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      <text>10.3.3.1 Hyperbolic and trigonometric functions</text>
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      <text><![CDATA[<h3>Hyperbolic and trigonometric functions</h3>
<p>We have seen previously that \[\qquad e^{i\theta}=\cos\theta +i\sin\theta\] It follows from this that \[\qquad e^{-i\theta}=\cos (-\theta)+i\sin (-\theta)=\cos\theta -i\sin\theta\] Now if we add these two relations together we obtain \[\qquad \cos\theta ={e^{i\theta}+e^{-i\theta}\over {2}}\] whereas if we subtract the second from the first we have\[\qquad \sin\theta ={e^{i\theta}-e^{-i\theta}\over {2i}}\] These new relations are reminiscent of the hyperbolic functions introduced in a previous quiz. There we defined \(\cosh x\) and \(\sinh x\) in terms of the exponential function: \[\qquad \cosh x ={e^{x}+e^{-x}\over {2}}\qquad \sinh x ={e^{x}-e^{-x}\over {2}}\] In fact, if we replace \(x\) by \(i\theta\) in these last two equations we obtain \[\qquad \cosh (i\theta) ={e^{i\theta } +e^{-i\theta}\over 2}\equiv \cos \theta\qquad \text{and}\qquad \sinh (i\theta) ={e^{i\theta } -e^{-i\theta}\over 2}\equiv i\sin \theta\] Although, by our notation, we have implied that both \(x\) and \(\theta\) are real quantities in fact these expressions for \(\cosh\) and \(\sinh\) in terms of \(\cos\) and \(\sin\) are quite general.<br></p>
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<h4>Key Point</h4>
<p> If \(z\) is any complex number then \[\cosh(iz) \equiv \cos z\quad\text{and}\quad\sinh(iz) \equiv i\sin z\] Equivalently, replacing \(z\) by \(iz\): \[\cosh z \equiv \cos(iz) \quad\text{and}\quad i\sinh z \equiv \sin(iz)\]</p></div>]]></text>
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