cot x and intercepts with the x-axis

[NE2000]

So...



when tan x tends to infinity, cot x tends to 0. If we used our general ideas of reciprocating graphs then we would find that half pi is an x-intercept. But if we sub it in:



But then:



So what's the answer?

[shinny]

I think the reason behind the discrepancy is you jumping steps. Your first definition of cot was correct (that is, ). However, on your next one, you jumped straight to the shortcut definition of which works in most cases as you can just derive that from . However, looking at it from this way, we see that we can't use that last step if as the would be undefined and of course then you won't be able to 'move' it up the top as the denominator of the whole fraction will be undefined. So basically, it's undefined and not zero.

[NE2000]

I think the reason behind the discrepancy is you jumping steps. Your first definition of cot was correct (that is, ). However, on your next one, you jumped straight to the shortcut definition of which works in most cases as you can just derive that from . However, looking at it from this way, we see that we can't use that last step if as the would be undefined and of course then you won't be able to 'move' it up the top as the denominator of the whole fraction will be undefined. So basically, it's undefined and not zero.

OK cool I sort of get where you're coming from (although I think you accidentally said cos(90) is undefined (as opposed to 0). So when you graph cot x then would you put the 'x-intercepts' as open circles? Do the examiners look for that?

[shinny]

I think the reason behind the discrepancy is you jumping steps. Your first definition of cot was correct (that is, ). However, on your next one, you jumped straight to the shortcut definition of which works in most cases as you can just derive that from . However, looking at it from this way, we see that we can't use that last step if as the would be undefined and of course then you won't be able to 'move' it up the top as the denominator of the whole fraction will be undefined. So basically, it's undefined and not zero.

OK cool I get where you're coming from. So when you graph cot x then would you put the 'x-intercepts' as open circles? Do the examiners look for that?

Yep they do (at least that's what my spesh teacher told me). It makes physical sense anyway if you just use a tan graph as a point of reference to draw its reciprocal.

[shinny]

So...
 \frac {1}{undef} = undef[/tex]

I think this is the problem. Informally speaking tan(pi/2)=infinity. hence cot(pi/)=1/infinity. But 1/infinity is (informally) zero.
More formally however, tan(pi/2) is undefined, however lim(x-->+pi/2)(tanx) is defined. So whenever we merely write tan(pi/2) we're really informally refering to the limit (which does exist).

So formalising ur first argument:

lim(x-->+pi/2)(cotx)=1/(lim(x-->+pi/2)(tanx))=0

So basically, if u want to ignore the limit notation then treat tan(pi/2) as infinity and use the "fact" 1/infinity=0 when evaluating such things.

An example of such would be to find the integral from 0 to infinity of e^(-x)dx

Where exactly are you heading with that though? It seems your conclusion was made in "use the "fact" 1/infinity=0 when evaluating such things" but isn't that the wrong conclusion to make anyway? should be undefined. Maybe we should get some Mao-ie power in here because my knowledge on the technicalities of maths theory isn't really that great and like I said, what I said before is just my take on the reason for the original discrepancy by NE2000.

[kamil9876]

So...
 \frac {1}{undef} = undef[/tex]

I think this is the problem. Informally speaking tan(pi/2)=infinity. hence cot(pi/)=1/infinity. But 1/infinity is (informally) zero.
More formally however, tan(pi/2) is undefined, however lim(x-->+pi/2)(tanx) is defined. So whenever we merely write tan(pi/2) we're really informally refering to the limit (which does exist).

So formalising ur first argument:

lim(x-->+pi/2)(cotx)=1/(lim(x-->+pi/2)(tanx))=0

So basically, if u want to ignore the limit notation then treat tan(pi/2) as infinity and use the "fact" 1/infinity=0 when evaluating such things.

An example of such would be to find the integral from 0 to infinity of e^(-x)dx

Where exactly are you heading with that though? It seems your conclusion was made in "use the "fact" 1/infinity=0 when evaluating such things" but isn't that the wrong conclusion to make anyway? should be undefined. Maybe we should get some Mao-ie power in here because my knowledge on the technicalities of maths theory isn't really that great and like I said, what I said before is just my take on the reason for the original discrepancy by NE2000.

Yep sorry. I did however mention in my post that tan(pi/2) is undefined, we do agree. I guess I was sort of thinking about certain practical cases that if tan(pi/2) does pop up it's ussually as a limit/assymptote etc. Hence same as cot(pi/2). The "fact" i referred to was a fact about limits but stated informally just like tan(pi/2) is sometimes stated informally rather than a limit. Hence when stated informally, treat it as a 'limit in disguise' and hence use other limit facts in disguise. I guess it all boils down to the application of tan(pi/2) since it won't appear in purely math questions but It might in more application based( i cbf making up examples now)
Sorry for my original post, I regret making it.

[shinny]

Well I don't think the application of limits is what we need here since we're actually looking for the actual value (if it exists, which I guess we've made a consensus on that it doesn't) at the x-intercepts; not the value it approaches from either side. But whatever, I think this question is resolved now unless someone can give a better reason to explain NE2000's original question.

[kamil9876]

Haha yea. I guess maybe I went a bit too off topic since probably the original question was only about pure trig given this time of the year. I just wanted to show that the two different answers that NE2000 got(undef and 0) are not that different, and in fact the same if we speak of limits, and that adds some clarification maybe(probably more useful in applications). Him asking this question sort of sparked my rant as I sometimes get a bit too enthusiastic about limits. haha epsilson-delta ftw.

[kamil9876]

Find the area bounded by lines x=pi/4, x=pi/2, y=0 and the curve y=d/dx(e^cotx)

Finally found an example without using the word 'limit'

[Mao]

oh, hi there, sorry, been away.

What shinny said is partially incorrect. Without going into the definition of sin, cos and tan in terms of the power series, is not a 'short-cut' way of defining the cotangent. The way to approach this problem is with limits as kamil has shown, because tan(pi/2) isn't actually defined.

Though I am not willing to bet my life on this, I am pretty certain is defined.

[shinny]

Though I am not willing to bet my life on this, I am pretty certain is defined.

So does that mean open circles at the intercepts or not? I've always been told to include them.

EDIT: And wait, when you say problem...which problem? Finding the value of ? Or did you mean the reason between the two different answers obtained by NE2000?

[Mao]

I've never seen open circles at intercepts for cot. And I am unsure about the validity of the first method used by NE2000, but I really don't know enough about this to give an authoritative answer. I suppose we leave that to Ahmad, Neobeo, humph or others who actually know these things as opposed to an amateur like me =]

[shinny]

Mao...amateur...WHUT!? What the hell does that make the rest of VN? =P

Well anyway, hopefully someone sheds some light on this.

[kamil9876]

Though I am not willing to bet my life on this, I am pretty certain is defined.

Yea, i resorted to googling before my first post as I wasn't sure myself. I knew that if it was defined it should be 0 and if it wasn't defined well then that would just be a trivial/arbitrary/convention. You mentioned power series definition, I remember one site defining it via e raised to some complex numbers with some messy fractions and operations. They seemed to say that such a definition was synonymous to 1/tan(z) as the expression was derived from taking the recirprocal of tan(z). If defined as 1/tan(z), then cot(pi/2) wouldnt exist as this is a composite function i.e: first z is put into tan, then tan(z) is spat out and reciprocated, because tan(pi/2) is undefined then the first thing that was spat out doesnt exist and hence there was nothing to reciprocate and so it is ultimately undefined. However wikipedia put up various definitions and one of such was adjacent over opposite (analogous to cos(x)/sin(x)) and in this case it would be defined.

I guess if u assign superiority to the power series expansion, then you would be saying that the technically true definition was the first one i mentioned (complex exponents) as I'm expecting that this does involve power series since that is where complex exponents come from.

http://mathworld.wolfram.com/Cotangent.html

[kamil9876]

actually i just subbed in pi/2 into that expression found on the link provided and found that it was 0/(-2)=0