Revision:Hyperbolic Functions

Register

About Us | Help | Sign in

Create New Article | Editing Help

Advanced Search

Edit \| Hide
Welcome \| Study Help \| A Levels \| Applications, UCAS etc. \| University Discussion \| Careers \| General Discussion \| Have Your Say \| Health & Relationships \| Debate & Discussion \| Technology \| Sport \| Entertainment \| Chat \| Ask A Moderator

Revision:Hyperbolic Functions

From The Student Room

Jump to:navigation, search

1 Hyperbolic Functions
2 Graphs of Hyperbolic Functions
3 Hyperbolic Identities
4 Osborn's Rule
5 Inverse Hyperbolic Functions and their logarithmic forms
6 Graphs of Inverse Hyperbolic Functions

Hyperbolic Functions

These are functions that have strong similarities to Trigonometric functions. They are combined functions of e^x and e^{-x}. They are called Hyperbolic Functions.

$sinhx = \frac{1}{2}(e^{x}-e^{-x})$

$coshx = \frac{1}{2}(e^{x}+e^{-x})$

$tanhx = \frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{e^{2x}-1}{e^{2x}+1}$

$sechx = \frac{1}{coshx} = \frac{2}{e^x+e^{-x}}$

$cosechx = \frac{1}{sinhx} = \frac{2}{e^x-e^{-x}}$

$cothx = \frac{1}{tanhx} = \frac{e^{2x}+1}{e^{2x}-1}$

Graphs of Hyperbolic Functions

Hyperbolic Identities

cosh^2x - sinh^2x = 1

1 - tanh^2x = sech^2x

coth^2x - 1 = cosech^2x

$sinh(A \pm B) = sinhAcoshB \pm coshAsinhB$

$cosh(A \pm B) = coshAcoshB \pm sinhAsinhB$

$tanh(A \pm B) = \frac{tanhA \pm tanhB}{1 \pm tanhAtanhB}$

sinh2x = 2sinhxcoshx

cosh2x = cosh^2x + sinh^2x = 2cosh^2x - 1 = 1 + 2sinh^2x

$tanh2x = \frac{2tanhx}{1+tanh^2x}$

Osborn's Rule

In a trigonometric identity you can replace each trigonometric function by the corresponding hyperbolic function to form the corresponding identity but you must change the sign of the every product of two sines.

For example

cos2x = cos^2x - sin^2x