• Revision: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

 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})


 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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