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<title>Small Angle Approximations</title>
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<h1>Small Angle Approximations</h1>
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<p>When the angle θ (in <a href="../geometry/radians.html">radians</a>) is small we can use these <b>approximations </b>for <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a>:</p>
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<div class="center large">sin θ ≈ θ</div>
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<div class="center large">cos θ ≈ 1 − <span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span></div>
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<div class="center large">tan θ ≈ θ</div>
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<!-- cos THT APR 1 - THT^2/2 -->
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<p>If we are very daring we can use <b>cos θ ≈ 1 </b></p>
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<p> Let's see some values! (Note: values are approximate)</p>
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<h3>sin θ ≈ θ</h3>
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<div class="simple">
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<table class="center">
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<tbody>
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<tr>
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<th>θ (radians)</th>
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<th>sin θ</th>
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<th>θ <i>−</i> sin θ</th>
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</tr><tr>
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<td>0</td>
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<td>0</td>
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<td>0</td></tr>
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<tr>
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<td>0.01</td>
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<td>0.0099998</td>
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<td>0.0000002</td></tr>
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<tr>
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<td>0.1</td>
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<td>0.0998</td>
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<td>0.0002</td></tr>
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<tr>
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<td>0.2</td>
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<td>0.1987</td>
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<td>0.0013</td></tr>
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<tr>
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<td>0.5</td>
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<td>0.4794</td>
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<td>0.0206</td></tr>
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<tr>
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<td>1</td>
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<td>0.8415</td>
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<td>0.1585</td></tr>
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</tbody></table>
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</div>
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<p>Perfect at zero, really good at 0.01, good at 0.1, and can be useful up to 0.5 if you aren't fussy.</p>
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<p>
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</p><h3>cos θ ≈ 1</h3>
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<p>Can we simply use <b>1</b> to approximate cos θ?</p>
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<div class="simple">
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<table class="center">
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<tbody>
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<tr><th>θ (radians)</th>
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<th>cos θ</th>
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<th>1 <i>−</i> cos θ</th>
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</tr>
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<tr>
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<td>0</td>
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<td>1</td>
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<td>0</td></tr>
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<tr>
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<td>0.01</td>
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<td>0.99995</td>
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<td>0.00005</td></tr>
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<tr>
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<td>0.1</td>
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<td>0.995</td>
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<td>0.005</td></tr>
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<tr>
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<td>0.2</td>
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<td>0.9801</td>
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<td>0.0199</td></tr>
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<tr>
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<td>0.5</td>
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<td>0.8776</td>
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<td>0.1224</td></tr>
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<tr>
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<td>1</td>
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<td>0.5403</td>
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<td>0.4597</td></tr>
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</tbody></table>
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</div>
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<p>Well yes we can, but only for very small angles.
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</p>
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<h3>cos θ ≈ 1 − <span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span></h3>
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<p>So let us try the better version of 1 − <span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span>
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:
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</p>
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<div class="simple">
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<table class="center">
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<tbody>
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<tr><th>θ (radians)</th>
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<th>cos θ</th>
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<th>1 − <span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span></th> <th>(1−<span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span>) − cos θ</th>
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</tr>
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<tr>
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<td>0</td>
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<td>1</td>
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<td>1</td>
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<td>0</td></tr>
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<tr>
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<td>0.01</td>
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<td>0.9999500004</td>
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<td>0.99995</td>
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<td>-0.0000000004</td></tr>
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<tr>
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<td>0.1</td>
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<td>0.9950042</td>
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<td>0.995</td>
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<td>-0.0000042</td></tr>
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<tr>
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<td>0.2</td>
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<td>0.980067</td>
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<td>0.98</td>
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<td>-0.000067</td></tr>
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<tr>
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<td>0.5</td>
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<td>0.8776</td>
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<td>0.875</td>
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<td>-0.0026</td></tr>
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<tr>
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<td>1</td>
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<td>0.5403</td>
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<td>0.5</td>
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<td>-0.0403</td></tr>
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</tbody></table>
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</div><p>Wow, that is a big improvement!</p>
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<h3>tan θ ≈ θ</h3>
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<div class="simple">
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<table class="center">
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<tbody>
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<tr><th>θ (radians)</th>
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<th>tan θ</th>
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<th>tan θ − θ</th>
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</tr>
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<tr>
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<td>0</td>
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<td>0</td>
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<td>0</td></tr>
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<tr>
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<td>0.01</td>
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<td>0.0100003</td>
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<td>-0.0000003</td></tr>
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<tr>
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<td>0.1</td>
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<td>0.1003</td>
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<td>-0.0003</td></tr>
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<tr>
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<td>0.2</td>
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<td>0.2027</td>
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<td>-0.0027</td></tr>
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<tr>
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<td>0.5</td>
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<td>0.5463</td>
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<td>-0.0463</td></tr>
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<tr>
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<td>1</td>
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<td>1.5574</td>
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<td>-0.5574</td></tr>
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</tbody></table>
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</div><p>Not too bad for small values, right?</p>
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<h2>Taylor Series</h2>
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<p>Did you see the magical improvement for cos when we went from <b>1</b> to <b>1 − <span class="intbl"><em>θ<sup>2</sup></em><strong>2</strong></span></b> ?</p>
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<p>The secret is the <a href="taylor-series.html">Taylor Series</a> expansion of cos:</p>
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<p class="center"><b>cos x</b> = 1 − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> − ...</p>
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<p>So ... we can use more terms if want more accuracy!</p>
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<p>Likewise we can improve sine:</p>
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<p class="center"><b>sin x</b> = x − <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>5</sup></em><strong>5!</strong></span> − ...</p>
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<p>Or <b>tan</b>, or other functions like <i><b>e</b></i><b><sup>x</sup></b></p>
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<div class="example">
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<h3>Example: you are stuck on an island without a calculator. Calculate sine of 20 degrees.</h3>
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<p>Degrees? But we need to use radians!</p>
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<p>Let us estimate as best we can:</p>
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<p class="center">20 × <span class="intbl"><em><span class="times">π</span></em><strong>180</strong></span> = <span class="intbl"><em><span class="times">π</span></em><strong>9</strong></span> ≈ 3.1416 × 0.11... ≈ 0.35 radians </p>
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<p>Now, using just one extra term:</p>
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<p><b>sin x</b> = x − <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> ...</p>
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<p><b>sin x</b> ≈ 0.35 − <span class="intbl"><em>0.35<sup>3</sup></em><strong>3!</strong></span> ≈ 0.35 − <span class="intbl"><em>0.35*0.35*0.35</em><strong>6</strong></span> <b>≈ 0.3428</b> (after much effort) </p>
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<p>(Later when you get home you use a calculator to get <b>sin(20°) = 0.3420201...</b>, not bad!)</p>
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</div>
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<h2>Uses</h2>
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<p>These approximations are very useful in <b>astronomy </b>where many angles are very small.</p>
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<p>Also in some areas of engineering and optics too.</p>
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<p><br></p>
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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="taylor-series.html">Taylor Series</a>
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<a href="index.html">Algebra Index</a>
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