Posts: 2865
Threads: 24
Joined: 31st May 2014
Reputation:
23
cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 06:27
Ok, another issue related to division by 0, this time in the context of trigonometry.
Since cot(x) is, as far as I know, the same as cos(x)/sin(x), then:
cot(pi/2) = cos(pi/2) / sin(pi/2) = 0/1 = 0 in radians
But cot(x) has also been equated to 1/tan(x) from what I've read, but if that's the case, then:
cot(pi/2) = 1/tan(pi/2) = 1/undefined = undefined?
So there's a contradiction here. In the case of pi/2, cos(x)/sin(x) != 1/tan(x). What gives? Is the answer found in calculus itself (with the limits and all)? Or is it a bit misleading to say that cot(x) = 1/tan(x)?
Posts: 13984
Threads: 114
Joined: 13th January 2014
Reputation:
102
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 06:56
Well, the limit of tan(x) as x approaches pi/2 is infinity.
So, effectively, you have 1/infinity = 0, and therefore no contradiction.
"There remain four irreducible objections to religious faith: that it wholly misrepresents the origins of man and the cosmos, that because of this original error it manages to combine the maximum servility with the maximum of solipsism, that it is both the result and the cause of dangerous sexual repression, and that it is ultimately grounded on wishthinking." ~Christopher Hitchens, god is not Great
>Please Consider Helping out a Beautiful Golden Retriever Just by Shopping!!!<
 It seriously doesn't cost you anything extra, you just buy things at Amazon and Amazon donates...
Posts: 2865
Threads: 24
Joined: 31st May 2014
Reputation:
23
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 07:12
(This post was last modified: 7th November 2016, 07:13 by Grandizer. )
(7th November 2016, 06:56)SteelCurtain Wrote: Well, the limit of tan(x) as x approaches pi/2 is infinity.
Would that work with the limit from the right being negative infinity?
And I always thought the limit of a function as x approached a certain value was not the same as the value of a function at that value?
Posts: 13984
Threads: 114
Joined: 13th January 2014
Reputation:
102
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 07:21
Yep. 1/inf = zero as well.
Limits are good for approximating the value of a function that never actually reaches a value but gets closer and closer without ever getting there.
There technically is no value of tan(pi/2) because the tangent function never reaches pi/2, rather it asymptotically approaches it, with the value going to infinity.
So when you evaluate it within a mathematical expression, you can use the limit.
"There remain four irreducible objections to religious faith: that it wholly misrepresents the origins of man and the cosmos, that because of this original error it manages to combine the maximum servility with the maximum of solipsism, that it is both the result and the cause of dangerous sexual repression, and that it is ultimately grounded on wishthinking." ~Christopher Hitchens, god is not Great
>Please Consider Helping out a Beautiful Golden Retriever Just by Shopping!!!<
 It seriously doesn't cost you anything extra, you just buy things at Amazon and Amazon donates...
Posts: 26869
Threads: 187
Joined: 9th August 2014
Reputation:
139
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 07:43
You might want to check out mapping to a Riemann sphere. It handles zero and infinity in a different way, I think you'd find it interesting.
Posts: 26869
Threads: 187
Joined: 9th August 2014
Reputation:
139
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 07:53
(This post was last modified: 7th November 2016, 07:59 by robvalue. )
Also 1/0 = lim(n> inf)[1/n] = 0
Whoops forget that.
But 0/0 is undefined.
Posts: 17527
Threads: 105
Joined: 19th January 2014
Reputation:
88
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 07:55
(This post was last modified: 7th November 2016, 08:00 by Alex K. )
(7th November 2016, 06:27)Irrational Wrote: Ok, another issue related to division by 0, this time in the context of trigonometry.
Since cot(x) is, as far as I know, the same as cos(x)/sin(x), then:
cot(pi/2) = cos(pi/2) / sin(pi/2) = 0/1 = 0 in radians
But cot(x) has also been equated to 1/tan(x) from what I've read, but if that's the case, then:
cot(pi/2) = 1/tan(pi/2) = 1/undefined = undefined?
So there's a contradiction here. In the case of pi/2, cos(x)/sin(x) != 1/tan(x). What gives? Is the answer found in calculus itself (with the limits and all)? Or is it a bit misleading to say that cot(x) = 1/tan(x)?
You're absolutely right in principle. At the points where cos is zero, this way of writing cot doesn't work. Often, one still writes the shorthand cot = 1/tan, and at isolated points where that is undefined, but the limit exists (for example for x > pi/2) one takes it to mean the limit, which is
lim_(x>pi/2) 1/tan(x) = 0
But strictly speaking 1/tan doesn't work there.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
Posts: 2865
Threads: 24
Joined: 31st May 2014
Reputation:
23
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 08:21
(7th November 2016, 07:55)Alex K Wrote: (7th November 2016, 06:27)Irrational Wrote: Ok, another issue related to division by 0, this time in the context of trigonometry.
Since cot(x) is, as far as I know, the same as cos(x)/sin(x), then:
cot(pi/2) = cos(pi/2) / sin(pi/2) = 0/1 = 0 in radians
But cot(x) has also been equated to 1/tan(x) from what I've read, but if that's the case, then:
cot(pi/2) = 1/tan(pi/2) = 1/undefined = undefined?
So there's a contradiction here. In the case of pi/2, cos(x)/sin(x) != 1/tan(x). What gives? Is the answer found in calculus itself (with the limits and all)? Or is it a bit misleading to say that cot(x) = 1/tan(x)?
You're absolutely right in principle. At the points where cos is zero, this way of writing cot doesn't work. Often, one still writes the shorthand cot = 1/tan, and at isolated points where that is undefined, but the limit exists (for example for x > pi/2) one takes it to mean the limit, which is
lim_(x>pi/2) 1/tan(x) = 0
But strictly speaking 1/tan doesn't work there.
So one has to be a little careful with using the formula cot(x) = 1/tan(x) then. I guess another question derived from this is what does it mean for an answer to be "undefined"? When I graphed both y = cos(x)/sin(x) and y = 1/tan(x) in Desmos, the two graphs looked virtually equal, and I didn't see any breaks in either graphs at any x around pi/2 no matter how far I zoomed in. But if it is true there are no breaks in the graph at around that point, then how this means undefined is not exactly undefined?
Rob, I will look into into that one once I'm done reviewing Trig and Calculus.
Posts: 23210
Threads: 186
Joined: 18th April 2014
Reputation:
72
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 10:55
(7th November 2016, 07:53)robvalue Wrote: Also 1/0 = lim(n> inf)[1/n] = 0
Whoops forget that.
But 0/0 is undefined.
That's why our administration has moved aggressively to secure our borders more, by hiring a record number of new border guards, by deporting twice as many criminal aliens as ever before, by cracking down on illegal hiring, by barring welfare benefits to illegal aliens . . . The jobs they hold might otherwise be held by citizens or legal immigrants. The public service they use impose burdens on our taxpayers. . . . it is wrong and ultimately self defeating for a nation of immigrants to permit the kind of abuse of our immigration laws we have seen in recent years and we must do more to stop it.
William Jefferson Clinton
Posts: 23210
Threads: 186
Joined: 18th April 2014
Reputation:
72
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 10:56
That's why our administration has moved aggressively to secure our borders more, by hiring a record number of new border guards, by deporting twice as many criminal aliens as ever before, by cracking down on illegal hiring, by barring welfare benefits to illegal aliens . . . The jobs they hold might otherwise be held by citizens or legal immigrants. The public service they use impose burdens on our taxpayers. . . . it is wrong and ultimately self defeating for a nation of immigrants to permit the kind of abuse of our immigration laws we have seen in recent years and we must do more to stop it.
William Jefferson Clinton
