• Revision:Hyperbolic Functions

Contents

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

 cosh2x = cosh^2x + sinh^2x

Inverse Hyperbolic Functions and their logarithmic forms

As with trigonometric functions, you also get inverse hyperbolic functions. The function arsinh(x) is the inverse function of the function sinh(x).

 y = arsinhx

 x = sinhy

 x = (e^y - e^{-y})/2

 x = (e^{2y} - 1)/(2e^y)

 e^{2y} - 2x e^y - 1 = 0


 e^y is the positive root of previous quadratic equation. So we have:

 e^y = x + \sqrt{x^2+1}

 y = ln(x + \sqrt{x^2+1})

From this we have

 arsinhx = ln( x + \sqrt{x^2 + 1})


Similarly:

 arcoshx = ln (x + \sqrt{x^2-1})

 artanhx = \frac{1}{2}ln \left(\frac{1+x}{1-x}\right)

Graphs of Inverse Hyperbolic Functions

Try Learn together, TSR's study area

35,665
revision notes

39,264
mindmaps

39,659
crosswords

15,195
quizzes

create
a study planner

thousands
of discussions


Today on TSR
Poll
Wake up and smell the...
Study resources

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE