Welcome to our community

Be a part of something great, join today!

cot (0)

  • Thread starter
  • Banned
  • #1

xyz_1965

Member
Jul 26, 2020
81
I know that cot (0) = 1/tan (0) = 1/0.

Most textbooks show that cot (0) = undefined.

Can I also say that cot (0) = positive infinity?

Is there a difference between 1/0 is undefined and 1/0 is positive infinity?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I would stick with calling \(\displaystyle \cot(0)\) undefined like any other division by zero until you are studying limits.
 
  • Thread starter
  • Banned
  • #3

xyz_1965

Member
Jul 26, 2020
81
I would stick with calling \(\displaystyle \cot(0)\) undefined like any other division by zero until you are studying limits.
By limits do you mean taking limits at positive and negative infinity?
 
  • Thread starter
  • Banned
  • #4

xyz_1965

Member
Jul 26, 2020
81
I would stick with calling \(\displaystyle \cot(0)\) undefined like any other division by zero until you are studying limits.
Does the same thing apply to csc (0°)?
In other words, csc (0°) = 1/sin (0°) = 1/0, which is undefined not positive infinity.
 
  • Thread starter
  • Banned
  • #5

xyz_1965

Member
Jul 26, 2020
81
I would stick with calling \(\displaystyle \cot(0)\) undefined like any other division by zero until you are studying limits.
Originally I meant to type cot (0°) not cot (0) but you understood right away.
 
  • Thread starter
  • Banned
  • #6

xyz_1965

Member
Jul 26, 2020
81
Mark,

Here is Jason completing a chart of trig function values. Jason said cot (0°) = positive infinity = csc (0°) = positive infinity.


I am going to put the idea of limits on hold as you suggested but please watch the video clip.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,739
Mark,

Here is Jason completing a chart of trig function values. Jason said cot (0°) = positive infinity = csc (0°) = positive infinity.


I am going to put the idea of limits on hold as you suggested but please watch the video clip.
Note that $\cot(-0.01^\circ)$ is pretty far negative - nowhere near positive infinity.
So saying that $\cot 0^\circ$ is positive infinity is wrong, but we might say it is infinity.
Note that the guy in the video does not say positive infinity, but instead he refers to just infinity, which he writes as $\infty$.

Just for fun, we have basically the following choices here:
  1. $\cot 0^\circ$ is $\text{undefined}$, which is correct with respect to the real numbers ($\mathbb R$), and avoids confusion with advanced concepts.
  2. $\cot 0^\circ=\infty$, which is correct with respect to the Real projective line ($\mathbb R\cup \{\infty\}$). In this case there is no distinction between $-\infty$ and $+\infty$. They are just $\infty$.
  3. $\cot 0^\circ = +\infty$ or $\cot 0^\circ =-\infty$, which are both wrong in this particular case, but they are with respect to the Hyperreal numbers (${}^*\mathbb R$), which includes $-\infty$, $+\infty$, and also many other infinities and infinitesimals.
 
  • Thread starter
  • Banned
  • #8

xyz_1965

Member
Jul 26, 2020
81
Note that $\cot(-0.01^\circ)$ is pretty far negative - nowhere near positive infinity.
So saying that $\cot 0^\circ$ is positive infinity is wrong, but we might say it is infinity.
Note that the guy in the video does not say positive infinity, but instead he refers to just infinity, which he writes as $\infty$.

Just for fun, we have basically the following choices here:
  1. $\cot 0^\circ$ is $\text{undefined}$, which is correct with respect to the real numbers ($\mathbb R$), and avoids confusion with advanced concepts.
  2. $\cot 0^\circ=\infty$, which is correct with respect to the Real projective line ($\mathbb R\cup \{\infty\}$). In this case there is no distinction between $-\infty$ and $+\infty$. They are just $\infty$.
  3. $\cot 0^\circ = +\infty$ or $\cot 0^\circ =-\infty$, which are both wrong in this particular case, but they are with respect to the Hyperreal numbers (${}^*\mathbb R$), which includes $-\infty$, $+\infty$, and also many other infinities and infinitesimals.
I will stick to the basic and just use the word undefined at this beginning stage of trigonometry.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Does the same thing apply to csc (0°)?
In other words, csc (0°) = 1/sin (0°) = 1/0, which is undefined not positive infinity.
Yes.
Originally I meant to type cot (0°) not cot (0) but you understood right away.
0 degrees and 0 radians are the same thing.
 

xyz_1965

Member
Jul 26, 2020
81
Yes.


0 degrees and 0 radians are the same thing.
You said:

"0 degrees and 0 radians are the same."

How silly of me to forget this basic fact.