exp, trig, hyp
What can we do with Euler’s Formula?
\[ e^{i t} = \cos (t) + i \, \sin (t) \]
Substituting \(-t\) for \(t\) gives \(e^{i (-t)} = \cos (-t) + i \, \sin (-t) = \cos (t) - i \, \sin (t)\), because \(\cos (t)\) is an even function and \(\sin (t)\) is an odd function.
That gives us
\[ e^{i t} + e^{-i t} = 2 \cos (t) \]
\[ \frac {e^{i t} + e^{-i t}}{2} = \cos (t) \]
It also gives us
\[ e^{i t} - e^{-i t} = 2 i \sin (t) \]
\[ \frac {e^{i t} - e^{-i t}}{2 i} = \sin (t) \]
We have also seen that
\[ \cosh (t) = \frac {e^t + e^{-t}}{2} \, \text { and } \, \sinh (t) = \frac {e^t - e^{-t}}{2} \]
This tells us that
\[ \cos (t) = \frac {e^{i t} + e^{-i t}}{2} = \cosh (i t) \]
\[ \sin (t) = \frac {e^{i t} - e^{-i t}}{2 i} = -i \sinh (i t) \]
If we work with Complex numbers, then trigonometric and hyperbolic functions are expressible in terms of exponential functions. They are different ways of expressing the same structure.
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More Examples of the Complex Bridge