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<h1 align="center">The Law of Cosines</h1>
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<p> </p>
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<p>For any triangle:</p>
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<table border="0" align="center">
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<tr align="center">
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<td><img src="images/triangle-sides-angles.svg" alt="triangle angles A,B,C and sides a,b,c" /></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>C</b> is the angle opposite side c</p>
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</td>
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</tr>
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</table>
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<p><b>The Law of Cosines</b> (also called the <b>Cosine Rule</b>) says:</p>
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<p class="center large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C)</p>
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<p>It helps us solve some triangles. Let's see how to use it.</p>
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<div class="example">
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<h3>Example: How long is side "c" ... ? </h3>
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<p align="center"><img src="images/trig-cosruleex1.gif" alt="trig cos rule example" /></p>
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<p align="center">We know angle C = 37º, and sides a = 8 and b = 11</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>The Law of Cosines</b> says:</span><span class="right">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C)</span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">c<sup>2</sup> = 8<sup>2</sup> + 11<sup>2</sup> − 2 × 8 × 11 × cos(37º)</span></div>
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<div class="row"><span class="left">Do some calculations:</span><span class="right">c<sup>2</sup> = 64 + 121 − 176 × 0.798…</span></div>
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<div class="row"><span class="left">More calculations:</span><span class="right">c<sup>2</sup> = 44.44...</span></div>
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<div class="row"><span class="left">Take the square root:</span><span class="right">c = √44.44 = <b>6.67</b> to 2 decimal places</span></div>
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</div>
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<p align="center" class="large">
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<br />
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Answer: c = 6.67</p>
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</div>
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<h2>How to Remember</h2>
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<p>How can you remember the formula?</p>
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<p>Well, it helps to know it's the <a href="../pythagoras.html">Pythagoras Theorem</a> with something extra so it works for all triangles:</p>
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<div class="tbl">
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<div class="row"><span class="left">Pythagoras Theorem:<br>
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(only for Right-Angled Triangles)</span><span class="right"><span class="large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></span></div>
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<div class="row"><span class="left">Law of Cosines:<br>
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(for all triangles)</span><span class="right"><span class="large">a<sup>2</sup> + b<sup>2</sup> <span class="style1">− 2ab cos(C)</span> = c<sup>2</sup></span></span></div>
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</div>
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<p>So, to remember it:</p>
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<ul>
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<li>think "<b>abc</b>": <b>a</b><sup>2</sup> + <b>b</b><sup>2</sup> = <b>c</b><sup>2</sup>, </li>
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<li>then a <b>2</b>nd "<b>abc</b>":<span class="style1"> <b>2ab</b> cos(<b>C</b>)</span>, </li>
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<li>and put them together: <b>a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C) = c<sup>2</sup></b></li>
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</ul>
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<h2>When to Use</h2>
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<p>The Law of Cosines is useful for finding:</p>
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<div class="bigul">
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<ul>
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<li>the third side of a triangle when we know <b>two sides and the angle between</b> them (like the example above)</li>
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<li>the angles of a triangle when we know <b>all three sides</b> (as in the following example)</li>
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</ul>
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</div>
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<div class="example">
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<h3>Example: What is Angle "C" ...?</h3>
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<p align="center"><span class="example2"><img src="images/trig-cosruleex4.gif" alt="trig cos rule example" /></span></p>
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<p>The side of length "8" is opposite angle <i><b>C</b></i>, so it is side <b>c</b>. The other two sides are <b>a</b> and <b>b</b>.</p>
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<p>Now let us put what we know into <b>The Law of Cosines</b>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C)</span></div>
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<div class="row"><span class="left">Put in a, b and c:</span><span class="right">8<sup>2</sup> = 9<sup>2</sup> + 5<sup>2 </sup>− 2 × 9 × 5 × cos(C)</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">64 = 81 + 25<sup> </sup>− 90 × cos(C)</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">Subtract 25 from both sides:</span><span class="right">39 = 81<sup> </sup>− 90 × cos(C)</span></div>
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<div class="row"><span class="left">Subtract 81 from both sides:</span><span class="right">−42 = <sup> </sup>−90 × cos(C)</span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right">−90 × cos(C) = −42 <sup> </sup></span></div>
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<div class="row"><span class="left">Divide both sides by −90:</span><span class="right">cos(C) = 42/90</span></div>
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<div class="row"><span class="left">Inverse cosine:</span><span class="right">C = cos<sup>−1</sup>(42/90)</span></div>
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<div class="row"><span class="left">Calculator:</span><span class="right">C = <b>62.2°</b> (to 1 decimal place)</span></div>
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</div>
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</div>
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<h2>In Other Forms</h2>
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<h3>Easier Version For Angles</h3>
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<p>We just saw how to find an angle when we know three sides. It took quite a few steps, so it is easier to use the "direct" formula (which is just a rearrangement of the <span class="large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C) </span> formula). It can be in either of these forms:</p>
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<p class="center large">cos(C) = <span class="intbl">
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<em>a<sup>2</sup> + b<sup>2</sup> − c<sup>2</sup></em>
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<strong>2ab</strong>
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</span></p>
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<p class="center large">cos(A) = <span class="intbl">
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<em>b<sup>2</sup> + c<sup>2</sup> − a<sup>2</sup></em>
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<strong>2bc</strong>
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</span></p>
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<p class="center large">cos(B) = <span class="intbl">
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<em>c<sup>2</sup> + a<sup>2</sup> − b<sup>2</sup></em>
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<strong>2ca</strong>
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</span></p>
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<div class="example">
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<h3>Example: Find Angle "C" Using The Law of Cosines (angle version)</h3>
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<p align="center"> <img src="images/trig-sssex1.gif" width="180" height="137" alt="triangle SSS" /></p>
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<p>In this triangle we know the three sides:</p>
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<ul>
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<li>a = 8, </li>
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<li>b = 6 and </li>
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<li>c = 7. </li>
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</ul>
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<p>Use The Law of Cosines (angle version) to find angle <b>C</b> :</p>
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<div class="tbl">
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<div class="row"><span class="left">cos C</span><span class="right">= (a<sup>2</sup> + b<sup>2</sup> − c<sup>2</sup>)/2ab </span></div>
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<div class="row"><span class="left"> </span><span class="right">= (8<sup>2</sup> + 6<sup>2</sup> − 7<sup>2</sup>)/2×8×6 </span></div>
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<div class="row"><span class="left"> </span><span class="right">= (64 + 36 − 49)/96 </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 51/96 </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 0.53125</span></div>
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<div class="row"><span class="left">C</span><span class="right">= cos<sup>−1</sup>(0.53125)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= <b>57.9°</b> to one decimal place</span></div>
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</div>
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</div>
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<p align="center"> </p>
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<h3>Versions for a, b and c</h3>
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<p>Also, we can rewrite the <span class="large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C) </span> formula into <span class="large">a<sup>2</sup>=</span> and <span class="large">b<sup>2</sup>=</span> form.</p>
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<p> Here are all three:</p>
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<p class="center large">a<sup>2</sup> = b<sup>2</sup> + c<sup>2</sup> − 2bc cos(A)</p>
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<p class="center large">b<sup>2</sup> = a<sup>2</sup> + c<sup>2</sup> − 2ac cos(B)</p>
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<p class="center large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C) </p>
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<p>But it is easier to remember the "<b>c<sup>2</sup></b>=" form and change the letters as needed !</p>
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<p>As in this example:</p>
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<div class="example">
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<h3>Example: Find the distance "z"</h3>
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<p align="center"><img src="images/trig-cosruleex3.gif" alt="trig cos rule example" /></p>
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<p>The letters are different! But that doesn't matter. We can easily substitute x for a, y for b and z for c</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C)</span></div>
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<div class="row"><span class="left">x for a, y for b and z for c</span><span class="right">z<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> − 2xy cos(Z)</span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">z<sup>2</sup> = 9.4<sup>2</sup> + 6.5<sup>2</sup> − 2×9.4×6.5×cos(131º)</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right">z<sup>2</sup> = 88.36 + 42.25 − 122.2 × (−0.656...)</span></div>
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<div class="row"><span class="left"> </span><span class="right">z<sup>2</sup> = 130.61 + 80.17...</span></div>
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<div class="row"><span class="left"> </span><span class="right">z<sup>2</sup> = 210.78...</span></div>
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<div class="row"><span class="left"> </span><span class="right">z = √210.78... = <b>14.5</b> to 1 decimal place. </span></div>
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</div>
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<p align="center" class="large">Answer: z = 14.5 </p>
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<p>Did you notice that cos(131º) is negative and this changes the last sign in the calculation to <span class="large">+</span> (plus)? The cosine of an obtuse angle is always negative (see <a href="../geometry/unit-circle.html">Unit Circle</a>).</p>
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</div>
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<p> </p>
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</script> </div>
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<div class="related"> <a href="trig-sine-law.html">The Law of Sines</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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