# 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

Science Advisor
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,
$\sin^{-1}\theta=\csc\theta$

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving