public-notes/Hyperbolic Trig.md

#Math #Trig

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.

Definition in terms of a hyperbola

https://www.desmos.com/calculator/ixmjpfmukk

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

b = \frac {B\sqrt{B^2 - 1}} 2 -\int _1^B \sqrt {x^2 - 1} dx \\
= \frac {B\sqrt{B^2 - 1}} 2 - \frac {B\sqrt {B^2 - 1} - \ln(B + \sqrt {B^2 - 1})} 2\\
= \frac {\ln(B + \sqrt {B^2 - 1})} 2

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

a = \ln(B + \sqrt {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