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<h1 align="center">Hyperbolic Functions</h1>
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<p>The two basic hyperbolic functions are "sinh" and "cosh":</p>
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<h3>Hyperbolic Sine:</h3>
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<p class="center larger">sinh(x) = <span class="intbl">
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<em>e<sup>x</sup> − e<sup>−x</sup></em>
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<strong>2</strong></span><!-- <p>They (quite amazingly) have many similarities to the trigonometric functions <a href="../sine-cosine-tangent.html">sin, cos and tan</a>.</p>
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--> </p>
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<p class="center"><i>(pronounced "shine")</i></p>
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<h3>Hyperbolic Cosine:</h3>
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<p class="center larger">cosh(x) = <span class="intbl"> <em>e<sup>x</sup> + e<sup>−x</sup></em> <strong>2</strong> </span></p>
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<p class="center"><i>(pronounced "cosh")</i></p>
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<p> </p>
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<p>They use the <a href="function-exponential.html">natural exponential function</a> <b>e<sup>x</sup> </b></p>
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<p>And are not the same as sin(x) and cos(x), but a little bit similar:</p>
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<p class="center"><img src="images/function-sinh-sin.svg" alt="sinh vs sin function" /><br>
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<i>sinh vs sin</i></p>
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<p class="center"><img src="images/function-cosh-cos.svg" alt="cosh vs cos function" /><br>
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<i>cosh vs cos</i></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../calculus/images/rope-bridge.jpg" width="300" height="226" alt="rope bridge" /></p><h2>Catenary</h2>
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<p>One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains. </p>
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<p>A hanging cable forms a curve called a <b>catenary</b> defined using the <b>cosh</b> function:</p>
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<p class="center larger">f(x) = a cosh(x/a)</p>
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<p>Like in this example from the page <a href="../calculus/arc-length.html">arc length</a> :</p>
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<p><img src="../calculus/images/catenary-1.gif" width="480" height="131" alt="catenary graph" /></p>
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<h2>Other Hyperbolic Functions</h2>
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<p>From <b>sinh</b> and <b>cosh</b> we can create:</p>
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<h3> Hyperbolic tangent "tanh" <span class="center"><i>(pronounced "than")</i></span>:</h3>
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<p class="center larger">tanh(x) = <span class="intbl"> <em>sinh(x)</em> <strong>cosh(x)</strong> </span> = <span class="intbl"> <em>e<sup>x</sup> − e<sup>−x</sup></em> <strong>e<sup>x</sup> + e<sup>−x</sup></strong> </span></p>
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<p class="center"><img src="images/function-tanh-tan.svg" alt="tanh vs tan function" /><br>
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<i>tanh vs tan</i></p>
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<p> </p>
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<h3>Hyperbolic cotangent:</h3>
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<p class="center larger">coth(x) = <span class="intbl"> <em>cosh(x)</em> <strong>sinh(x)</strong></span> = <span class="intbl"> <em>e<sup>x</sup> + e<sup>−x</sup></em> <strong>e<sup>x</sup> − e<sup>−x</sup></strong></span></p>
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<p> </p>
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<h3>Hyperbolic secant:</h3>
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<p class="center larger">sech(x) = <span class="intbl"> <em>1</em> <strong>cosh(x)</strong></span> = <span class="intbl"> <em>2</em> <strong>e<sup>x</sup> + e<sup>−x</sup></strong></span></p>
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<p> </p>
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<h3>Hyperbolic cosecant "csch" or "cosech": </h3>
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<p class="center larger">csch(x) = <span class="intbl"> <em>1</em> <strong>sinh(x)</strong> </span> = <span class="intbl"> <em>2</em> <strong>e<sup>x</sup> − e<sup>−x</sup></strong> </span></p>
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<h2>Why the Word "Hyperbolic" ?</h2>
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<p>Because it comes from measurements made on a <a href="../geometry/hyperbola.html">Hyperbola</a>:</p>
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<p class="center"><img src="../geometry/images/hyperbolic-functions.svg" alt="tanh vs tan function" /></p>
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So, just like the <a href="../sine-cosine-tangent.html">trigonometric</a> functions relate to a circle, the hyperbolic functions relate to a hyperbola.
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<h2>Identities</h2>
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<ul>
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<li>sinh(−x) = −sinh(x)</li>
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<li>cosh(−x) = cosh(x)</li>
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</ul>
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<p>And</p>
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<ul>
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<li>tanh(−x) = −tanh(x)</li>
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<li>coth(−x) = −coth(x)</li>
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<li>sech(−x) = sech(x)</li>
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<li>csch(−x) = −csch(x)</li>
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</ul>
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<h2>Odd and Even</h2>
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<p>Both <b>cosh</b> and <b>sech</b> are <a href="../algebra/functions-odd-even.html">Even Functions</a>, the rest are Odd Functions.</p>
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<h2>Derivatives</h2>
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<p><a href="../calculus/derivatives-introduction.html">Derivatives</a> are:</p>
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<p class="center larger"><span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span> sinh(x) = cosh(x)</p>
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<p class="center larger"><span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span> cosh(x) = sinh(x)<span class="intbl"></span></p>
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<p class="center larger"><span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span> tanh(x) = 1 − tanh<sup>2</sup>(x) </p>
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<p> </p>
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<div class="related"><a href="functions-common.html">Common Functions Reference</a><a href="index.html">Sets Index</a></div>
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