Understanding Inverse Trig Notations: Clarifying tan-1x and tan-2x

[Rose Garden]

I just want to know, what tan^-1x means to you guys? does it mean arctan or cot?
What about tan^-2x? Isn't that cot²x? Shouldn't the notation tan^-2x be banned to avoid misinterpreting it as an inverse trig function?

 

[micromass]

tan^-1(x) means arctan to me. But it's a very unfortunate notation. That's why I always use arctan atan or bgtan instead of tan^-1...

 

[dextercioby]

It's indeed an unfortunate notation. However, LaTeX language has the function $\arctan$ but I couldn'f find one for <arccotangent>, even though $\cot$ exists. Unfortunately, the very friendly Mathematica software uses the misleading notation.

http://www.wolframalpha.com/input/?i=Integrate+1/(1+x^2)+

And the inverse hyperbolic functions are not defined in LaTeX, apparently.

 

[uart]

I just want to know, what tan^-1x means to you guys? does it mean arctan or cot?
What about tan^-2x? Isn't that cot²x? Shouldn't the notation tan^-2x be banned to avoid misinterpreting it as an inverse trig function?

Hi Rose Garden. Yes I would interpret $\tan^{-1}x$ as arctan(x), while I would interpret $\tan^{-2}x$ as $\cot^2x$.

In most cases hopefully the context would make it clear what function one was using, but in general I'd say that you should interpret $\tan^{n}x$ as $(\tan \, x)^n$ for all values of $n$ except $n=-1$ which should be treated as arctan (and similar applies to the other trig functions).

BTW. If ever you wish to use the "-1" notation to mean reciprocal with a trig function just use the bracketed notation as for example :

$${\rm cosec} \, x = (\sin x)^{-1}$$

 

[Mark44]

As is usually the case, context means a lot. When you're dealing with a variable, x^-1 means the reciprocal of x, of 1/x.

When you're dealing with functions, however, an exponent of -1 usually means the inverse of the function. So tan^-1(x) means arctan(x), not 1/tan(x).

 

[Rose Garden]

thanks guys I think I'm getting the picture now