tangent question: What is value of 1/tan(pi*x) when x = 1/2? Is it 0?

[Harry_the_cat]

What is the value of $\displaystyle \frac{1}{\tan(\pi x)}$ when $\displaystyle x = \frac{1}{2}$ ? Is it 0?

I know that $\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}$ tells us it is 0, but does that definition hold if $\displaystyle \cos \theta =0$.

 

[skeeter]

yes …

$$\dfrac{\cos\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right)} = \dfrac{0}{1} = 0$$

 

[Harry_the_cat]

yes …

$$\dfrac{\cos\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right)} = \dfrac{0}{1} = 0$$

Yes I can see that, but what happens when you plug $\displaystyle x=\frac{1}{2}$ into $\displaystyle \tan(\pi x)$ ?

 

[topsquark]

What is the value of $\displaystyle \frac{1}{\tan(\pi x)}$ when $\displaystyle x = \frac{1}{2}$ ? Is it 0?

I know that $\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}$ tells us it is 0, but does that definition hold if $\displaystyle \cos \theta =0$.

$cot(\pi/2) = \dfrac{cos(\pi/2)}{sin(\pi/2)} = \dfrac{0}{1} = 0$

But
$\dfrac{1}{tan(\pi/2)} = \dfrac{ \, \, 1\,\, }{ \dfrac{sin(\pi/2)}{cos(\pi/2)} } = \dfrac{ \, \, 1\,\, }{ \dfrac{1}{0} }$
does not exist.

-Dan

 

[skeeter]

evaluate: 1/tan(pi/2) - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

 

[Harry_the_cat]

Do those little grey dots on the x-intercepts on Desmos represent discontinuities?

 

[skeeter]

Wolfram graph looks continuous

graph y = 1/tan(pi*x) - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

 

[Dr.Peterson]

Do those little grey dots on the x-intercepts on Desmos represent discontinuities?

They should, but unfortunately they represent intersections. (There are no dots when this equation isn't selected.) Both Desmos and Wolfram Alpha apparently don't recognize the issue.

Similarly, I've seen sources that define cot(x) as 1/tan(x), which is wrong.

 

[Harry_the_cat]

They should, but unfortunately they represent intersections. (There are no dots when this equation isn't selected.) Both Desmos and Wolfram Alpha apparently don't recognize the issue.

Similarly, I've seen sources that define cot(x) as 1/tan(x), which is wrong.

and also $\displaystyle \tan(x)$ as $\displaystyle \frac{\sin(x)}{\cos(x)}$ ??

 

[skeeter]

They should, but unfortunately they represent intersections. (There are no dots when this equation isn't selected.) Both Desmos and Wolfram Alpha apparently don't recognize the issue.

Similarly, I've seen sources that define cot(x) as 1/tan(x), which is wrong.

First time I’ve heard that … how should cotangent be defined, then?

 

[Dr.Peterson]

and also $\displaystyle \tan(x)$ as $\displaystyle \frac{\sin(x)}{\cos(x)}$ ??

No, that's perfectly valid, as is $\displaystyle \cot(x)=\frac{\cos(x)}{\sin(x)}$.

It can also be properly defined as x/y on the unit circle, or as $\tan(\pi-x)$. All of these have the correct domain.

It is only valid to define $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ if you specify that you are working in the projectively extended reals, as I've seen stated in one place. Over the reals, it gets the wrong domain.

I've been looking for the sources I'd seen that gave that latter equation as a definition (rather than an identity that is valid when both sides are defined), but can't currently find it except in one dictionary. I suspect it is not uncommon for students to read the identity and think of it as a definition.

 

[skeeter]

discussion at stack exchange ...

Is $\cot(x)=\frac{1}{\tan(x)}$?

I just found in some place say that $\cot(x)=\frac{1}{\tan(x)}$. If you think about it as a function they are definitely not same. but if you think about as a relation between angle in a right angle

math.stackexchange.com

 

[Steven G]

A necessary condition for a fraction to equal 0 is if the numerator is 0. The op's numerator is 1 which is never 0.

 

[Harry_the_cat]

Interesting discussion. My brain hurts. Thanks. Still not sure about it all!!

 

[Steven G]

Interesting discussion. My brain hurts. Thanks. Still not sure about it all!!

View attachment 35567

1/(1/0) = 1/undefined = undefined.

1/tanx = cotx iff x is in the domain for both functions.

 

[apple2357]

It is only valid to define $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ if you specify that you are working in the projectively extended reals, as I've seen stated in one place. Over the reals, it gets the wrong domain.

Can i check what you mean by 'projectively extended reals'? Is this the real line but also includes infinity somehow?

 

[Dr.Peterson]

Can i check what you mean by 'projectively extended reals'? Is this the real line but also includes infinity somehow?

Yes, you can. Just Google "projectively extended reals". You'll find a variety of explanations, such as this.