A calculus of the absurd

15.3 Relationship to trig functions

Previously (in Section 14.7.2) we showed two very useful identities

\begin{align} & \cos (x) = \frac {e^{ix} + e^{-ix}}{2} \\ & \sin (x) = \frac {e^{ix} - e^{-ix}}{-2i} \end{align}

If, in either of these equations we set $x=i\theta $ we can write $\sin (i\theta )$ and $\cos (i\theta )$ in terms of $\sinh (x)$ and $\cosh (x)$, respectively.

\begin{align} \sin (i\theta ) & = \frac {e^{i(i\theta )} - e^{-i(i\theta )}}{-2i} \\ & = \frac {e^{-\theta } - e^{\theta }}{-2i} \\ & = i\frac {e^\theta - e^{-\theta }}{2} \text { multiplying by $\frac {i}{i}=1$} \\ & = i\sinh (\theta ) \end{align}

\begin{align} \cos (i\theta ) & = \frac {e^{i(i\theta )} + e^{-i(i\theta )}}{2} \\ & = \frac {e^{-\theta } + e^{\theta }}{2} \\ & = \cosh (\theta ) \end{align}