Hyperbolic Functions

Who Needs Them?

Chapter 6.9

The gentle arc of a suspension bridge or the

complex geometry of Einstein’s

Theory of Relativity provide us with real-life

examples of Hyperbolic Functions!

Basic Definitions…

• The hyperbolic functions are built out of the exponential function:

• We can form other hyperbolic functions:

sinh( ) cosh( )2 2

x x x xe e e ex x

sinh( ) 2tanh( )cosh( )

2

x x

x x

x x x x

e ex e e

xx e e e e

Contrasting Circular Functions with Hyperbolic Functions

• Sine and Cosine are bounded, periodic functions

• Sinh and Cosh are unbounded

• Sine and Cosine are related to the UNIT CIRCLE – Hyperbolic functions are related to the UNIT HYPERBOLA!

Review Identities

• Pg 165 – Basic Relations

• Pg 166-167 - Derivative Patterns

Compare and Contrast• The hyperbolic functions, their inverses and derivatives

often appear very similar to the circular functions (sin, cos etc)…

• So – how do I remember all this stuff?• You don’t! Just be sure to understand that these are a

new set of functions with some important similarities and differences with the ordinary circular functions. You will learn some of the easier patterns but in general it is fair to rely on Tables or Computer Algebra Systems.

1 12 2

1 1tanh ( ) , tan ( )

1 1x xD x D x

x x

Samples…

• Pg 167: 7, 11, 17, 21, 31