How Do Hyperbolic Functions Relate to Trigonometric Functions?

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SUMMARY

The discussion centers on the relationships between hyperbolic functions and trigonometric functions, specifically demonstrating that cosh(z) = cos(iz) and the corresponding relationship for sinh(z). Participants explored the derivatives and integrals of these functions, confirming that d/dx(cosh(z)) = sinh(z) and d/dx(sinh(z)) = cosh(z). Additionally, they validated the identity cosh²(z) - sinh²(z) = 1 and discussed the integral of 1/sqrt(1+x²) equating to arcsinh(x) through substitution x = sinh(z).

PREREQUISITES
  • Understanding of hyperbolic functions, specifically cosh(z) and sinh(z)
  • Knowledge of derivatives and integrals in calculus
  • Familiarity with trigonometric identities and their hyperbolic counterparts
  • Ability to perform substitutions in integrals
NEXT STEPS
  • Study the derivation of hyperbolic functions from exponential definitions
  • Learn about the properties and applications of hyperbolic identities
  • Explore advanced integration techniques, particularly involving substitutions
  • Investigate the relationship between hyperbolic and trigonometric functions in greater depth
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and advanced algebra, as well as anyone interested in the applications of hyperbolic functions in physics and engineering.

  • #31
Oblio said:
It seems wrong that I can say
the integral of 1/cosh is 1/sinh

you need to change the dx... if x = sinhz, what is dx?
 
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  • #32
coshz..
 
  • #33
Oblio said:
coshz..

yes dx = coshz*dz.

so what do you get from the integral?
 
  • #34
do i just do the integral of each the numerator and denominator?
 
  • #35
either way i don't see how i won't just get 1/1 = 1...
 
  • #36
Oblio said:
either way i don't see how i won't just get 1/1 = 1...

Right... you get the integral of dz, which is just z. z = arcsinh(x), since x = sinh(z).
 
  • #37
learningphysics said:
Right... you get the integral of dz, which is just z. z = arcsinh(x), since x = sinh(z).

I had accidentally switched some z's for x's... that's a bad thing. lol

I get it.
The only thing that seems odd now, is
WHY can one just say x=sinhz? Was that shown in the question somehow already that I'm not seeing?
 
  • #38
Oblio said:
I had accidentally switched some z's for x's... that's a bad thing. lol

I get it.
The only thing that seems odd now, is
WHY can one just say x=sinhz? Was that shown in the question somehow already that I'm not seeing?

I agree with you that it is odd the order we did it...

The better way to approach is to let z = arcsinh(x) (so we're introducing a new variable z... x is already given.)... and then from there saying x = sinh(z). I think that makes more sense.
 
  • #39
So, am I correct in thinking though, that this is only true SINCE x =sinhx.
I mean, they made it a 'Hint', but that definition of x is completely necessary to solve it, isn't it?

Without the hint, it couldn't be done?
 
  • #40
I guess I mean, it should written that the statement is true WHEN x =sinhz. (made a typo above i see)
 
  • #41
agree or am i missing something still? lol
 
  • #42
Oblio said:
So, am I correct in thinking though, that this is only true SINCE x =sinhx.
I mean, they made it a 'Hint', but that definition of x is completely necessary to solve it, isn't it?

Without the hint, it couldn't be done?

No, that definition wasn't necessary...

We need to define z = arcsinh(x). we're not defining x... x is already given... we define z = arcsinh(x), and then we get x = sinh(z). but this isn't a definition... just a relationship... z is the variable being defined.

given any variable... say u...

I can simply define v = arcsinh(u). now I have defined v not u... but u = sinh(v), because of the way v has been defined.
 
  • #43
Oblio said:
I guess I mean, it should written that the statement is true WHEN x =sinhz. (made a typo above i see)

z is defined in such a way that x = sinhz... in other words we create a variable z that is defined as arcsinh(x).

so z is being defined, not x.
 
  • #44
Oblio said:
So, am I correct in thinking though, that this is only true SINCE x =sinhx.
I mean, they made it a 'Hint', but that definition of x is completely necessary to solve it, isn't it?

Without the hint, it couldn't be done?

Ok, switch x for z.
Was it necessary for the question to define z?
 
  • #45
Oblio said:
Ok, switch x for z.
Was it necessary for the question to define z?

The question didn't have to define it... the question suggested it as a hint to help you solve the problem...

Even if the question didn't define it... you could define a variable as arcsinh(x), to help you solve the problem.

The question is simply prove that arcsin(x) = integral(...)

The z = arcsinh(x) is only a hint to solve the problem... the question is complete even without that hint.
 

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