Rearrangeing Inverse Hyperbolic functions

In summary, the conversation is about a person struggling with a mathematical problem and seeking help from others. They discuss the rearranging of an expression involving coshy^2 and sinhy^2, with one person providing a simplified solution and the other person trying to understand the book's solution.
  • #1
Observables
15
0
Hi,

My brain is not working today. So can someone please tell me what I am doing wrong.

(^2 = squared)

coshy^2 - sinhy^2 = 1, how do I rearrange this for coshy^2

I keep getting: coshy^2 = 1 + Sinhy^2

The book that I'm looking at has it this way: coshy^2 = Sinhy^2 + 1

Thanks

Obs
 
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  • #2
Time for brain overhaul. Both expressions evaluate the same.
 
  • #3
Observables said:
Hi,

My brain is not working today. So can someone please tell me what I am doing wrong.

(^2 = squared)

coshy^2 - sinhy^2 = 1, how do I rearrange this for coshy^2

I keep getting: coshy^2 = 1 + Sinhy^2

The book that I'm looking at has it this way: coshy^2 = Sinhy^2 + 1

Thanks

Obs

Hey observables and welcome to the forums.

The easiest way to find if the book is right (ie coshy^2 - sinhy^2 = 1) is to just use the definition of sinhy and coshy.

sinhy = 1/2 (e^x - e^-x)
coshy = 1/2(e^x + e^-x)

so sinhy^2 = 1/4 [e^2x - 2 + e^-2x)
coshy^2 = 1/4(e^2x + 2 - e^-2x)

coshy^2 - sinhy^2 = 1/2 - (-1/2) = 1
 
  • #4
SteamKing said:
Time for brain overhaul. Both expressions evaluate the same.

Hi,

I realized that, but I couldn't figure out how the book was doing the rearranging. I've been differentiating a few Inverse Hyperbolic functions, and I thought the reason that the book rearranged differently to me was that somewhere down the line it would become clear why it was done.

Obs.
 
  • #5
chiro said:
Hey observables and welcome to the forums.

The easiest way to find if the book is right (ie coshy^2 - sinhy^2 = 1) is to just use the definition of sinhy and coshy.

sinhy = 1/2 (e^x - e^-x)
coshy = 1/2(e^x + e^-x)

so sinhy^2 = 1/4 [e^2x - 2 + e^-2x)
coshy^2 = 1/4(e^2x + 2 - e^-2x)

coshy^2 - sinhy^2 = 1/2 - (-1/2) = 1

Hi Chiro,

Thanks for the info.

Obs.
 
  • #6
Gezzz breakfast time must be confusing at your place. Trying to decide whether to have "eggs and toast" or "toast and eggs". It must keep you occupied for hours.
 

1. What are inverse hyperbolic functions?

Inverse hyperbolic functions are mathematical functions that are the inverse of hyperbolic functions. They are used to find the input value (x) that would produce a given output value (y) in a hyperbolic function. Inverse hyperbolic functions are denoted by "arc" or "ar" before the name of the corresponding hyperbolic function, such as arsinh, arcosh, and artanh.

2. How are inverse hyperbolic functions different from regular inverse functions?

Regular inverse functions have a one-to-one relationship between the input and output values, while inverse hyperbolic functions have a many-to-one relationship. This means that multiple input values can produce the same output value in inverse hyperbolic functions. Additionally, inverse hyperbolic functions are defined for a specific range of values, unlike regular inverse functions which can be defined for any input value.

3. What is the purpose of rearranging inverse hyperbolic functions?

Rearranging inverse hyperbolic functions can help simplify equations and make them easier to solve. It can also help in finding the input value (x) when the output value (y) is known, which can be useful in applications such as calculus and physics.

4. What are the common techniques used for rearranging inverse hyperbolic functions?

The most common techniques for rearranging inverse hyperbolic functions involve using algebraic manipulations, trigonometric identities, and logarithmic properties. These techniques can help convert inverse hyperbolic functions into their corresponding hyperbolic functions or simplify the expressions to make them easier to solve.

5. Can inverse hyperbolic functions be graphed?

Yes, inverse hyperbolic functions can be graphed, just like regular hyperbolic functions. However, since inverse hyperbolic functions have a many-to-one relationship, their graphs may not pass the vertical line test. This means that a vertical line can intersect the graph at more than one point, making them not functions in the strict sense. However, they are still useful in many mathematical applications and can represent real-world phenomena.

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