Why is sinh-1(x) not equal to 1/sinh(x)?

[Stephanus]

Dear PF Forum.
I saw once that sinh^-1(x) is arcsinh(x). The reverse of sinh
Why not sinh^-1(x) is 1/sinh(x)?
While x^-1 = 1/x
Is it just a 'convention' between mathematician?

Thanks.

 

[Mentallic]

Dear PF Forum.
I saw once that sinh^-1(x) is arcsinh(x). The reverse of sinh
Why not sinh^-1(x) is 1/sinh(x)?
While x^-1 = 1/x
Is it just a 'convention' between mathematician?

Thanks.

For the same reason that [itex]\sin^{-1}(x)=\arcsin(x)[/itex]. It's merely a convention. Meanwhile, [itex]1/\sin(x)[/itex] is instead given the function name [itex]\csc(x)[/itex] and similarly, [itex]1/\sinh(x)=csch(x)[/itex].

 

[Stephanus]

Wow, that fast. Thanks!
So what is 1/sin(x)? No power?
And what is sin^-2(x)? 1/sin²(x)?

 

[Stephanus]

Meanwhile, [itex]1/\sin(x)[/itex] is instead given the function name [itex]\csc(x)[/itex]

Csc, cosecant? If it's cosecant than it's ##Arcsin(\sqrt{1-sin^2(x)})##

 

[Mentallic]

Wow, that fast. Thanks!
So what is 1/sin(x)? No power?

Right, you wouldn't represent 1/sin(x) with a power. You'd leave it as such or replace it with csc(x).

And what is sin^-2(x)? 1/sin²(x)?

1/sin²(x) doesn't appear often enough to warrant much criticism about how it should be denoted. I would always leave it in that form, but if you're unhappy with it or have other reasons to change it, the most obvious choice is to go with csc²(x), but never make it sin^-2(x) because that just causes confusion.

Csc, cosecant? If it's cosecant than it's ##Arcsin(\sqrt{1-sin^2(x)})##

How so?

[tex]\csc(x)=\frac{1}{\sin(x)}[/tex]
while
[tex]\arcsin(\sqrt{1-\sin^2(x)})=\arcsin(\sqrt{\cos^2(x)})=\arcsin(|cos(x)|)[/tex]

 

[Stephanus]

Right, you wouldn't represent 1/sin(x) with a power. You'd leave it as such or replace it with csc(x).1/sin²(x) doesn't appear often enough to warrant much criticism about how it should be denoted. I would always leave it in that form, but if you're unhappy with it or have other reasons to change it, the most obvious choice is to go with csc²(x), but never make it sin^-2(x) because that just causes confusion.How so?

[tex]\csc(x)=\frac{1}{\sin(x)}[/tex]
while
[tex]\arcsin(\sqrt{1-\sin^2(x)})=\arcsin(\sqrt{\cos^2(x)})=\arcsin(|cos(x)|)[/tex]

The answer 'convention' in previous post is enough. It's just that in SR forum, someone says ##T = \frac{c}{a} sinh^{-1}(\frac{at}{c})## I calculate it using ##T = \frac{c}{a * sinh(\frac{at}{c})}##. I'm having trouble accepting sinh^-1 is not 1/sinh. Once you said, it's a 'convention', I let that go.
Thanks.

 

[Mentallic]

You're welcome. Whenever you see a trig function with a -1 power, always think of the inverse and not the reciprocal.

 

[wabbit]

This convention is not an oddity though, it is related to the fact that the natural, generally defined operation between functions is composition rather than multplication. ##f^{-1}## is the inverse of ##f## under the composition operation, not under multiplication, and in the same way ##f^n## designates ##f## iterated ##n## times. Of course it can conflict with usage of multiplicative exponent sometimes, but generally the composition interpretation is the default one (polynomial functions are the main exception I guess).

 

[Stephanus]

This convention is not an oddity though, it is related to the fact that the natural, generally defined operation between functions is composition rather than multplication. ##f^{-1}## is the inverse of ##f## under the composition operation, not under multiplication, and in the same way ##f^n## designates ##f## interated ##n## times.

Okay, it's a matter of "favor"? In variable it's the power of division, right?

Btw can I ask here?

Proper acceleration in Relativity.
We can go further and define the proper acceleration of the particle by
##A^{\mu} = \frac{dU^{\mu}}{d\tau}##

What does this symbol ##\mu## mean?

 

[wabbit]

Okay, it's a matter of "favor"? In variable it's the power of division, right?

Of multiplication rather. ##x^{-1}## is the multiplicative inverse of ##x##, defined by the equation ##x^{-1}×x=1##, while ##f^{-1}## is the composition inverse of ##f##, defined by the equation ##f^{-1}\circ f=Id## (##1## and ##Id## being the identity element of the corresponding operation).

But yes, it is a matter of context / usage, the exponent notation generally refers to some operation iterated or inverted, but which operation is implied can be ambiguous.

Btw can I ask here?
What does this symbol ##\mu## mean?

Ah, completely unrelated usage : ) It's just an index here (corresponding to the components of the vector), not an operation.

 

[Stephanus]

Of multiplication rather. ##x^{-1}## is the multiplicative inverse of ##x##, defined by the equation ##x^{-1}×x=1##,

Yes. I understand completely

##f^{-1}\circ f=Id## (##1## and ##Id##.

Yes.

Ah, completely unrelated usage : ) It's just an index here (corresponding to the components of the vector), not an operation.

Ah, I see. Thanks.

 

[jack476]

My math professor rants about this endlessly =D

Basically, the -1 superscript is just an unfortunate convention of writing for "inverse" that we somehow got stuck with. So sinh^-1(x) should be read "inverse hyperbolic sine of x" (sometimes sinh(x) will be pronounced like "Cinch of x") rather than "1 divided by the hyperbolic sine of x". Personally though I very much prefer to write arcsin(x) because 1.) words like "Arcsine" and "Arctangent" just sound so pretty and 2.) they avoid any possible ambiguity.

 

[Stephanus]

My math professor rants about this endlessly =D

Basically, the -1 superscript is just an unfortunate convention of writing for "inverse" that we somehow got stuck with. So sinh^-1(x) should be read "inverse hyperbolic sine of x" (sometimes sinh(x) will be pronounced like "Cinch of x") rather than "1 divided by the hyperbolic sine of x". Personally though I very much prefer to write arcsin(x) because 1.) words like "Arcsine" and "Arctangent" just sound so pretty and 2.) they avoid any possible ambiguity.

Yeah I (being a perfectionist) have trouble with this term either. But who I am to protest.

...they avoid any possible ambiguity

That's why I choose computer programming. No ambiguity in programming language

 

[micromass]

That's why I choose computer programming. No ambiguity in programming language

 

[Stephanus]

So, what is
1. "2" + "2" -> ??
2. "2" + "3" -> ??
3. What's the difference between NaN and NaP?
4. Why [1,2,3] + 2 is False? I see no comparison in [1,2,3] + 2
5. Why [1.2.3] + 4 is True? Are No 4 and No 5 binary operators?
6. Is [1,2,3] = 1 and 2 and 3?
7. Why 2 / (2 - (3/2 + 1/2)) = NaN.00000013. NaN with point? I see that 2/(2-(3/2+1/2) = 2/0 = NaN
8. Why + 2 = 12?
9. Why 2 + 2 = Done?
10. Shouldn't Range (1,5) -> (1,2,3,4,5) not (1,4,3,4,5)?
Care to tell me how it works?

 

[Svein]

Care to tell me how it works?

xkcd - complicated math humor.

 

[Stephanus]

xkcd - complicated math humor.

I think I didn't get the joke

 

[HallsofIvy]

That's alright- it was a bad joke!

By the way, you titled this thread "negative power of a function". When a "-1" is used to indicate the "inverse function", as in "[itex]f^{-1}[/itex], that is not considered a "power".

 

[Stephanus]

That's alright- it was a bad joke!

Yeah. Like saying "Achilles can't catch the turtle no matter how fast he runs"

By the way, you titled this thread "negative power of a function". When a "-1" is used to indicate the "inverse function", as in "[itex]f^{-1}[/itex], that is not considered a "power".

Now you tell me after all the calculation that I make using 1/sinh(x) instead of arcinh(X)