# cot (0)

#### xyz_1965

##### Member
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

Staff member
I would stick with calling $$\displaystyle \cot(0)$$ undefined like any other division by zero until you are studying limits.

#### xyz_1965

##### Member
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?

#### xyz_1965

##### Member
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.

#### xyz_1965

##### Member
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.

#### xyz_1965

##### Member
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
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.

• MarkFL and xyz_1965

#### xyz_1965

##### Member
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.

• Klaas van Aarsen

#### MarkFL

Staff member
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

#### xyz_1965

##### Member
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.