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292 lines
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HTML
292 lines
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HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/trig-finding-side-right-triangle.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:02:45 GMT -->
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<title>Finding a Side in a Right-Angled Triangle</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Finding a Side in a Right-Angled Triangle</h1>
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<h2>Find a Side when we know another Side and Angle</h2>
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<p>We can find an unknown side in a <a href="../right_angle_triangle.html">right-angled triangle</a> when we know:</p>
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<ul>
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<li>one length, and</li>
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<li>one angle (apart from the right angle, that is).</li>
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</ul>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-2example2.gif" alt="Ship Anchor Triangle" width="242" height="170" /></p>
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<h3>Example: Depth to the Seabed</h3>
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<p>The ship is anchored on the seabed.</p>
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<p>We know:</p>
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<ul>
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<li>the cable length (30 m), and</li>
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<li>the angle the cable makes with the seabed</li>
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</ul>
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<p> So we should be able to find the depth!</p>
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<p><b>But How?</b></p>
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<div style="clear:both"></div>
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</div>
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<p align="center"><span class="large">The answer is to use <a href="../sine-cosine-tangent.html">Sine, Cosine <i>or</i> Tangent</a>! </span></p>
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<h3>But Which One?</h3>
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<p>Which one of <b>Sine, Cosine or Tangent</b> to use?</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /></p> <p>To find out which, first we give <b>names</b> to the sides:</p>
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<ul>
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<li><b>Adjacent</b> is adjacent (next to) to the angle,<br />
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</li>
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<li><b>Opposite</b> is opposite the angle, <br />
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</li>
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<li>and the longest side is the <b>Hypotenuse</b>.</li>
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</ul>
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<div style="clear:both"></div>
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<p>Now, for <b>the side we already know</b> and the <b>side we are trying to find</b>, we use the first letters of their names and the phrase <a href="sohcahtoa.html">"SOHCAHTOA"</a> to decide which function:</p>
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<table border="0" align="center">
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<tr>
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<td width="100"><div align="left"><b><i>SOH...</i></b></div></td>
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<td width="20"> </td>
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<td><b>S</b>ine: sin(θ) = <b>O</b>pposite / <b>H</b>ypotenuse</td>
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</tr>
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<tr>
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<td><div align="center"><b><i>...CAH...</i></b></div></td>
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<td> </td>
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<td><b>C</b>osine: cos(θ) = <b>A</b>djacent / <b>H</b>ypotenuse</td>
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</tr>
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<tr>
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<td><div align="right"><b><i>...TOA</i></b></div></td>
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<td> </td>
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<td><b>T</b>angent: tan(θ) = <b>O</b>pposite / <b>A</b>djacent</td>
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</tr>
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</table>
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<p>Like this:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-2example2.gif" alt="Ship Anchor Triangle" width="242" height="170" /><br>
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<img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /></p>
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<h3>Example: Depth to the Seabed (Continued)</h3>
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<p> </p>
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<p>Find the <b>names</b> of the two sides we are working on: </p>
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<ul>
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<li>the side we know is the <b>Hypotenuse</b></li>
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<li>the side we want to find is <b>Opposite</b> the angle (check for yourself that "d" is opposite the angle 39°)</li>
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</ul><div style="clear:both"></div>
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<p>Now use the first letters of those two sides (<b>O</b>pposite and<b> H</b>ypotenuse) and the phrase "SOHCAHTOA" which gives us "<b>SOH</b>cahtoa", which tells us we need to use <b>Sine</b>:</p>
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<p class="center larger"><b>S</b>ine: sin(θ) = <b>O</b>pposite / <b>H</b>ypotenuse</p>
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<p> </p>
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<p>Now put in the values we know:</p>
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<p class="center larger">sin(39°) = d / 30</p>
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<p> </p>
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<p>And solve that equation!</p>
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<p>But how do we calculate <b>sin(39°)</b> ... ?</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" width="118" height="75" /></td>
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<td><p>Use your calculator. <br>
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Type in 39 and then use the "sin" key. <br>
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That's easy!</p></td>
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</tr>
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</table>
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<p align="center" class="larger">sin(39°) = 0.6293... </p>
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<p> </p>
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<p>So now we have:</p>
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<p align="center"> <span class="larger">0.6293... = d / 30</span></p>
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<p> </p>
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<p>Now we rearrange it a little bit, and solve:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">0.6293... = d / 30</span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">d / 30 = 0.6293... </span></div>
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<div class="row"><span class="left">Multiply both sides by 30:</span><span class="right">d = 0.6293... x 30 </span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">d = <b>18.88</b> to 2 decimal places</span></div>
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</div>
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<p align="center" class="large"> <span class="larger">The depth the anchor ring lies beneath the hole is <b>18.88 m</b></span> </p>
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</div>
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<h2>Step By Step</h2>
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<p>These are the four steps to follow: </p>
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<ul>
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<li><b>Step 1</b> Find the names of the two sides we are using, one we are trying to find and one we already know, out of Opposite, Adjacent and Hypotenuse.</li>
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</ul>
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<ul>
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<li><b>Step 2</b> Use SOHCAHTOA to decide which one of Sine, Cosine <b>or</b> Tangent to use in this question.</li>
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</ul>
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<ul>
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<li><b>Step 3</b> For Sine write down Opposite/Hypotenuse, for Cosine write down Adjacent/Hypotenuse <b>or</b> for Tangent write down Opposite/Adjacent. One of the values is the unknown length.</li>
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</ul>
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<ul>
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<li><b>Step 4</b> Solve using your calculator and your skills with <a href="introduction-multiply.html">Algebra</a>.</li>
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</ul>
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<h2>Examples</h2>
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<p>Let’s look at a few more examples: </p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-flagpole.gif" width="265" height="147" alt="trig example airplane 1000, 60 degrees" /></p>
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<h3>Example: find the height of the plane. </h3>
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<p>We know the distance to the plane is 1000<br>
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And the angle is 60° </p>
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<p>What is the plane's height?</p>
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<p>Careful! The <b>60°</b> angle is at the top, so the "h" side is <b>Adjacent</b> to the angle!</p>
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<ul>
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<li><b>Step 1</b> The two sides we are using are <b>A</b>djacent (h) and <b>H</b>ypotenuse (1000).</li>
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</ul>
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<ul>
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<li><b>Step 2</b> SOH<b>CAH</b>TOA tells us to use <b>C</b>osine.</li>
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</ul>
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<ul>
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<li><b>Step 3</b> Put our values into the Cosine equation:
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<p class="larger center">cos 60° = Adjacent / Hypotenuse <br>
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= h / 1000</p>
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</li>
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</ul>
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<ul>
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<li><b>Step 4</b> Solve:</li>
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</ul>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><b>cos 60° = h/1000</b></span></div>
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<div class="row"><span class="left">Swap:</span><span class="right"><b>h/1000</b> = <b>cos 60°</b></span></div>
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<div class="row"><span class="left">Calculate cos 60°:</span><span class="right"><b>h/1000</b> = <b>0.5</b></span></div>
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<div class="row"><span class="left">Multiply both sides by 1000:</span><span class="right"><b>h</b> = <b>0.5 x 1000 </b></span></div>
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<div class="row"><span class="left"> </span><span class="right"><b>h = 500</b></span></div>
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</div>
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<p class="center larger"> The height of the plane = <b>500 meters</b></p>
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</div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/triangle-7-y-35.gif" width="120" height="154" alt="triangle 7, y and 35 degrees" /></p>
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<h3>Example: Find the length of the side <b>y</b>:</h3>
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<ul>
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<li><b>Step 1</b> The two sides we are using are <b>O</b>pposite (y) <br>
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and <b>A</b>djacent (7).</li>
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</ul>
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<ul>
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<li><b>Step 2</b> SOHCAH<b>TOA</b> tells us to use <b>T</b>angent.</li>
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</ul>
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<ul>
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<li><b>Step 3</b> Put our values into the tangent function: <p class="larger center">tan 53° = Opposite/Adjacent <br>
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= y/7</p></li>
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</ul>
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<ul>
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<li><b>Step 4</b> Solve:</li>
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</ul>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">tan 53° = y/7</span></div>
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<div class="row"><span class="left">Swap:</span><span class="right">y/7 = tan 53°</span></div>
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<div class="row"><span class="left">Multiply both sides by 7:</span><span class="right">y = 7 tan 53°</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">y = 7 x 1.32704...</span></div>
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<div class="row"><span class="left"> </span><span class="right">y = <b>9.29</b> (to 2 decimal places)</span></div>
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</div>
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<p class="center larger">Side y = <b>9.29</b></p>
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</div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-tent.gif" width="171" height="172" alt="trig tower 70 m and angle 68 degrees" /></p><h3>Example: Radio Mast</h3>
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<p>There is a mast that is 70 meters high. </p>
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<p>A wire goes to the top of the mast at an angle of 68°.</p>
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<p>How long is the wire?</p>
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<ul>
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<li><b>Step 1</b> The two sides we are using are <b>O</b>pposite (70) and <b>H</b>ypotenuse (w).</li>
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</ul>
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<ul>
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<li><b>Step 2</b> <b>SOH</b>CAHTOA tells us to use <b>S</b>ine.</li>
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</ul>
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<ul>
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<li><b>Step 3</b> Write down:
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<p class="larger center">sin 68° = 70/w</p></li>
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</ul>
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<ul>
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<li><b>Step 4</b> Solve:</li>
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</ul>
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<p>The unknown length is on the bottom (the denominator) of the fraction! </p>
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<p>So we need to follow a slightly different approach when solving:</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 68° = 70/w </span></div>
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<div class="row"><span class="left">Multiply both sides by w:</span><span class="right">w × (sin 68°) = 70 </span></div>
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<div class="row"><span class="left">Divide both sides by "sin 68°":</span><span class="right">w = 70 / (sin 68°)</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">w = 70 / 0.9271...</span></div>
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<div class="row"><span class="left"> </span><span class="right">w = 75.5 m (to 1 place)</span></div>
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</div>
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<p class="center larger"> The length of the wire = <b>75.5 m</b></p>
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</div>
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<p> </p>
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<div class="questions">
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||
<script type="text/javascript">getQ(258, 1504, 1505, 1506, 1507, 2346, 2347, 2348, 3935, 248);</script>
|
||
|
||
|
||
</div>
|
||
|
||
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||
<div class="related"> <a href="trig-sine-law.html">The Law of Sines</a> <a href="trig-cosine-law.html">The Law of Cosines</a> <a href="trig-solving-triangles.html">Solving Triangles</a> <a href="trigonometry-index.html">Trigonometry Index</a> <a href="index.html">Algebra Index</a> </div>
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