Author Topic: cot(x) = 0 => 1 = 0 (Read 962 times) Tweet Share

/0 · « **on:** June 15, 2008, 05:47:28 pm »

$\cot{x} = 0$

$\Rightarrow \frac{1}{\tan{x}} = 0$

$\therefore \ 1 = 0$

OH SHI-

I got this problem when trying to find the stationary points of $y=\log_{e}(\sin{x})$ . I know $\cot{x} = 0$ where $\tan{x}= \infty$ , but why does this happen?

Ahmad · « **Reply #1 on:** June 15, 2008, 06:01:08 pm »

$\cot x = \frac{1}{\tan x}$ when $\tan x \ne 0$ , otherwise it's $\frac{\cos x}{\sin x}$

Pandemonium · « **Reply #2 on:** June 15, 2008, 06:27:18 pm »

... otherwise it's cosx/sinx?
no, that just doesn't make sense, you'd still get 1/0 which is as undefined as 1=0

/0 · « **Reply #3 on:** June 15, 2008, 06:33:57 pm »

Quote from: Ahmad on June 15, 2008, 06:01:08 pm

$\cot x = \frac{1}{\tan x}$ when $\tan x \ne 0$ , otherwise it's $\frac{\cos x}{\sin x}$

So what you're saying is... for $\frac{1}{\tan{x}} = 0$ , if $\tan{x}\neq 0$ , then $\cot{x}$ is defined... but the equation must be incorrect.

So... what value must $\tan{x}$ take to satisfy $\frac{1}{\tan{x}} = 0$ ? It can't be positive or negative, so the only alternative must be $\tan{x} =0$ ...

Solving for x in $\tan{x} =0$ , you get values which, upon being substituted back into the original equation, do not work... weird.

This is a contradiction.

Ahmad · « **Reply #4 on:** June 15, 2008, 07:01:52 pm »

Sorry guys, I originally had $\tan x \ne 0$ when I really meant $\cos x \ne 0$ .

This is what I mean,

$\cot \frac{\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = \frac{0}{1} = 0$

However if you tried with the definition $\cot x = \frac{1}{\tan x}$ you'd get:

$\cot \frac{\pi}{2} = \frac{1}{\tan \frac{\pi}{2}} = \frac{1}{\left( \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} \right)} = \frac{1}{(\frac{1}{0})}$ which is non-sense.

Whenever you use the relationship $\cot x = \frac{1}{\tan x}$ you're automatically assuming that $x \ne \frac{\pi}{2} + k\cdot\pi, k \in \mathbb{Z}$ in other words, $\cos x \ne 0$

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Author Topic: cot(x) = 0 => 1 = 0 (Read 962 times) Tweet Share

/0

cot(x) = 0 => 1 = 0

Ahmad

Re: cot(x) = 0 => 1 = 0

Pandemonium

Re: cot(x) = 0 => 1 = 0

/0

Re: cot(x) = 0 => 1 = 0

Ahmad

Re: cot(x) = 0 => 1 = 0

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