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<h1 class="center">Magic Hexagon for Trig Identities</h1>
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<p> </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>This hexagon is a special <b>diagram</b><br>
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to help you remember some <a href="trigonometric-identities.html">Trigonometric Identities</a></td>
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<td> </td>
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<td><img src="images/magic-hexagon.svg" alt="magic hexagon" height="133" width="178"></td>
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</tr>
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</tbody></table>
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<p>Sketch the diagram when you are struggling with trig identities ... it may help you! Here is how:</p>
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<h2>Building It: The Quotient Identities</h2>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p>Start with:</p>
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<p class="center"><b>tan(x) = sin(x) /</b> <b>cos(x)</b></p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><span class="center"><i><img src="images/tsk-tsk.gif" alt="tsk tsk" height="54" width="51"></i></span></td>
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<td><span class="center"><i>To help you remember<br>
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think "tsc !"</i></span></td>
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</tr>
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</tbody></table></td>
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<td> </td>
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<td><img src="images/magic-hexagon-2.svg" alt="magic hexagon tan(x) = sin(x) / cos(x)" height="133" width="178"></td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td>
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<p>Then add:</p>
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<ul>
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<li>cot (which is <b>co</b>tangent) on the opposite<br>
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side of the hexagon to tan</li>
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<li>csc (which is <b>co</b>secant) next, and</li>
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<li>sec (which is secant) last</li>
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</ul></td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"><img src="images/magic-hexagon.svg" alt="magic hexagon" height="133" width="178"></td>
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</tr>
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<tr>
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<td colspan="3" align="center"><i>To help you remember: the "co" functions are all on the right</i></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p><b>OK, we have now built our hexagon, what do we get out of it?</b></p>
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<p>Well, we can now follow "around the clock" (either direction) to get all the "Quotient Identities":</p>
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<div style="float:left; margin: 0 10px 5px 0;">
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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 style="text-align:center;"><b>Clockwise</b></td>
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</tr>
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<tr>
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<td>
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<ul>
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<li>tan(x) = sin(x) / cos(x)</li>
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<li>sin(x) = cos(x) / cot(x)</li>
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<li>cos(x) = cot(x) / csc(x)</li>
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<li>cot(x) = csc(x) / sec(x)</li>
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<li>csc(x) = sec(x) / tan(x)</li>
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<li>sec(x) = tan(x) / sin(x)</li>
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</ul> </td>
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</tr>
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</tbody></table>
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</div>
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<div style="float:right; margin: 0 0 5px 10px;">
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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 style="text-align:center;"><b>Counterclockwise</b></td>
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</tr>
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<tr>
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<td>
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<ul>
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<li>cos(x) = sin(x) / tan(x)</li>
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<li>sin(x) = tan(x) / sec(x)</li>
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<li>tan(x) = sec(x) / csc(x)</li>
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<li>sec(x) = csc(x) / cot(x)</li>
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<li>csc(x) = cot(x) / cos(x)</li>
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<li>cot(x) = cos(x) / sin(x)</li>
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</ul></td>
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</tr>
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</tbody></table>
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</div>
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<div style="clear:both"></div>
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<h2>Product Identities</h2>
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<p>The hexagon also shows that a function <b>between</b> any two functions is equal to them multiplied together
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(if they are opposite each other, then the "1" is between them):</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><span class="center"><img src="images/magic-hexagon-prod1.svg" alt="magic hexagon tan(x)cos(x) = sin(x)" style="max-width:100%" height="135" width="279"></span></td>
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<td> </td>
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<td><span class="center"><img src="images/magic-hexagon-prod2.svg" alt="magic hexagon tan(x)cot(x) = 1" style="max-width:100%" height="132" width="268"></span></td>
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</tr>
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<tr style="text-align:center;">
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<td><span class="center"> Example:<b><br>
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tan(x)cos(x) = sin(x)</b></span></td>
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<td> </td>
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<td><span class="center"> Example:<br>
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<b>tan(x)cot(x) = 1</b></span></td>
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</tr>
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</tbody></table>
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<p>Some more examples:</p>
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<ul>
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<li>sin(x)csc(x) = 1</li>
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<li>tan(x)csc(x) = sec(x)</li>
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<li>sin(x)sec(x) = tan(x)</li>
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</ul>
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<h2>But Wait, There is More!</h2>
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<p>You can also get the "Reciprocal Identities", by going "through the 1"</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/magic-hexagon-3.svg" alt="magic hexagon sin(x) = 1/csc(x)" height="" width=""></td>
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<td> </td>
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<td>Here you can see that <b>sin(x) = 1 / csc(x)</b></td>
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</tr>
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</tbody></table>
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<p>Here is the full set:</p>
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<ul>
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<li>sin(x) = 1 / csc(x)</li>
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<li>cos(x) = 1 / sec(x)</li>
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<li>cot(x) = 1 / tan(x)</li>
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<li>csc(x) = 1 / sin(x)</li>
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<li>sec(x) = 1 / cos(x)</li>
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<li>tan(x) = 1 / cot(x)</li>
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</ul>
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<h2>Bonus!</h2>
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<p>AND we also get these co-function identities:</p>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/magic-hexagon-cofunc-deg.svg" alt="magic hexagon sin(x) = cos(90-x), tan(x) = cot(90-x), sec(x) = csc(90-x), " height="118" width="300"></p>
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<p>Examples:</p>
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<ul>
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<li>sin(30°) = cos(60°)</li>
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<li>tan(80°) = cot(10°)</li>
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<li>sec(40°) = csc(50°)</li>
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</ul>
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<div style="clear:both"></div>
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<p><br></p>
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<p>Or, if you prefer, in <a href="../geometry/radians.html">radians</a>:</p>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/magic-hexagon-cofunc-rad.svg" alt="magic hexagon sin(x) = cos(pi/2-x), tan(x) = cot(pi/2-x), sec(x) = csc(pi/2-x), " height="118" width="296"></p>
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<p>Examples:</p>
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<ul>
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<li>sin(0.1<span class="times">π</span>) = cos(0.4<span class="times">π</span>)</li>
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<li>tan(<span class="times">π</span>/4) = cot(<span class="times">π</span>/4)</li>
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<li>sec(<span class="times">π</span>/3) = csc(<span class="times">π</span>/6)</li>
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</ul>
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<div style="clear:both"></div>
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<h2>Double Bonus: The Pythagorean Identities</h2>
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<p>The <a href="../geometry/unit-circle.html">Unit Circle</a> shows us that</p>
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<p class="center"><b>sin<sup>2 </sup>x + cos<sup>2</sup> x = 1</b></p>
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<p>The magic hexagon can help us remember that, too, by going clockwise around any of these three triangles:</p>
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<p class="center"><img src="images/magic-hexagon-5.svg" alt="magic hexagon sin^2(x) + cos^2(x)=1" height="" width=""></p>
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<p>And we have:</p>
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<ul>
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<li>sin<sup>2</sup>(x) + cos<sup>2</sup>(x) = 1</li>
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<li>1 + cot<sup>2</sup>(x) = csc<sup>2</sup>(x)</li>
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<li>tan<sup>2</sup>(x) + 1 = sec<sup>2</sup>(x)</li>
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</ul>
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<div style="clear:both"></div>
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<p>You can also travel counterclockwise around a triangle, for example:</p>
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<ul>
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<li>1 − cos<sup>2</sup>(x) = sin<sup>2</sup>(x)</li>
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</ul>
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<h2>Triple Bonus: Quadrants Positive</h2>
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<p>It can also help us remember which <a href="trig-four-quadrants.html">quadrants</a> each function is positive in.</p>
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<p class="center" style="vertical-align:middle;"><img src="images/trig-graph-quadrants.svg" style="vertical-align:middle;" alt="trig graph 4 quadrants" height="" width="">
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<img src="images/magic-hexagon-quads.svg" style="vertical-align:middle;" alt="magic hexagon quadrants positive" height="" width=""></p>
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<p><br></p>
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<p class="larger center">I hope this helps you!</p>
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<p> </p>
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<div class="questions">7908, 7911, 7912, 7915, 7916, 7917, 7918, 7921, 7944, 7946</div>
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<div class="related">
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<a href="trigonometric-identities.html">Trigonometric Identities</a>
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<a href="index.html">Algebra Index</a>
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