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# x=cot(x)? Watch

1. Question is as follows:

Using the behaviour of the cot function near a singularity, solve a quadratic equation to find the 15th positive solution of x = cot(x)

Hope someone can give me a tip here. Would love to know what my quadratic should look like!
2. cot(x) = 1/tanx

Re-arrange good fellow
3. I suspect a series expansion is involved. For instance, . I'm fairly certain that this line of investigation will not yield an exact solution though...
4. Still no idea as to how to find, specifically, the 15th positive solution? I have made an a priori estimate of 14*Pi..
5. Well, I'm going to guess you've noticed that in each interval there's exactly one solution. (You may have to prove this.) So you're looking for a solution in the interval . Note, however, that cot is periodic, so asking for the 16th positive solution is the same as asking for the 1st positive solution of .
6. What context is this question in? I don't think you can find a solution in closed form, but you could certainly use iteration. Think, in relation to the locations of the singularities of the graph , how you can find an upper and lower bound for the 15th positive root of the equation . It would definitely help you to draw the graphs of and on the same axes.
7. Right, so you need to approximate it, not find the exact solution; and you need to use MAPLE (both these bits of information would have been useful for you to put in your first post).

What numerical methods do you know?
8. Apologies. I am certainly not an expert in Numerical Methods.

But I have a few notes to go on, albeit sketchy.

EDIT: If anybody can help with the other parts of this piece of work (as shown in my above link.. there should be an image of the questions in there) then it would be greatly appreciated.

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Updated: December 13, 2010
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