Question

is cot(x) defined as 1/tan(x) or cos(x)/sin(x)?
I was trying to solve the equation
csc(x) + cot(x) = 1 in [0,2pi] and it turns out the solution depends on how. If cot(x) = 1/tan(x), then there is no solution. But if cot(x) = cos(x)/sin(x), then pi/2 is the solution.

Answer 1

type. *depends on how cot(x) is defined*

Answer 2

Hmm...well...

\[\large \frac{1}{\tan(x)} = \frac{1}{\frac{sin(x)}{cos(x)}} = \frac{cos(x)}{sin(x)}\]

So they are essentially equivalent

Answer 3

can be used either way.

Answer 4

well, they equivalent except at certain points.

Answer 5

like which ones?

Answer 6

if cot(x) = 1/tan(x), then the domain is cos(x) ≠ 0 and sin ≠ 0 because 1/tan(x) = 1/(sin(x)/cos(x)). However, if cot(x) = cos(x)/sin(x), then the domain is sin(x) ≠ 0.

Answer 7

that will make a difference when determining the solutions

Answer 8

as I already stated, one has no solution while the other has pi/2 in [0,2pi]

Answer 9

tan(x) is defined for x=0, so that should make it clear that you want to use sinx/cosx, not 1/(cosx/sinx), lest you think that is undefined as that would be 1/1/0
same for this situation

Answer 10

is cot(pi/2) defined?

Answer 11

so tan(x) = cos(x) / sin(x)? It has to be one or the other

Answer 12

cotx=cosx/sinx

Answer 13

=1/tanx
all is equivalent

Answer 14

according to you, "if cot(x) = 1/tan(x), then the domain is cos(x) ≠ 0 and sin ≠ 0 because 1/tan(x)" so if that is true, then cot(pi/2) should be undefined, yet it is not....

Answer 15

uhm... makes sense. I guess I was comparing cot(x) to sec(x) and csc(x).
I mean sec(x) = 1/cos(x) and csc(x) = 1/sin(x), so I should makes sense that cot(x) = 1/tan(x). But then again, maybe the definition that cot(x) = cos(x)/sin(x) is just for convenience?

Answer 16

they are equivalent expressions in every way, cotx=1/tanx=1/(sinx/cosx)

the fallacy in your reasoning is that this means that at x=pi/2 we get
1/(0/1)=1/0
but that's not true... if it were, we could always say that 0=1/(1/0)=undefined
you've just overthought it a bit, which is often a good thing in math ;)

Answer 17

hehe thank you for the replies :3

Answer 18

welcome :)