Hyperbolic Functions

Another question arises if we are familiar with the HYPERBOLIC FUNCTIONS

$$\hbox{\fbox{ ${\displaystyle \cosh x \; \equiv \; {1\over2} \left( e^x \; + \; e^{-x} \right) }$ } }$$

$$\hbox{\fbox{ ${\displaystyle \sinh x \; \equiv \; {1\over2} \left( e^x \; - \; e^{-x} \right) }$ } }$$

These are so similar to the definitions of the $\sin$ and $\cos$ in terms of complex exponentials that we suspect a connection between $\cosh$ and $\cos$ that is deeper than just the fact that the names are so similar (which should of course have made us suspicious in the first place). I will leave it as a (trivial) exercise for the reader to show that

${\displaystyle \cos \theta = \cosh(i\theta) \qquad \qquad i \sin \theta = \sinh(i\theta) }$
 
${\displaystyle \cosh x = \cos(ix) \qquad \qquad i \sinh x = \sin(ix) }$.