# Negative power of function

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1. Jul 31, 2015

### 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.

2. Jul 31, 2015

### Mentallic

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

3. Jul 31, 2015

### Stephanus

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

4. Jul 31, 2015

### Stephanus

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

5. Jul 31, 2015

### Mentallic

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?

$$\csc(x)=\frac{1}{\sin(x)}$$
while
$$\arcsin(\sqrt{1-\sin^2(x)})=\arcsin(\sqrt{\cos^2(x)})=\arcsin(|cos(x)|)$$

6. Jul 31, 2015

### Stephanus

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.

7. Jul 31, 2015

### Mentallic

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

8. Jul 31, 2015

### 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).

9. Jul 31, 2015

### Stephanus

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

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

10. Jul 31, 2015

### wabbit

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.

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

Last edited: Jul 31, 2015
11. Jul 31, 2015

### Stephanus

Yes. I understand completely
Yes.
Ah, I see. Thanks.

12. Jul 31, 2015

### jack476

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.

13. Jul 31, 2015

### Stephanus

Yeah I (being a perfectionist) have trouble with this term either. But who I am to protest.
That's why I choose computer programming. No ambiguity in programming language

14. Aug 1, 2015

### micromass

15. Aug 1, 2015

### 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?

16. Aug 1, 2015

### Svein

xkcd - complicated math humor.

17. Aug 1, 2015

### Stephanus

I think I didn't get the joke

18. Aug 1, 2015

### 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 "$f^{-1}$, that is not considered a "power".

19. Aug 1, 2015

### Stephanus

Yeah. Like saying "Achilles can't catch the turtle no matter how fast he runs"
Now you tell me after all the calculation that I make using 1/sinh(x) instead of arcinh(X)