Hyperbolic Functions

  • Thread starter hex.halo
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  • #1
hex.halo
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Homework Statement



Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

Homework Equations





The Attempt at a Solution



I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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[itex]cos(x+y)=cosxcosy-sinxsiny[/itex]

Osborn said that when you have the product of two sines, you replace the sines with sinh and a negative sign.

so therefore [itex]cosh(x+y)=cosxcosy-(-sinhxsinhy)[/itex]
 
  • #3
exk
119
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rock.freak missed the h in his last bit for the 2 cos terms on the rhs.
[itex]
cosh(x+y)=coshxcoshy-(-sinhxsinhy)
[/itex]
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
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970

Homework Statement



Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity

Homework Equations





The Attempt at a Solution



I can use Osborns rule to find the hyperbolic equivilent of the identity, however, I don't understand how I am to prove this identity...

Okay, you already know that cosh(x+ y)= cosh(x)cosh(y)+ sinh(x)sinh(y). Now replace cosh(x) by [itex](e^x+ e^{-x})/2[/itex], replace sinh(x)= [itex](e^x- e{-x})/2[/tex], the corresponding things for cosh(y) and sinh(y) and do the algebra. What do you get when you multiply
[tex]\frac{e^x+ e^{-x}}{2}\frac{e^y+ e^{-y}}{2}[/tex]
 
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