Why are cosh(x) and sinh(x) defined with denominator of 2?

Question

Looking at graphs of $2sinh(x)$ and $e^x$ they match perfectly in the right half of $xy$ plane. Wouldn't it make more sense to define $sinh(x)$ as $sinh(x)=e^x-e^{-x}$?

Answer 1

That might be useful if that definition was the only thing $\sinh$ was good for. However, there are many properties that necessitate the $\frac12$. Some algebraic properties are $$ e^x = \sinh x + \cosh x\\ \cosh^2 x - \sinh^2x = 1\\ $$ In addition, there are the algebraic-geometric properties, connecting $\sinh$ and $\cosh$ to the hyperbola $x^2-y^2 = 1$ as tightly as $\sin$ and $\cos$ are connected to the circle $x^2+y^2 = 1$. All of these are nice enough that we can stand to live with a $\frac12$ factor without too many issues.

Answer 2

For each function $f$, we can define an even and odd part: $$f_\text{e}(x)=\frac{f(x)+f(-x)}{2} \\ f_\text{o}(x)=\frac{f(x)-f(-x)}{2} $$ so that $f(x)=f_\text{e}(x)+f_\text{o}(x)$. If we apply this to $x\mapsto e^x$, we get $$e^x=\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2} $$ so $\cosh$ and $\sinh$ are, respectively, the even and odd parts of $e^x$.