Hyperbolic Functions

Hyperbolic Functions

Let's see the basics of the hyperbolic functions (sinh x, coshx): definition, property, and the proof of the property.

Definition

sinh x = (e^x - e^-x)/2, cosh x = (e^x + e^-x)/2

Hyperbolic sine:
sinh x is defined as (ex - e-x) / 2.

Hyperbolic cosine:
cosh x is defined as (ex + e-x) / 2.

Property

cosh^2 x - sinh^2 x = 1

Recall that
the point on a unit circle, x2 + y2 = 1,
is (cos x, sin x).

Point on a unit circle (using sine and cosine)

Similiarly,
the point on a unit hyperbola, x2 - y2 = 1,
is (cosh x, sinh x).

This is why the letter 'h' is added.

So cosh2 x - sinh2 x = 1.

Proof

Hyperbolic Functions: Proof of the Property

cosh x = (ex + e-x) / 2
sinh x = (ex - e-x) / 2

Then cosh2 x - sinh2 x = [(ex + e-x) / 2]2 - [(ex - e-x) / 2]2.

Square of a sum

Square of a difference

Cancel the dark gray terms.

So cosh2 x - sinh2 x = 1.