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<title>Sohcahtoa: Sine, Cosine, Tangent</title>
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<h1 align="center">Sohcahtoa</h1>
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<p align="center">Sohca...<i>what?</i> Just an <b>easy way to remember</b> how <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a> work:</p>
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<table border="0" align="center" cellpadding="5">
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<tr>
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<td><div align="center"><b><i>Soh...</i></b></div></td>
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<td nowrap><div align="center"><b>S</b>ine = <b>O</b>pposite / <b>H</b>ypotenuse</div></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 nowrap><div align="center"><b>C</b>osine = <b>A</b>djacent / <b>H</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td><div align="center"><b><i>...toa</i></b></div></td>
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<td nowrap><div align="center"><b>T</b>angent = <b>O</b>pposite / <b>A</b>djacent</div></td>
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</tr>
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</table>
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<h2>Right Triangle</h2>
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<p>OK, let's see what this is all about.</p>
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<p>Firstly, the names <b>Opposite, Adjacent and Hypotenuse</b> come from the <a href="../right_angle_triangle.html">right triangle</a>: </p>
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<div align="center">
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<img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /></div><br />
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<div class="bigul">
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<ul>
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<li>"Opposite" is opposite to the angle θ</li>
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<li>"Adjacent" is adjacent (next to) to the angle θ</li>
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<li>"Hypotenuse" is the long one</li>
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</ul>
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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/adjacent-opposite-hypotenuse-rot.svg" alt="examples of Opposite, Adjacent and Hypotenuse" /></p>
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<p><b>Adjacent</b> is always next to the angle</p>
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<p>And <b>Opposite</b> is opposite the angle</p>
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</div>
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<h2>Sine, Cosine and Tangent</h2>
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<p> And <b>Sine</b>, <b>Cosine</b> and <b>Tangent</b> are the three main functions in <a href="trigonometry.html">trigonometry</a>. </p>
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<p>They are often shortened to <b>sin</b>, <b>cos</b> and <b>tan</b>.<br />
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</p>
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<p>The calculation is simply <b>one side of a right angled triangle divided by another side</b> ... we just have to know which sides, and that is where "sohcahtoa" helps.<br />
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<br />
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For a triangle with an angle <b><i>θ</i></b> , the functions are calculated this way:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td><div align="right">Sine:</div></td>
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<td><div align="center"><b><i>soh...</i></b></div></td>
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<td><div align="center"><b>s</b>in(<i>θ</i>) = <b>o</b>pposite / <b>h</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td><div align="right">Cosine:</div></td>
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<td><div align="center"><b><i>...cah...</i></b></div></td>
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<td><div align="center"><b>c</b>os(<i>θ</i>) = <b>a</b>djacent / <b>h</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td><div align="right">Tangent:</div></td>
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<td><div align="center"><b><i>...toa</i></b></div></td>
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<td><div align="center"><b>t</b>an(<i>θ</i>) = <b>o</b>pposite / <b>a</b>djacent</div></td>
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</tr>
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</table>
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</div>
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<p> </p>
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<div class="example">
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<h3 class="Larger">Example: what are the sine, cosine and tangent of 30° ?</h3>
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<p>A 30° triangle has a hypotenuse (the long side) of length <b>2</b>, an opposite side of length <b>1</b> and an adjacent side of
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<b>√3</b>, like this:<br />
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</p>
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<div class="center"> <img src="../images/triangle-30.gif" alt="30° triangle" width="298" height="216" /> </div>
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<p>Now we know the lengths, we can calculate the functions:<br />
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</p>
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<div class="simple">
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<table border="0" align="center" cellpadding="3">
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<tr>
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<td><div align="center"><b>Sine</b></div></td>
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<td align="center"><b><i>soh...</i></b></td>
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<td><div align="center">sin(30°) = <span class="intbl"><em>1</em><strong>2</strong></span> = 0.5</div></td>
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</tr>
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<tr>
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<td><div align="center"><b>Cosine</b></div></td>
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<td align="center"><b><i>...cah...</i></b></td>
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<td><div align="center">cos(30°) = <span class="intbl"><em>1.732...</em><strong>2</strong></span> = 0.866...</div></td>
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</tr>
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<tr>
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<td><div align="center"><b>Tangent</b></div></td>
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<td align="center"><b><i>...toa</i></b></td>
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<td><div align="center">tan(30°) = <span class="intbl"><em>1</em><strong>1.732...</strong></span> = 0.577...</div></td>
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</tr>
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</table>
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</div>
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<p>(get your calculator out and check them!) </p>
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</div>
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<h2>How to Remember</h2>
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<p>I find "sohcahtoa" easy to remember ... but here are other ways if you like:</p><ul>
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<li><b>S</b>ailors <b>O</b>ften <b>H</b>ave <b>C</b>urly <b>A</b>uburn <b>H</b>air <b>T</b>ill <b>O</b>ld <b>A</b>ge. </li>
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<li><b>S</b>ome <b>O</b>ld <b>H</b>orses <b>C</b>an <b>A</b>lways <b>H</b>ear <b>T</b>heir <b>O</b>wners <b>A</b>pproach.</li>
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<li><b>S</b>ome <b>O</b>ld <b>H</b>en <b>C</b>aught <b>A</b>nother <b>H</b>en <b>T</b>aking <b>O</b>ne <b>A</b>way.</li>
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</ul>
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<h2>Practice Here:</h2>
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<iframe src="../geometry/itriangle-q.html" scrolling="no" style="width:402px; height:362px; overflow:hidden; margin:auto; display:block; border: none;"></iframe>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(728, 1496, 729, 1497, 2364, 2365, 2366, 2367, 2368, 2369);</script>
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</div>
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<div class="related">
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<a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a>
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<a href="trigonometry-index.html">Trigonometry Index</a> </div>
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