Continuity of cot(x)

Identity · 2009-02-23 18:27:20

Nobody seems to be able to answer this question, not even my math teacher.

Is cot(x) continuous at its intercepts?

Argument 1: Yes, because

Argument 2: No, because

, which is undefined.

What is correct?

Last edited by Identity (2009-02-23 18:27:31)

Eigma · 2009-02-23 23:25:13

The graph of tangent function approaches infinity at π/2 or 90 deg. Since the length and direction of cotangent graph are determined by its intersection with the tangent graph at a certain point, the cotangent graph starts with very large values for very small positive angles and decreases to 0 at 90 deg then approaches negative ∞ as it approaches 180 deg. Therefore, cotangent is asymptotic only at 180 deg. and its multiples.

Identity · 2009-02-23 23:32:38

I don't understand why that makes

invalid. Also I don't understand your argument from a 'shape' perspective. I am already familiar with the shape of the cotangent graph.

Last edited by Identity (2009-02-23 23:32:56)

TheDude · 2009-02-24 00:52:58

It would depend on the formal definition of cotangent. For example, consider the function

Obviously this is equal to 1/(x+1) which is defined at x = 1, but we still say that f(x) is not continuous at x = 1 because, in it's formal definition, it is not defined at that point.

In this case I would say that cotangent is continuous at x = pi/2. I believe that the definition based on triangles is the most formal definition of the trigonometric functions, which would mean cot(x) = adjacent / opposite, which is defined for pi/2. The definition that cot(x) = 1/tan(x) is the shortcut, and is not entirely accurate as you pointed out. For that matter, cot(x) = cos(x) / sin(x) is also a shortcut.

This is all assuming that the basic definitions of trigonometric functions are based on ratios of the sides of triangles. If that's not their formal definitions then I could be wrong.

Eigma · 2009-02-24 00:57:03

I should say that cot π/2 = 1/tan(π/2) = 1/((sin π/2)/(cos π/2)) = (cos π/2)/(sin π/2) = 0 , which makes argument 1 more valid.

JaneFairfax · 2009-02-24 01:19:04

Last edited by JaneFairfax (2009-02-24 11:57:08)

JaneFairfax · 2009-02-24 01:22:04

Identity · 2009-02-24 19:44:47

Thankyou!

Math Is Fun Forum

#1 2009-02-23 18:27:20

Continuity of cot(x)

#2 2009-02-23 23:25:13

Re: Continuity of cot(x)

#3 2009-02-23 23:32:38

Re: Continuity of cot(x)

#4 2009-02-24 00:52:58

Re: Continuity of cot(x)

#5 2009-02-24 00:57:03

Re: Continuity of cot(x)

#6 2009-02-24 01:19:04

Re: Continuity of cot(x)

#7 2009-02-24 01:22:04

Re: Continuity of cot(x)

#8 2009-02-24 19:44:47

Re: Continuity of cot(x)

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