Post by: NE2000 on February 19, 2009, 04:57:26 pm
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?
Post by: shinny on February 19, 2009, 06:37:10 pm
Post by: NE2000 on February 19, 2009, 06:39:33 pm
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?
Post by: shinny on February 19, 2009, 06:44:05 pm
I think the reason behind the discrepancy is you jumping steps. Your first definition of cot was correct (that is,
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.
Post by: shinny on February 19, 2009, 09:59:43 pm
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?
Post by: kamil9876 on February 19, 2009, 11:25:33 pm
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.
Post by: shinny on February 19, 2009, 11:36:16 pm
Post by: kamil9876 on February 19, 2009, 11:50:26 pm
Post by: kamil9876 on February 20, 2009, 09:48:45 am
Finally found an example without using the word 'limit' ;)
Post by: Mao on February 20, 2009, 08:21:39 pm
What shinny said is partially incorrect. Without going into the definition of sin, cos and tan in terms of the power series,
Though I am not willing to bet my life on this, I am pretty certain
Post by: shinny on February 20, 2009, 08:25:34 pm
Though I am not willing to bet my life on this, I am pretty certainis 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
Post by: Mao on February 20, 2009, 09:14:24 pm
Post by: shinny on February 20, 2009, 09:20:52 pm
Well anyway, hopefully someone sheds some light on this.
Post by: kamil9876 on February 20, 2009, 09:27:01 pm
Though I am not willing to bet my life on this, I am pretty certain
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
Post by: kamil9876 on February 20, 2009, 09:31:33 pm
Post by: NE2000 on February 21, 2009, 12:23:59 pm
See initially when reciprocating graphs I got taught that as y --> infinity, then 1/y ---> 0 and therefore the x-intercepts of the reciprocal graph occur when there is an asymptote on the y-axis. But then the problem that arises is that can we consider infinity as a number (without applying limits) which has the property of 1/infinity = 0?
For many of the reciprocal graphs I've drawn, y = 0 is an asymptote. Eg for y =x^2 where the reciprocal is just the basic truncus. While y = x^2 doesn't have an asymptote, as x--->infinity, y ---> infinity and hence for the reciprocal graph we say y----> 0 and that y = 0 is the asymptote. Here there is no crossing the asymptote and it is pretty clear that no matter what y will not equal zero. So if we are sketching the reciprocal of tan x, or even things like log x, why should they have x-intercepts at the asymptote?? This confuses me a bit.
As for limits, I think that can be deceptive because even if we have y = x, x is an element of R\{2}. Then the limits would tell us that as x ---> 2, then y ---> 2, but we would still put an open circle there because we know that x can't actually = 2. I thought limits in such cases were mainly to test differentiability but then it's just the start of the spesh course and I don't know half the stuff taught in spesh yet (let alone uni stuff which some people on this forum have covered).
Post by: shinny on February 21, 2009, 01:16:25 pm
Post by: Mao on February 21, 2009, 05:05:09 pm
if
Post by: Mao on February 21, 2009, 05:09:15 pm
(http://upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Circle-trig6.svg/338px-Circle-trig6.svg.png)
I hope that's convincing enough, as
Post by: /0 on February 21, 2009, 07:04:46 pm
QED
Post by: shinny on February 21, 2009, 07:16:19 pm
Post by: NE2000 on February 21, 2009, 07:36:43 pm
Calculator says it is 0, therefore it is 0.
QED
lol
Post by: TrueTears on February 21, 2009, 07:51:17 pm
Calculator says it is 0, therefore it is 0.
QED
roflmao /0
Post by: kamil9876 on February 21, 2009, 09:10:11 pm
Calculator says it is 0, therefore it is 0.
QED
http://answers.yahoo.com/question/index?qid=20080708165540AA4aeqH
LOL, guess it all depends on the opinion of the manufacturer ;p
Post by: mystikal on February 21, 2009, 09:57:24 pm
Post by: /0 on February 24, 2009, 11:31:21 pm
heys mao how did u make the cool picture? what program you used?
It's from Wikipedia
Also, I just noticed, in Essential Specialist Maths it says
"The cotangent function is defined as: