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Revision:Hyperbolic Functions

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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

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