(November 7, 2016 at 7:55 am)Alex K Wrote:(November 7, 2016 at 6:27 am)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.
