What is the relation to $\sinh{x},\cosh{x}$ and $\sin{x},\cos{x}$

Question

I've learned what $\sinh{x},\cosh{x}$ (the hyperbolic trig functions) are defined as formula, but how is it related to $\sin{x},\cos{x}?$
The only thing I've noticed is that $\cosh^2(x)-\sinh^2(x)=1.$

Answer 1

They’re related by Euler’s formula. Since $e^{ix}=\cos x+i \sin x$ we have $e^{-ix}=\cos x-i \sin x$. This reveals,

$$\cosh (ix)=\cos x$$

$$\sinh (ix)=i \sin x$$