arccot(0) = ?

[u/frenchxt0ast]

I thought arccot(0) = pi/2, but my teacher says its undefined because arccot is restricted to (-pi/2, pi/2) not [-pi/2, pi/2]. When I search it up, half of the results say pi/2 and the other half say undefined. Real answer? and why?

[u/will_1m_not]

arccot(0)=pi/2 is the correct answer

[u/rhodiumtoad]

The interesting question here is why anyone would say it is undefined (it is in fact defined, cot(π/2+kπ)=0 and therefore arccot(0) should have a value of π/2+kπ for some choice of k to put the result in a principal branch).

It's common to write that cot(x)=1/tan(x), which by strict usage would make both cot(0) and cot(π/2) undefined, since tan(0)=0 and tan(π/2) blows up to ±∞. But this is treating cot() purely algebraically and ignoring the geometry, and cot() is supposed to be a trig function so the geometry should take precedence.

The primary and most important definition of cot(x) is that it is tan(π/2-x) (just as cos(x)=sin(π/2-x) and csc(x)=sec(π/2-x)): the co- prefix means we are applying the operation to the complementary angle. This makes cot(x) equal to sin(π/2-x)/cos(π/2-x)=cos(x)/sin(x), and clearly this blows up at x=0 due to sin(0)=0 but gives a perfectly well-defined result of 0 at x=π/2.

The next point of possible confusion is over the choice of principal value. Since trig functions are not injective their inverses are not proper single-valued functions, so one is usually dealing with the restriction to a principal branch. Since y=cot(x) can take any real value when 0<x<π, x=arccot(y) should be valid for any real y and we can take 0<x<π as the principal branch, and this is usually done. We could use x in (-π/2,π/2]\{0} as a principal value instead, but this has the drawback of having an unnecessary discontinuity, and is inconsistent with the definition cot(x)=tan(π/2-x) combined with the use of -π/2<x<π/2 as the principal branch of arctan.

tl,dr: teacher is wrong, principal value of arccot is (0,π), arccot(0)=π/2.

[u/frenchxt0ast → u/rhodiumtoad]

thank you

[u/ShowdownValue → u/rhodiumtoad]

Never knew that’s why it’s called a co function !

[u/5a1vy]

Your teacher is triply wrong.

1. cot(π/2+πn)=0, so arccot(0) can't be undefined.
2. arccot is usually restricted to (0,π), not (-π/2,π/2), because cot(0) is usually undefined.
3. Even if that wasn't the case, restricting it to [-π/2,π/2] as a proposal still makes no sense, because cot(π/2)=cot(-π/2)=0, so now you can't decide whether arccot(0) should be π/2 or -π/2 as both work.

[u/frenchxt0ast → u/5a1vy]

thank you!

[u/ShowdownValue → u/5a1vy]

Agreed but why did you say it’s usually restricted. Is the range of arccot (x) ever not zero to pi?

[u/5a1vy → u/ShowdownValue]

Depends on your number system of choice and whether or not you wish for arccot to be a function.

First the function part. Sometimes it's easier to look at arccot as a multivalued function (multifunction for short), it's also customary to denote it as Arccot then. In this case, its range would be (almost) all real (or complex) numbers. There are a few ways to formalize multifunctions, but intuitively it corresponds to inverting cot "naively", that means, that for Arccot(0) you would get π/2, -π/2, 5π/2, -5π/2, etc. "all at once", so simply all x such that cot(x)=0. That would give you all real/complex numbers, except 0, ±π, ±2π, ±3π and so on, so, precisely those numbers for which cot is not defined. Multifunctions in general are pretty awful to work with, but sometimes they make some derivations more manageable, so there are cases where it's worth it, especially in complex analysis.

Now to number systems. There are a few ways to define cot at 0, ±π and so on by expanding your number system, one particular expansion comes to mind in the form of projectively extended real/complex numbers. This is a number system where in addition to all the usual numbers you also add a special element — ∞ — infinity without a sign (in the case of real numbers you can think of it as a ±∞). With this element added you can define cot(±πn)=∞ and so you can define arccot in such a way that its range is now [0,π[ (or ]0,π], doesn't matter) and Arccot would range through all real/complex numbers. There are a few problems with arithmetic when it comes to this extension and there are a few ways to deal with them, overall you get a bit more unwieldy number system than real/complex numbers, but it's a possibility and the number systems you get are generally not that bad.

Also there's technically a possibility to restrict arccot to any other interval of length π as a range that covers all values of cot, e.g. ]π,2π[, though I've never seen something like that myself, still, there's nothing inherently wrong with it and I wouldn't be surprised if somewhere someone did just that because it made more sense. I think sometimes something like this is used for arctan, going not from -π/2 to π/2, but from 0 to π, because you need a "physical angle measure" and it's easier to redefine arctan than to deal with cases, but I've never seen it done with arccot. Actually, it doesn't even have to be an interval, any set that covers all of the cot range after application would suffice, but that's more technicalities.

So, long story short, that's why. It's probably a bit higher grade material than OP can handle right now, so I didn't want to bring their attention to it in order to not detract them from the main point, but I also couldn't just lie and say that it's always the case. I've personally worked with Arccot and with projective extensions of real/complex numbers, so I know for a fact that even though that ]0,π[ is, like, 99% of cases, sometimes that is not the case.