Definition

Definition in terms of $e$

We define $cosh$ and $sinh$ to be the even and odd parts of $e^{x}$ respectively:

cosh x = \frac{e ^{x} + e ^{- x}}{2} sinh x = \frac{e ^{x} - e ^{- x}}{2}

Note this gives us:

sinh x + cosh x = e^{x}

similar to Euler’s Formula for circular trig functions.

Know that the geometric definition of $cosh$ is that $B = cosh 2 b$ , where $b$ is the blue area. To find $b$ , we can use:

b = \frac{B B ^{2} - 1}{2} - \int_{1}^{B} x^{2} - 1 d x = \frac{B B ^{2} - 1}{2} - \frac{B B ^{2} - 1 - ln ( B + B ^{2} - 1 )}{2} = \frac{ln ( B + B ^{2} - 1 )}{2}

Now let $a = 2 b = - ln (B + B^{2} - 1)$ . Now we can solve for $B$ in terms of $a$ to define $cosh$ :

a = ln (B + B^{2} - 1) B = \frac{e ^{a} + e ^{- a}}{2} cosh x = \frac{e ^{x} + e ^{- x}}{2}

Now using the fact $cosh$ and $sinh$ lie on a hyperbola (can be proved algebraically) we get:

sinh x = \frac{e ^{x} - e ^{- x}}{2}