15.3 Relationship to trig functions ‣ Chapter 15 Hyperbolic functions ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd‣ Chapter 15 Hyperbolic functions ‣ Part I A Level mathematics (and further mathematics) ‣ A calculus of the absurd

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

	$\displaystyle\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$		(15.25)
	$\displaystyle\sin(x)=\frac{e^{ix}-e^{-ix}}{-2i}$		(15.26)

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.

$\displaystyle\sin(i\theta)$	$\displaystyle=\frac{e^{i(i\theta)}-e^{-i(i\theta)}}{-2i}$	(15.27)
	$\displaystyle=\frac{e^{-\theta}-e^{\theta}}{-2i}$	(15.28)
	$\displaystyle=i\frac{e^{\theta}-e^{-\theta}}{2}\text{ multiplying by $\frac{i}% {i}=1$}$	(15.29)
	$\displaystyle=i\sinh(\theta)$	(15.30)

$\displaystyle\cos(i\theta)$	$\displaystyle=\frac{e^{i(i\theta)}+e^{-i(i\theta)}}{2}$	(15.31)
	$\displaystyle=\frac{e^{-\theta}+e^{\theta}}{2}$	(15.32)
	$\displaystyle=\cosh(\theta)$	(15.33)