cot(x) = cos(x)/sin(x) = 1/tan(x)?

[GrandizerII]

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)?

[SteelCurtain]

Well, the limit of tan(x) as x approaches pi/2 is infinity.

So, effectively, you have 1/infinity = 0, and therefore no contradiction.

[GrandizerII]

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?

[SteelCurtain]

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.

[robvalue]

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.

[robvalue]

Also 1/0 = lim(n-> inf)[1/n] = 0

Whoops forget that.

But 0/0 is undefined.

[Alex K]

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.

[GrandizerII]

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.

[vorlon13]

Also 1/0 = lim(n-> inf)[1/n] = 0

Whoops forget that.

But 0/0 is undefined.

[vorlon13]