-law.html">The Law of Cosines</a>
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<a href="trig-solving-triangles.html">Solving Triangles</a>
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<a href="trigonometry-index.html">Trigonometry Index</a>
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<a href="index.html">Algebra Index</a>
|
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</div>
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