Our server costs ~$56 per month to run. Please consider donating or becoming a Patron to help keep the site running. Help us gain new members by following us on Twitter and liking our page on Facebook!
Current time: 23rd February 2017, 14:43

Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
cot(x) = cos(x)/sin(x) = 1/tan(x)?
#1
cot(x) = cos(x)/sin(x) = 1/tan(x)?
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)?
Reply
#2
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
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 wish-thinking." ~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 Kroger and Kroger donates...|

Reply
#3
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
(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?
Reply
#4
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
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 wish-thinking." ~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 Kroger and Kroger donates...|

Reply
#5
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
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.
Feel free to send me a private message.
Please visit my website here! It's got lots of information about atheism/theism and support for new atheists.

Index of useful threads and discussions
Index of my best videos
Quickstart guide to the forum
Reply
#6
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
Also 1/0 = lim(n-> inf)[1/n] = 0

Whoops forget that.

But 0/0 is undefined.
Feel free to send me a private message.
Please visit my website here! It's got lots of information about atheism/theism and support for new atheists.

Index of useful threads and discussions
Index of my best videos
Quickstart guide to the forum
Reply
#7
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
(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

Reply
#8
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
(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.
Reply
#9
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
(7th November 2016, 07:53)robvalue Wrote: Also 1/0 = lim(n-> inf)[1/n] = 0

Whoops forget that.

But 0/0 is undefined.

Panic Panic Panic
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.


















Reply
#10
RE: cot(x) = cos(x)/sin(x) = 1/tan(x)?
[Image: Divide_by_Zero_by_Nastyguy.jpg]
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.


















Reply




Users browsing this thread: 1 Guest(s)