Cot theta graph

[Skelly4444]

If Cot theta is defined as 1/tan theta, how is the graph of cot theta permitted to go straight through zero?

As Tan approaches 90 degrees, the values obviously become extremely large, but tan 90 is undefined.

So how can we have the reciprocal of an undefined value and allow the graph of Cot theta to pass through zero at the 90 degree point?

 

[Dr.Peterson]

If Cot theta is defined as 1/tan theta, how is the graph of cot theta permitted to go straight through zero?

As Tan approaches 90 degrees, the values obviously become extremely large, but tan 90 is undefined.

So how can we have the reciprocal of an undefined value and allow the graph of Cot theta to pass through zero at the 90 degree point?

That is not the definition of the cotangent! It's just an identity that applies when both are defined. (Unfortunately, I have seen some sources wrongly present it as if it were a definition.)

What definition were you really given for the cotangent? I'd use the unit circle to define it.

 

[Skelly4444]

I watched a Youtube lecture and was given the definition as 1/Tan theta.

He then went on to say that as Tan approached 90 degrees, you will be dividing 1 by an extremely large number and therefore it will tend towards zero. My argument is that it will never get to zero though, so how can Cot 90 be simply defined as zero?

You can't say that Cot theta is 1/Tan theta in one sentence and then imply that 1/Tan 90 is zero on a graph.

I'm still puzzled to be honest.

 

[Deleted member 4993]

I watched a Youtube lecture and was given the definition as 1/Tan theta.

Can you provide a reference to that video?

 

[Skelly4444]

When I paste the link in, it errors and states that the video is not available.
If you type in Cot theta graph into Youtube and fairly high up on the search results, just look out for an EXAM SOLUTIONS lecture and that's the one I'm referring to.

In that lecture, he specifically states that as the graph approaches 90 degrees, we are dividing by a huge positive number which tends towards zero.
This still doesn't explain how the graph is permitted to seamlessly travel through the zero point and continue on its merry way?

Still confused!

 

[topsquark]

When I paste the link in, it errors and states that the video is not available.
If you type in Cot theta graph into Youtube and fairly high up on the search results, just look out for an EXAM SOLUTIONS lecture and that's the one I'm referring to.

In that lecture, he specifically states that as the graph approaches 90 degrees, we are dividing by a huge positive number which tends towards zero.
This still doesn't explain how the graph is permitted to seamlessly travel through the zero point and continue on its merry way?

Still confused!

It doesn't "seamlessly" travel through 0. However I will admit that this comment is not usually made at a first introduction to the cotangent graph. We should be putting a small circle around the point (0, 0) to reflect this.

-Dan

 

[blamocur]

I watched a Youtube lecture and was given the definition as 1/Tan theta.

He then went on to say that as Tan approached 90 degrees, you will be dividing 1 by an extremely large number and therefore it will tend towards zero. My argument is that it will never get to zero though, so how can Cot 90 be simply defined as zero?

You can't say that Cot theta is 1/Tan theta in one sentence and then imply that 1/Tan 90 is zero on a graph.

I'm still puzzled to be honest.

One can say that the limit of [imath]\frac{1}{\tan \theta}[/imath] is 0 when [imath]\theta \rightarrow 90^\circ[/imath]. But a more practical and less confusing definition is [imath]\cot \theta = \frac{\cos\theta}{\sin\theta}[/imath], which is equivalent to your definition everywhere where [imath]\tan\theta[/imath] is defined. Even better, use the "unit circle" definitions.

 

[Dr.Peterson]

The video reference is below:

See https://www.freemathhelp.com/forum/threads/media-youtube-not-displaying.131869/
It can be viewed here.

As I have said, he is starting from incorrect definitions. These functions are not defined as reciprocals! Rather, the definition of the cotangent is x/y for a point on the terminal ray of the given angle. Alternatively, it is [imath]\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta}[/imath].

See here:

How do triangles, quadrants, and the unit circle relate?

You know that similar triangles have the same trigonometric ratios. Now learn the connection with the unit circle.

www.purplemath.com

https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet_Reduced.pdf

A slightly different version is given here:

Trigonometric functions - Wikipedia

en.wikipedia.org