# Why does the definition of cosh and sinh contain division by 2?

1. Feb 2, 2010

### Juwane

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

In the above definition, why there is division by 2? Is it there so that when x=0, y could be 1?

2. Feb 2, 2010

### dextercioby

The 1/2 comes from the related function <cos>. The definition of <cos> in terms of exponentials has a 1/2. To relate the hyperbolic functions in the simplest way to their circular counterparts, the 1/2 must be in the definition of <cosh/ch>.

3. Feb 2, 2010

### HallsofIvy

There is a thread under "General Math" about even and odd functions in which I addressed exactly this point!
They are also, by the way, the "fundamental solutions" to the differential equation $d^2y/dt^2= y$. That is, cosh(x) is the solution to that equation such that y(0)= 1 and y'(0)= 0 and sinh(x) is the solution such that y(0)= 0, y'(1)= 0. If y is a solution to that differential equation with y(0)= A, y'(0)= B, then y(x)= A cosh(x)+ B sinh(x)- the coefficients are just the value of y and its derivative at 0.