From Euler's identity one may obtain that, $$\sin x=\dfrac{e^{ix}-e^{-ix}}{2i}$$
$$\cos x=\dfrac{e^{ix}+e^{-ix}}{2}$$
However, it looks quite same to the hyperbolic functions such as $$\sinh x=\dfrac{e^x-e^{-x}}{2}$$ $$\cosh x=\dfrac{e^x+e^{-x}}{2}$$ where the imaginary unit, $i$, is omitted.
Now my question is, what's the interconnection between them?
One may answer $\sinh x=-i\sin ix$ or, $\cosh x=\cos ix$ but that doesn't help me to see why it's true. Or even the bigger question what was the necessity to introduce hyperbolic functions?
I expect you to help me with this.
P.S. I'm in college. I am seeking for intuition rather than tons of formal theorem.
Best Answer
Just plug the argument $ix$:
$$\sin ix=\frac{e^{i(ix)}-e^{-i(ix)}}{2i}=\frac{e^{-x}-e^{x}}{2i}=i\sinh x.$$
The trigonometric and hyperbolic functions are two specializations of the more general complex exponential
$$e^z=e^{x+iy}$$ that is an essential function in calculus.
If you want an explanation that does not involve complex numbers appear explicitly, a true source of trigonometric and hyperbolic function is found in the differential equations
$$y''+y=0$$ (such as that of the oscillations of the pendulum) and $$y''-y=0$$ (found to explain the shape of the catenary).
But if you are not familiar with these concepts and their usefulness in many branches of mathematics and physics, this explanation might be vain.