How to write powers of inverse trigonometric functions?

[Kuhan]

Does $(\sin^{-1}\theta)^2 =\sin^{-2}\theta$ ?

 

[alberto7]

I think we use $\arcsin$ for the functional inverse of $\sin$ and $\csc$ for its multiplicative inverse, instead of $\sin^{-1}$, in order to avoid this confusion. As $\sin^n$ is multiplicative for $n>0$, I would say that $\sin^{-n}=\csc^n$. If one needed to write many times the functional iterates and inverses of $\sin$, I would recommend to use a notation like $\sin^{[n]}$ or $\sin^{\circ n}$, which I've found in papers dealing with iterated functions.

 

[chief10]

why would you even come across something like that?

would be easier to just use the inverse function mate.

 

[HallsofIvy]

To put it simply, "$sin^{-1}(x)$" for the inverse function is an unfortunate notation!

 

[Kuhan]

I think we use $\arcsin$ for the functional inverse of $\sin$ and $\csc$ for its multiplicative inverse, instead of $\sin^{-1}$, in order to avoid this confusion. As $\sin^n$ is multiplicative for $n>0$, I would say that $\sin^{-n}=\csc^n$. If one needed to write many times the functional iterates and inverses of $\sin$, I would recommend to use a notation like $\sin^{[n]}$ or $\sin^{\circ n}$, which I've found in papers dealing with iterated functions.

Thanks! now it makes sense. I used to use $\sin^{-1}\theta$ instead of $\arcsin\theta$ . They just aren't the same, I guess. Basically,
$\sin^{-1}\theta=\csc\theta$

 

[Kuhan]

http://mathworld.wolfram.com/InverseTrigonometricFunctions.html
they do use $\sin^{-1}\theta=\arcsin\theta$ in math! wow so confusing!

 

[Kuhan]

http://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/InverseTrig.aspx
answers the problem