Solving for c: One Solution in (0,1)?

  • Thread starter martint
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In summary, the conversation revolves around trying to prove that the given equation has precisely one solution in the interval (0,1) for all positive integer values of n. The individual has tried various approaches but has not been successful. They believe that a theorem or its consequences may be needed to provide a proof.
  • #1
martint
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Hello,

I've been trying to solve the following but with no luck its so frustrating!

if c= (n/pi) arccos [(n/pi)sin(pi/n)] + 2k(pi), where n is a positive integer,

how can I show that c has precisely one solution in (0,1)?

Thanks!
 
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  • #2
What have you done to try and solve it?

Perhaps you can start by plugging in some positive integers for n...
 
  • #3
that won't be a complete proof tho, need to show it holds for all values of n. Have tried breaking it down and looking at the limits of certain parts within the function, e.g sin (pi/n) lies between 0 and 1 for all n etc but this doesn't seem to get me anywhere as I get stuck when i reach the (n/pi) in front of the arccos!
 
  • #4
martint said:
that won't be a complete proof tho, need to show it holds for all values of n. Have tried breaking it down and looking at the limits of certain parts within the function, e.g sin (pi/n) lies between 0 and 1 for all n etc but this doesn't seem to get me anywhere as I get stuck when i reach the (n/pi) in front of the arccos!

You are right, it won't be a complete proof. But by trying and working out the first few integer values, you will have a better understanding as to why it will always be between 0 and 1. Then perhaps you can produce a proof by induction.
 
  • #5
i know that it lies between 0 and 1, but it is trying to show that there is only ONE solution between 0 and 1 that i haven't been able to achieve.I don't think it is possible by induction or contradiction but that a theorem or consequences of a theorem is required :confused:
 

1. What does it mean to solve for c in (0,1)?

Solving for c in (0,1) means finding the value of c that satisfies the given equation or inequality within the interval of 0 and 1.

2. How do I determine if there is only one solution in (0,1)?

If the equation or inequality has only one solution within the interval of 0 and 1, it means that there is only one value of c that makes the equation or inequality true.

3. Why is it important to specify the interval of (0,1) when solving for c?

The interval of (0,1) is important because it limits the possible values of c and ensures that the solution falls within a certain range. This allows for a more precise and accurate solution to the problem.

4. Can there be more than one solution in (0,1) when solving for c?

No, if the interval is specified as (0,1), there can only be one solution. If there are multiple values of c that satisfy the equation or inequality, they must all fall within the given interval.

5. What should I do if there is no solution in (0,1) for my equation or inequality?

If there is no solution within the specified interval, it means that the equation or inequality cannot be satisfied with any value of c within that range. You may need to revise your equation or consider a different interval to find a solution.

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