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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)?
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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.
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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?
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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.
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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.
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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.
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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.
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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.
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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.
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RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
7th November 2016, 10:56
If any email controversy impacted the election, it wasn't Russian hacking of the DNC and John Podesta's email, it was Hillary Clinton's decision to have her own email server and the subsequent FBI investigation which raised all the same questions Hillary Clinton has justifiably faced for decades about her trustworthiness and honesty.
